# Autoconverted from set.mm by xlat_mm.py and gen_set_mm.py param (PROP pax/prop () "") param (SET_MM_AX zfc/set_mm_ax (PROP) "") # set.mm - Version of 17-Oct-2006 # # PUBLIC DOMAIN # # This file (specifically, the version of this file with the above date) # has been released into the Public Domain per the Creative Commons Public # Domain Dedication. http://creativecommons.org/licenses/publicdomain/ # # Norman Megill - email: nm(at)alum(dot)mit(dot)edu # var (wff ph) var (wff ps) var (wff ch) var (wff th) var (wff ta) stmt (dummylink () (ph ps) ph) stmt (a1i () (ph) (-> ps ph)) stmt (a2i () ((-> ph (-> ps ch))) (-> (-> ph ps) (-> ph ch))) stmt (syl () ((-> ph ps) (-> ps ch)) (-> ph ch)) stmt (com12 () ((-> ph (-> ps ch))) (-> ps (-> ph ch))) stmt (a1d () ((-> ph ps)) (-> ph (-> ch ps))) stmt (a2d () ((-> ph (-> ps (-> ch th)))) (-> ph (-> (-> ps ch) (-> ps th)))) stmt (imim2 () () (-> (-> ph ps) (-> (-> ch ph) (-> ch ps)))) stmt (imim1 () () (-> (-> ph ps) (-> (-> ps ch) (-> ph ch)))) 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(A. x (-. 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(A. x ph)) (A. x (-. (A. x ph))))) stmt (hbe1 () () (-> (E. x ph) (A. x (E. x ph)))) stmt (ax6-2 () () (-> (-. (A. x ph)) (A. x (-. (A. x ph))))) stmt (modal-5 () () (-> (-. (A. x (-. ph))) (A. x (-. (A. x (-. ph)))))) stmt (modal-b () () (-> ph (A. x (-. (A. x (-. ph)))))) stmt (19.8a () () (-> ph (E. x ph))) stmt (19.2 () () (-> (A. x ph) (E. x ph))) stmt (19.3r () ((-> ph (A. x ph))) (<-> ph (A. x ph))) stmt (alcom () () (<-> (A. x (A. y ph)) (A. y (A. x ph)))) stmt (alnex () () (<-> (A. x (-. ph)) (-. (E. x ph)))) stmt (alex () () (<-> (A. x ph) (-. (E. x (-. ph))))) stmt (19.9r () ((-> ph (A. x ph))) (<-> ph (E. x ph))) stmt (19.9t () () (-> (A. x (-> ph (A. x ph))) (-> (E. x ph) ph))) stmt (19.9d () ((-> ps (A. x ps)) (-> ps (-> ph (A. x ph)))) (-> ps (-> (E. x ph) ph))) stmt (exnal () () (<-> (E. x (-. ph)) (-. 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(E. x ph)) (<-> (E! x (\/ ph ps)) (E! x ps)))) stmt (moexex () ((-> ph (A. y ph))) (-> (/\ (E* x ph) (A. x (E* y ps))) (E* y (E. x (/\ ph ps))))) stmt (moexexv ((ph y)) () (-> (/\ (E* x ph) (A. x (E* y ps))) (E* y (E. x (/\ ph ps))))) stmt (2moex () () (-> (E* x (E. y ph)) (A. y (E* x ph)))) stmt (2euex () () (-> (E! x (E. y ph)) (E. y (E! x ph)))) stmt (2eumo () () (-> (E! x (E* y ph)) (E* x (E! y ph)))) stmt (2eu2ex () () (-> (E! x (E! y ph)) (E. x (E. y ph)))) stmt (2moswap () () (-> (A. x (E* y ph)) (-> (E* x (E. y ph)) (E* y (E. x ph))))) stmt (2euswap () () (-> (A. x (E* y ph)) (-> (E! x (E. y ph)) (E! y (E. x ph))))) stmt (2exeu () () (-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E! x (E! y ph)))) stmt (2mo ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph z) (ph w) (ph v) (ph u)) () (<-> (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) (A. x (A. y (A. z (A. w (-> (/\ ph ([/] (cv z) x ([/] (cv w) y ph))) (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))) stmt (2mos ((w z) (ph z) (ph w) (x y) (ps x) (ps y) (x z) (w x) (y z) (w y)) ((-> (/\ (= (cv x) (cv z)) (= (cv y) (cv w))) (<-> ph ps))) (<-> (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))) (A. x (A. y (A. z (A. w (-> (/\ ph ps) (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))) stmt (2eu1 () () (-> (A. x (E* y ph)) (<-> (E! x (E! y ph)) (/\ (E! x (E. y ph)) (E! y (E. x ph)))))) stmt (2eu2 () () (-> (E! y (E. x ph)) (<-> (E! x (E! y ph)) (E! x (E. y ph))))) stmt (2eu3 () () (-> (A. x (A. y (\/ (E* x ph) (E* y ph)))) (<-> (/\ (E! x (E! y ph)) (E! y (E! x ph))) (/\ (E! x (E. y ph)) (E! y (E. x ph)))))) stmt (2eu4 ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (/\ (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))))) stmt (2eu5 ((x y) (x z) (w x) (y z) (w y) (w z) (ph z) (ph w)) () (<-> (/\ (E! x (E! y ph)) (A. x (E* y ph))) (/\ (E. x (E. y ph)) (E. z (E. w (A. x (A. y (-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w))))))))))) stmt (2eu6 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph z) (ph w) (ph v) (ph u)) () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E. z (E. w (A. x (A. y (<-> ph (/\ (= (cv x) (cv z)) (= (cv y) (cv w)))))))))) stmt (2eu7 () () (<-> (/\ (E! x (E. y ph)) (E! y (E. x ph))) (E! x (E! y (/\ (E. x ph) (E. y ph)))))) stmt (2eu8 () () (<-> (E! x (E! y (/\ (E. x ph) (E. y ph)))) (E! x (E! y (/\ (E! x ph) (E. y ph)))))) stmt (exists1 ((x y)) () (<-> (E! x (= (cv x) (cv x))) (A. x (= (cv x) (cv y))))) stmt (exists2 ((x y)) () (-> (/\ (E. x ph) (E. x (-. ph))) (-. (E! x (= (cv x) (cv x)))))) var (set t) stmt (axext ((x y) (x z) (y z)) () (E. z (-> (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y))) (= (cv x) (cv y))))) stmt (zfext2 ((x z) (w z) (w x) (y z) (w y)) () (-> (A. z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (= (cv x) (cv y)))) stmt (bm1.1 ((x y) (x z) (y z) (ph z)) ((-> ph (A. x ph))) (-> (E. x (A. y (<-> (e. (cv y) (cv x)) ph))) (E! x (A. y (<-> (e. (cv y) (cv x)) ph))))) var (class C) var (class D) var (class P) var (class Q) var (class R) var (class S) var (class T) var (class U) stmt (abid () () (<-> (e. (cv x) ({|} x ph)) ph)) stmt (hbab1 ((x y)) () (-> (e. (cv y) ({|} x ph)) (A. x (e. (cv y) ({|} x ph))))) stmt (hbab ((x z)) ((-> ph (A. x ph))) (-> (e. (cv z) ({|} y ph)) (A. x (e. (cv z) ({|} y ph))))) stmt (hbabd ((x z)) ((-> ph (A. x (A. y ph))) (-> ph (-> ps (A. x ps)))) (-> ph (-> (e. (cv z) ({|} y ps)) (A. x (e. (cv z) ({|} y ps)))))) stmt (dfcleq ((A x) (B x) (x y) (x z) (y z)) () (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. 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A C) (e. B C)))) stmt (eleq2 ((A x) (B x) (C x)) () (-> (= A B) (<-> (e. C A) (e. C B)))) stmt (eleq12 () () (-> (/\ (= A B) (= C D)) (<-> (e. A C) (e. B D)))) stmt (eleq1i () ((= A B)) (<-> (e. A C) (e. B C))) stmt (eleq2i () ((= A B)) (<-> (e. C A) (e. C B))) stmt (eleq12i () ((= A B) (= C D)) (<-> (e. A C) (e. B D))) stmt (eleq1d () ((-> ph (= A B))) (-> ph (<-> (e. A C) (e. B C)))) stmt (eleq2d () ((-> ph (= A B))) (-> ph (<-> (e. C A) (e. C B)))) stmt (eleq12d () ((-> ph (= A B)) (-> ph (= C D))) (-> ph (<-> (e. A C) (e. B D)))) stmt (eleq1a () () (-> (e. A B) (-> (= C A) (e. C B)))) stmt (eqeltr () ((= A B) (e. B C)) (e. A C)) stmt (eqeltrr () ((= A B) (e. A C)) (e. B C)) stmt (eleqtr () ((e. A B) (= B C)) (e. A C)) stmt (eleqtrr () ((e. A B) (= C B)) (e. A C)) stmt (eqeltrd () ((-> ph (= A B)) (-> ph (e. B C))) (-> ph (e. A C))) stmt (eqeltrrd () ((-> ph (= A B)) (-> ph (e. A C))) (-> ph (e. B C))) stmt (eleqtrd () ((-> ph (e. A B)) (-> ph (= B C))) (-> ph (e. A C))) stmt (eleqtrrd () ((-> ph (e. A B)) (-> ph (= C B))) (-> ph (e. A C))) stmt (syl5eqel () ((-> ph (e. A B)) (= C A)) (-> ph (e. C B))) stmt (syl5eqelr () ((-> ph (e. A B)) (= A C)) (-> ph (e. C B))) stmt (syl5eleq () ((-> ph (= A B)) (e. C A)) (-> ph (e. C B))) stmt (syl5eleqr () ((-> ph (= B A)) (e. C A)) (-> ph (e. C B))) stmt (syl6eqel () ((-> ph (= A B)) (e. B C)) (-> ph (e. A C))) stmt (syl6eqelr () ((-> ph (= B A)) (e. B C)) (-> ph (e. A C))) stmt (syl6eleq () ((-> ph (e. A B)) (= B C)) (-> ph (e. A C))) stmt (syl6eleqr () ((-> ph (e. A B)) (= C B)) (-> ph (e. A C))) stmt (cleqf ((A y) (B y) (x y)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (<-> (= A B) (A. x (<-> (e. (cv x) A) (e. (cv x) B))))) stmt (nelneq () () (-> (/\ (e. A C) (-. (e. B C))) (-. (= A B)))) stmt (nelneq2 () () (-> (/\ (e. A B) (-. (e. A C))) (-. (= B C)))) stmt (hblem ((A y) (x y) (x z) (y z)) ((-> (e. (cv y) A) (A. x (e. (cv y) A)))) (-> (e. (cv z) A) (A. x (e. 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(cv y) H) (A. x (e. (cv y) H))) (-> (e. (cv y) R) (A. x (e. (cv y) R))) (-> (e. (cv y) S) (A. x (e. (cv y) S))) (-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (-> (Isom H R S A B) (A. x (Isom H R S A B)))) stmt (isof1o ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y)) () (-> (Isom H R S A B) (:-1-1-onto-> H A B))) stmt (isorel ((x y) (A x) (A y) (B x) (B y) (R x) (R y) (S x) (S y) (H x) (H y) (C x) (C y) (D x) (D y)) () (-> (/\ (Isom H R S A B) (/\ (e. C A) (e. 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D A)) (= (" H (i^i A (" (`' R) ({} D)))) (i^i B (" (`' S) ({} (` H D))))))) stmt (isofrlem ((x y) (w x) (A x) (w y) (A y) (A w) (x z) (B x) (y z) (B y) (w z) (B z) (B w) (R x) (R y) (R w) (S x) (S y) (S z) (S w) (H x) (H y) (H z) (H w)) () (-> (Isom H R S A B) (-> (Fr S B) (Fr R A)))) stmt (isofr () () (-> (Isom H R S A B) (<-> (Fr R A) (Fr S B)))) stmt (isowe ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (R x) (R y) (S x) (S y) (S z) (S w) (H x) (H y) (H z) (H w)) () (-> (Isom H R S A B) (<-> (We R A) (We S B)))) stmt (f1oiso ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B v) (B u) (H x) (H y) (H z) (H w) (H v) (H u) (R x) (R y) (R z) (R w) (R v) (R u) (S v) (S u)) () (-> (/\ (:-1-1-onto-> H A B) (= S ({<,>|} z w (E.e. x A (E.e. y A (/\ (/\ (= (cv z) (` H (cv x))) (= (cv w) (` H (cv y)))) (br (cv x) R (cv y)))))))) (Isom H R S A B))) stmt (f1owe ((w z) (R z) (R w) (x y) (x z) (w x) (S x) (y z) (w y) (S y) (S z) (S w) (A z) (A w) (B z) (B w) (F x) (F y) (F z) (F w)) ((= R ({<,>|} x y (br (` F (cv x)) S (` F (cv y)))))) (-> (:-1-1-onto-> F A B) (-> (We S B) (We R A)))) stmt (f1oweALT ((w z) (v z) (u z) (f z) (R z) (v w) (u w) (f w) (R w) (u v) (f v) (R v) (f u) (R u) (R f) (x y) (x z) (w x) (v x) (u x) (f x) (S x) (y z) (w y) (v y) (u y) (f y) (S y) (S z) (S w) (S v) (S u) (S f) (A z) (A w) (A v) (A u) (A f) (B z) (B w) (B v) (B u) (B f) (F x) (F y) (F z) (F w) (F v) (F u) (F f)) ((= R ({<,>|} x y (br (` F (cv x)) S (` F (cv y)))))) (-> (:-1-1-onto-> F A B) (-> (We S B) (We R A)))) stmt (canth ((x y) (A x) (A y) (F x) (F y)) ((e. A (V))) (-. (:-onto-> F A (P~ A)))) stmt (ncanth () () (:-onto-> (I) (V) (P~ (V)))) stmt (iunon ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) ((e. A (V)) (e. B (V))) (-> (A.e. x A (e. B (On))) (e. (U_ x A B) (On)))) stmt (iinon ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) ((e. B (V))) (-> (/\ (A.e. x A (e. B (On))) (-. (= A ({/})))) (e. (|^|_ x A B) (On)))) stmt (tfrlem1 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (w x) (F x) (w y) (F y) (w z) (F z) (F w) (G x) (G y) (G z) (G w)) () (-> (e. A (On)) (-> (/\ (Fn F A) (Fn G A)) (-> (A.e. x A (/\ (= (` F (cv x)) (` B (|` F (cv x)))) (= (` G (cv x)) (` B (|` G (cv x)))))) (A.e. x A (= (` F (cv x)) (` G (cv x)))))))) stmt (tfrlem2 ((A w) (B w) (F w) (G w) (w x) (w y)) () (-> (/\ (Fn F A) (Fn G A)) (-> (/\ (e. (<,> (cv x) (cv y)) F) (e. (<,> (cv x) (cv z)) G)) (-> (e. A (On)) (-> (A. w (-> (e. A (On)) (-> (e. (cv w) A) (/\ (= (` F (cv w)) (` B (|` F (cv w)))) (= (` G (cv w)) (` B (|` G (cv w)))))))) (= (cv y) (cv z))))))) stmt (tfrlem3 ((x y) (f x) (g x) (f y) (g y) (f g) (x z) (y z) (g z) (G f) (G g) (G x) (G z)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))))) (= A ({|} g (E.e. z (On) (/\ (Fn (cv g) (cv z)) (A.e. y (cv z) (= (` (cv g) (cv y)) (` G (|` (cv g) (cv y)))))))))) stmt (tfrlem4 ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (x z) (F x) (y z) (F y) (f z) (F z) (F f) (G x) (G y) (g z) (G z) (G f) (G g)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (-> (e. (cv g) A) (Fun (cv g)))) stmt (tfrlem5 ((x y) (f x) (g x) (h x) (A x) (f y) (g y) (h y) (A y) (f g) (f h) (A f) (g h) (A g) (A h) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (f z) (F z) (v w) (u w) (f w) (F w) (u v) (f v) (F v) (f u) (F u) (F f) (G x) (G y) (g z) (h z) (G z) (g w) (h w) (G w) (G f) (G g) (G h) (g v) (h v) (g u) (h u)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (-> (/\ (e. (cv g) A) (e. (cv h) A)) (-> (/\ (e. (<,> (cv x) (cv u)) (cv g)) (e. (<,> (cv x) (cv v)) (cv h))) (= (cv u) (cv v))))) stmt (tfrlem6 ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (F x) (F y) (F f) (G x) (G y) (G f) (G g)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (Rel F)) stmt (tfrlem7 ((x y) (f x) (g x) (h x) (A x) (f y) (g y) (h y) (A y) (f g) (f h) (A f) (g h) (A g) (A h) (v x) (u x) (F x) (v y) (u y) (F y) (u v) (f v) (F v) (f u) (F u) (F f) (G x) (G y) (G f) (G g) (G h) (g v) (h v) (g u) (h u)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (Fun F)) stmt (tfrlem8 ((x y) (f x) (A x) (f y) (A y) (A f) (x z) (w x) (v x) (F x) (y z) (w y) (v y) (F y) (w z) (v z) (f z) (F z) (v w) (f w) (F w) (f v) (F v) (F f) (G x) (G y) (G z) (G w) (G f)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (Ord (dom F))) stmt (tfrlem9 ((x y) (f x) (A x) (f y) (A y) (A f) (x z) (F x) (y z) (F y) (f z) (F z) (F f) (G x) (G y) (G z) (G f)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (-> (e. (cv y) (dom F)) (= (` F (cv y)) (` G (|` F (cv y)))))) stmt (tfrlem10 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A)) (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F)))))))) (-> (e. (dom F) (On)) (Fn C (suc (dom F))))) stmt (tfrlem11 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A)) (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F)))))))) (-> (e. (dom F) (On)) (-> (e. (cv y) (suc (dom F))) (= (` C (cv y)) (` G (|` C (cv y))))))) stmt (tfrlem12 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A)) (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F)))))))) (-> (e. (dom F) (On)) (e. C A))) stmt (tfrlem13 ((x y) (f x) (C x) (f y) (C y) (C f) (x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A)) (= C (u. F ({} (<,> (dom F) (` G (|` F (dom F)))))))) (= (dom F) (On))) stmt (tfr1 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (Fn F (On))) stmt (tfr2 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f) (y z)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (-> (e. (cv z) (On)) (= (` F (cv z)) (` G (|` F (cv z)))))) stmt (tfr3 ((x y) (f x) (A x) (f y) (A y) (A f) (F x) (F y) (F f) (G x) (G y) (G f) (B x) (B y)) ((= A ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. A))) (-> (/\ (Fn B (On)) (A.e. x (On) (= (` B (cv x)) (` G (|` B (cv x)))))) (= B F))) stmt (tz7.44lem1 ((x y) (A x) (A y) (G x) (H y)) ((= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))))) (Fun G)) stmt (tz7.44-1 ((x y) (A x) (A y) (F x) (G x) (H y)) ((= G ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. 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(cv x) (On)) (= (` F (cv x)) (` G (|` F (cv x))))) (e. B (On))) (-> (Lim B) (= (` F B) (U. (" F B))))) stmt (dfrdg2 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (A x) (A y) (A z) (A f) (A g)) () (= (rec F A) (U. ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` ({<,>|} g z (\/\/ (/\ (= (cv g) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv g) ({/})) (Lim (dom (cv g))))) (= (cv z) (` F (` (cv g) (U. (dom (cv g))))))) (/\ (Lim (dom (cv g))) (= (cv z) (U. (ran (cv g))))))) (|` (cv f) (cv y))))))))))) stmt (rdgeq1 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (G x) (G y) (G z) (G f) (G g) (A x) (A y) (A z) (A f) (A g)) () (-> (= F G) (= (rec F A) (rec G A)))) stmt (rdgeq2 ((x y) (x z) (f x) (g x) (F x) (y z) (f y) (g y) (F y) (f z) (g z) (F z) (f g) (F f) (F g) (A x) (A y) (A z) (A f) (A g) (B x) (B y) (B z) (B f) (B g)) () (-> (= A B) (= (rec F A) (rec F B)))) stmt (hbrdg ((y z) (w y) (v y) (f y) (g y) (F y) (w z) (v z) (f z) (g z) (F z) (v w) (f w) (g w) (F w) (f v) (g v) (F v) (f g) (F f) (F g) (A y) (A z) (A w) (A v) (A f) (A g) (x y) (x z) (w x) (v x) (f x) (g x)) ((-> (e. (cv y) F) (A. x (e. (cv y) F))) (-> (e. (cv y) A) (A. x (e. (cv y) A)))) (-> (e. (cv y) (rec F A)) (A. x (e. (cv y) (rec F A))))) stmt (rdglem1 ((x y) (f x) (g x) (f y) (g y) (f g) (x z) (y z) (g z) (G f) (G g) (G x) (G z) (w y) (G y) (w z) (g w) (G w)) () (= ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y)))))))) ({|} g (E.e. z (On) (/\ (Fn (cv g) (cv z)) (A.e. w (cv z) (= (` (cv g) (cv w)) (` G (|` (cv g) (cv w)))))))))) stmt (rdglem2 ((x y) (x z) (w x) (y z) (w y) (w z) (A x) (A z) (A w) (H x) (H z) (H w)) () (= ({<,>|} x y (\/\/ (/\ (= (cv x) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv x) ({/})) (Lim (dom (cv x))))) (= (cv y) (` H (` (cv x) (U. (dom (cv x))))))) (/\ (Lim (dom (cv x))) (= (cv y) (U. (ran (cv x))))))) ({<,>|} z y (\/\/ (/\ (= (cv z) ({/})) (= (cv y) A)) (/\ (-. (\/ (= (cv z) ({/})) (Lim (dom (cv z))))) (= (cv y) (` H (` (cv z) (U. (dom (cv z))))))) (/\ (Lim (dom (cv z))) (= (cv y) (U. (ran (cv z))))))))) stmt (rdgfnon ((x y) (x z) (f x) (g x) (u x) (v x) (w x) (F x) (y z) (f y) (g y) (u y) (v y) (w y) (F y) (f z) (g z) (u z) (v z) (w z) (F z) (f g) (f u) (f v) (f w) (F f) (g u) (g v) (g w) (F g) (u v) (u w) (F u) (v w) (F v) (F w) (A x) (A y) (A z) (A f) (A g) (A u) (A v) (A w)) () (Fn (rec F A) (On))) stmt (rdgval ((x y) (x z) (f x) (g x) (u x) (v x) (w x) (F x) (y z) (f y) (g y) (u y) (v y) (w y) (F y) (f z) (g z) (u z) (v z) (w z) (F z) (f g) (f u) (f v) (f w) (F f) (g u) (g v) (g w) (F g) (u v) (u w) (F u) (v w) (F v) (F w) (A x) (A y) (A z) (A f) (A g) (A u) (A v) (A w)) () (-> (e. (cv g) (On)) (= (` (rec F A) (cv g)) (` ({<,>|} w z (\/\/ (/\ (= (cv w) ({/})) (= (cv z) A)) (/\ (-. (\/ (= (cv w) ({/})) (Lim (dom (cv w))))) (= (cv z) (` F (` (cv w) (U. (dom (cv w))))))) (/\ (Lim (dom (cv w))) (= (cv z) (U. (ran (cv w))))))) (|` (rec F A) (cv g)))))) stmt (rdg0 ((g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((e. A (V))) (= (` (rec F A) ({/})) A)) stmt (rdgsuc ((g z) (B z) (B g) (g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((e. B (On))) (= (` (rec F A) (suc B)) (` F (` (rec F A) B)))) stmt (rdglim ((g z) (B z) (B g) (g z) (w z) (F z) (g w) (F g) (F w) (A z) (A g) (A w)) ((e. B (On))) (-> (Lim B) (= (` (rec F A) B) (U. (" (rec F A) B))))) stmt (rdg0t ((A x) (F x)) () (-> (e. A C) (= (` (rec F A) ({/})) A))) stmt (rdgsuct () () (-> (e. B (On)) (= (` (rec F A) (suc B)) (` F (` (rec F A) B))))) stmt (rdgsucopab ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((-> (e. (cv z) A) (A. x (e. (cv z) A))) (-> (e. (cv z) B) (A. x (e. (cv z) B))) (-> (e. (cv z) D) (A. x (e. (cv z) D))) (= F (rec ({<,>|} x y (= (cv y) C)) A)) (-> (= (cv x) (` F B)) (= C D))) (-> (/\ (e. B (On)) (e. D R)) (= (` F (suc B)) D))) stmt (rdgsucopabn ((D z) (y z) (C y) (C z) (A z) (B z) (x y) (x z)) ((-> (e. (cv z) A) (A. x (e. (cv z) A))) (-> (e. (cv z) B) (A. x (e. (cv z) B))) (-> (e. (cv z) D) (A. x (e. 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(<,> (<,> A B) C) ({<<,>,>|} x y z ph)) th))) stmt (ssoprab2i ((ph w) (ps w) (x y) (x z) (w x) (y z) (w y) (w z)) ((-> ph ps)) (C_ ({<<,>,>|} x y z ph) ({<<,>,>|} x y z ps))) stmt (funoprab ((x y) (x z) (w x) (y z) (w y) (w z) (ph w)) ((E* z ph)) (Fun ({<<,>,>|} x y z ph))) stmt (fnoprab ((x y) (x z) (y z) (ph z)) ((-> ph (E! z ps))) (Fn ({<<,>,>|} x y z (/\ ph ps)) ({<,>|} x y ph))) stmt (fnoprab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((e. C (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (Fn F (X. A B))) stmt (dmoprab2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((e. C (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (= (dom F) (X. A B))) stmt (ffnoprval ((x y) (w x) (A x) (w y) (A y) (A w) (B x) (B y) (B w) (C x) (C y) (C w) (F x) (F y) (F w)) () (<-> (:--> F (X. A B) C) (/\ (Fn F (X. A B)) (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C)))))) stmt (eqfnoprval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z) (D z) (F x) (F y) (F z) (G x) (G y) (G z)) () (-> (/\ (Fn F (X. A B)) (Fn G (X. C D))) (<-> (= F G) (/\ (= (X. A B) (X. C D)) (A.e. x A (A.e. y B (= (opr (cv x) F (cv y)) (opr (cv x) G (cv y))))))))) stmt (fnoprval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w)) () (<-> (Fn F (X. A B)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y)))))))) stmt (foprval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (F x) (F y) (F z)) () (<-> (:--> F (X. A B) C) (/\ (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) (opr (cv x) F (cv y)))))) (A.e. x A (A.e. y B (e. (opr (cv x) F (cv y)) C)))))) stmt (oprabex ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((e. A (V)) (e. B (V)) (-> (/\ (e. (cv x) A) (e. (cv y) B)) (E* z ph)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) ph)))) (e. F (V))) stmt (oprabex2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C z)) ((e. A (V)) (e. B (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (e. F (V))) stmt (oprabex3 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (R x) (R y) (R z)) ((e. H (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) R)))))))))) (e. F (V))) stmt (oprabval ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((e. C (V)) (-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (-> (= (cv z) C) (<-> ch th)) (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E! z ph)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph)))) (-> (/\ (e. A R) (e. B S)) (<-> (= (opr A F B) C) th))) stmt (oprabvalig ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (-> (= (cv z) C) (<-> ch th)) (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E* z ph)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph)))) (-> (/\/\ (e. A R) (e. B S) (e. C D)) (-> th (= (opr A F B) C)))) stmt (oprabvali ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z) (S x) (S y) (S z) (ps x) (ch x) (ch y) (th x) (th y) (th z)) ((e. C (V)) (-> (= (cv x) A) (<-> ph ps)) (-> (= (cv y) B) (<-> ps ch)) (-> (= (cv z) C) (<-> ch th)) (-> (/\ (e. (cv x) R) (e. (cv y) S)) (E* z ph)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) R) (e. (cv y) S)) ph)))) (-> (/\ (e. A R) (e. B S)) (-> th (= (opr A F B) C)))) stmt (oprabval2gf ((u v) (u x) (u y) (u z) (A u) (v x) (v y) (v z) (A v) (x y) (x z) (A x) (y z) (A y) (A z) (B u) (B v) (B x) (B y) (B z) (C u) (C v) (C x) (C y) (C z) (u w) (G u) (G w) (D u) (D v) (D x) (D y) (D z) (R u) (v w) (R v) (w z) (R w) (R z) (S u) (S v) (S w) (S z) (w x) (w y)) ((-> (e. (cv w) G) (A. x (e. (cv w) G))) (-> (e. (cv w) S) (A. y (e. (cv w) S))) (-> (= (cv x) A) (= R G)) (-> (= (cv y) B) (= G S)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R))))) (-> (/\/\ (e. A C) (e. B D) (e. S H)) (= (opr A F B) S))) stmt (oprabval2g ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (w x) (G x) (G w) (w z) (R w) (R z) (w y) (S w) (S y) (S z)) ((-> (= (cv x) A) (= R G)) (-> (= (cv y) B) (= G S)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R))))) (-> (/\/\ (e. A C) (e. B D) (e. S H)) (= (opr A F B) S))) stmt (oprabval2 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (R z) (S y) (S z)) ((e. S (V)) (-> (= (cv x) A) (= R G)) (-> (= (cv y) B) (= G S)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) C) (e. (cv y) D)) (= (cv z) R))))) (-> (/\ (e. A C) (e. B D)) (= (opr A F B) S))) stmt (oprabval5 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (G x) (R z) (S y) (S z)) ((e. S (V)) (-> (= (cv x) A) (= R G)) (-> (= (cv y) B) (= G S)) (= F ({<<,>,>|} x y z (= (cv z) R)))) (-> (/\ (e. A C) (e. B D)) (= (opr A F B) S))) stmt (oprabval3 ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (R x) (R y) (R z) (S x) (S y) (S z) (S w) (S v) (S u) (S f)) ((e. S (V)) (-> (/\ (/\ (= (cv w) A) (= (cv v) B)) (/\ (= (cv u) C) (= (cv f) D))) (= R S)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. H H)) (e. (cv y) (X. H H))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) R)))))))))) (-> (/\ (/\ (e. A H) (e. B H)) (/\ (e. C H) (e. D H))) (= (opr (<,> A B) F (<,> C D)) S))) stmt (oprabval4g ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C z) (C w) (C v) (C u) (C f) (D w) (D v)) ((= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (-> (/\/\ (e. (cv x) A) (e. (cv y) B) (e. C D)) (= (opr (cv x) F (cv y)) C))) stmt (foprab2OLD ((u v) (u w) (u x) (u y) (u z) (A u) (v w) (v x) (v y) (v z) (A v) (w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B u) (B v) (B w) (B x) (B y) (B z) (t u) (t v) (t w) (t z) (C t) (C u) (C v) (C w) (C z) (t x) (t y) (D t) (D u) (D v) (D w) (D x) (D y) (F u) (F v) (F w)) ((e. C (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (<-> (A.e. x A (A.e. y B (e. C D))) (:--> F (X. A B) D))) stmt (foprrn () () (-> (/\/\ (:--> F (X. R S) C) (e. A R) (e. B S)) (e. (opr A F B) C))) stmt (elrnoprab ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C z) (D x) (D y) (D w) (D v) (D u) (F w) (F v) (F u)) ((e. C (V)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (<-> (e. D (ran F)) (E.e. x A (E.e. y B (= D C))))) stmt (oprvalex ((x y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (A x) (A y) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (V)) (e. B (V))) (e. ({|} z (E.e. x A (E.e. y B (= (cv z) (opr (cv x) F (cv y)))))) (V))) stmt (oprssdm ((x y) (S x) (S y) (F x) (F y)) ((-. (e. ({/}) S)) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (C_ (X. S S) (dom F))) stmt (ndmoprg () ((= (dom F) (X. S S))) (-> (/\ (e. B C) (-. (/\ (e. A S) (e. B S)))) (= (opr A F B) ({/})))) stmt (ndmoprcl ((A x) (B x) (F x) (S x)) ((= (dom F) (X. S S)) (-> (/\ (e. A S) (e. (cv x) S)) (e. (opr A F (cv x)) S)) (e. ({/}) S)) (e. (opr A F B) S)) stmt (ndmopr () ((e. B (V)) (= (dom F) (X. S S))) (-> (-. (/\ (e. A S) (e. B S))) (= (opr A F B) ({/})))) stmt (ndmoprrcl () ((e. B (V)) (= (dom F) (X. S S)) (-. (e. ({/}) S))) (-> (e. (opr A F B) S) (/\ (e. A S) (e. B S)))) stmt (ndmoprcom () ((e. B (V)) (= (dom F) (X. S S)) (e. A (V))) (-> (-. (/\ (e. A S) (e. B S))) (= (opr A F B) (opr B F A)))) stmt (ndmoprass () ((e. B (V)) (= (dom F) (X. S S)) (e. C (V)) (-. (e. ({/}) S))) (-> (-. (/\/\ (e. A S) (e. B S) (e. C S))) (= (opr (opr A F B) F C) (opr A F (opr B F C))))) stmt (ndmoprdistr () ((e. B (V)) (= (dom F) (X. S S)) (e. C (V)) (-. (e. ({/}) S)) (= (dom G) (X. S S))) (-> (-. (/\/\ (e. A S) (e. B S) (e. C S))) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C))))) stmt (ndmord () ((e. B (V)) (= (dom F) (X. S S)) (e. A (V)) (C_ R (X. S S)) (-. (e. ({/}) S)) (-> (/\/\ (e. A S) (e. B S) (e. C S)) (<-> (br A R B) (br (opr C F A) R (opr C F B))))) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B))))) stmt (ndmordi () ((e. A (V)) (= (dom F) (X. S S)) (C_ R (X. S S)) (-. (e. ({/}) S)) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B))))) (-> (br (opr C F A) R (opr C F B)) (br A R B))) stmt (caoprcl ((x y) (F x) (F y) (S x) (S y) (A x) (A y) (B x) (B y)) ((-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S))) (-> (/\ (e. A S) (e. B S)) (e. (opr A F B) S))) stmt (caoprcom ((x y) (F x) (F y) (A x) (A y) (B x) (B y)) ((e. A (V)) (e. B (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (= (opr A F B) (opr B F A))) stmt (caoprass ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr (opr A F B) F C) (opr A F (opr B F C)))) stmt (caoprcan ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. C (V)) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z))))) (-> (/\ (e. A S) (e. B S)) (-> (= (opr A F B) (opr A F C)) (= B C)))) stmt (caoprord ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((e. A (V)) (e. B (V)) (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y)))))) (-> (e. C S) (<-> (br A R B) (br (opr C F A) R (opr C F B))))) stmt (caoprord2 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (R x) (R y) (R z)) ((e. A (V)) (e. B (V)) (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y))))) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x)))) (-> (e. C S) (<-> (br A R B) (br (opr A F C) R (opr B F C))))) stmt (caoprord3 ((x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (R x) (R y) (R z)) ((e. A (V)) (e. B (V)) (-> (e. (cv z) S) (<-> (br (cv x) R (cv y)) (br (opr (cv z) F (cv x)) R (opr (cv z) F (cv y))))) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (e. D (V))) (-> (/\ (/\ (e. B S) (e. C S)) (= (opr A F B) (opr C F D))) (<-> (br A R C) (br D R B)))) stmt (caoprdistr ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (G x) (G y) (G z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z))))) (= (opr A G (opr B F C)) (opr (opr A G B) F (opr A G C)))) stmt (caopr32 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr (opr A F B) F C) (opr (opr A F C) F B))) stmt (caopr12 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr A F (opr B F C)) (opr B F (opr A F C)))) stmt (caopr31 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr (opr A F B) F C) (opr (opr C F B) F A))) stmt (caopr13 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr A F (opr B F C)) (opr C F (opr B F A)))) stmt (caopr4 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (e. D (V))) (= (opr (opr A F B) F (opr C F D)) (opr (opr A F C) F (opr B F D)))) stmt (caopr411 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (e. D (V))) (= (opr (opr A F B) F (opr C F D)) (opr (opr C F B) F (opr A F D)))) stmt (caopr42 ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (e. D (V))) (= (opr (opr A F B) F (opr C F D)) (opr (opr A F C) F (opr D F B)))) stmt (caoprdistrr ((x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (G x) (G y) (G z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x))) (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z))))) (= (opr (opr A F B) G C) (opr (opr A G C) F (opr B G C)))) stmt (caoprdilem ((x y) (x z) (H x) (y z) (H y) (H z) (x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (G y) (G z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x))) (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z)))) (e. D (V)) (e. H (V)) (= (opr (opr (cv x) G (cv y)) G (cv z)) (opr (cv x) G (opr (cv y) G (cv z))))) (= (opr (opr (opr A G C) F (opr B G D)) G H) (opr (opr A G (opr C G H)) F (opr B G (opr D G H))))) stmt (caoprlem2 ((x y) (x z) (H x) (y z) (H y) (H z) (R x) (R y) (R z) (x y) (x z) (F x) (y z) (F y) (F z) (A x) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (G x) (G y) (G z) (R x) (R y) (R z)) ((e. A (V)) (e. B (V)) (e. C (V)) (= (opr (cv x) G (cv y)) (opr (cv y) G (cv x))) (= (opr (cv x) G (opr (cv y) F (cv z))) (opr (opr (cv x) G (cv y)) F (opr (cv x) G (cv z)))) (e. D (V)) (e. H (V)) (= (opr (opr (cv x) G (cv y)) G (cv z)) (opr (cv x) G (opr (cv y) G (cv z)))) (e. R (V)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z))))) (= (opr (opr (opr (opr A G C) F (opr B G D)) G H) F (opr (opr (opr A G D) F (opr B G C)) G R)) (opr (opr A G (opr (opr C G H) F (opr D G R))) F (opr B G (opr (opr C G R) F (opr D G H)))))) stmt (caoprmo ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (S w) (S v) (A w) (A v) (B w) (B v) (F w) (F v) (x y) (x z) (F x) (y z) (F y) (F z) (S x) (S y) (S z) (A x) (A y) (A z) (B x) (B y) (B z)) ((e. A (V)) (e. B S) (= (dom F) (X. S S)) (-. (e. ({/}) S)) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (-> (e. (cv x) S) (= (opr (cv x) F B) (cv x)))) (E* w (= (opr A F (cv w)) B))) stmt (1stval ((x y) (A x) (A y)) () (= (` (1st) A) (U. (dom ({} A))))) stmt (2ndval ((x y) (A x) (A y)) () (= (` (2nd) A) (U. (ran ({} A))))) stmt (op1st () ((e. A (V))) (= (` (1st) (<,> A B)) A)) stmt (op2nd () ((e. A (V)) (e. B (V))) (= (` (2nd) (<,> A B)) B)) stmt (op1stg ((A x) (B x)) () (-> (e. A C) (= (` (1st) (<,> A B)) A))) stmt (op2ndg ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A C) (e. B D)) (= (` (2nd) (<,> A B)) B))) stmt (1stval2 ((x y) (A x) (A y)) () (-> (e. A (X. (V) (V))) (= (` (1st) A) (|^| (|^| A))))) stmt (2ndval2 ((x y) (A x) (A y)) () (-> (e. A (X. (V) (V))) (= (` (2nd) A) (|^| (|^| (|^| (`' ({} A)))))))) stmt (fo1st ((x y)) () (:-onto-> (1st) (V) (V))) stmt (fo2nd ((x y)) () (:-onto-> (2nd) (V) (V))) stmt (f1stres ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (:--> (|` (1st) (X. A B)) (X. A B) A)) stmt (1st2val ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` ({<<,>,>|} x y z (= (cv z) (cv x))) A) (` (1st) A))) stmt (2nd2val ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` ({<<,>,>|} x y z (= (cv z) (cv y))) A) (` (2nd) A))) stmt (df1st2 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (u x) (u y) (u z) (u w) (u v)) () (= ({<<,>,>|} x y z (= (cv z) (cv x))) (|` (1st) (X. (V) (V))))) stmt (elxp6 () () (<-> (e. A (X. B C)) (/\ (= A (<,> (` (1st) A) (` (2nd) A))) (/\ (e. (` (1st) A) B) (e. (` (2nd) A) C))))) stmt (elxp7 () () (<-> (e. A (X. B C)) (/\ (e. A (X. (V) (V))) (/\ (e. (` (1st) A) B) (e. (` (2nd) A) C))))) stmt (1st2nd () () (-> (/\ (Rel B) (e. A B)) (= A (<,> (` (1st) A) (` (2nd) A))))) stmt (elopabi ((x y) (A x) (A y) (ch x) (ch y)) ((-> (= (cv x) (` (1st) A)) (<-> ph ps)) (-> (= (cv y) (` (2nd) A)) (<-> ps ch))) (-> (e. A ({<,>|} x y ph)) ch)) stmt (csbopeq1a () () (-> (= (<,> (cv x) (cv y)) A) (= B ([_/]_ (` (1st) A) x ([_/]_ (` (2nd) A) y B))))) stmt (dfoprab3 ((ph w) (x y) (x z) (w x) (y z) (w y) (w z)) () (= ({<<,>,>|} x y z ph) ({<,>|} w z (/\ (e. (cv w) (X. (V) (V))) ([/] (` (1st) (cv w)) x ([/] (` (2nd) (cv w)) y ph)))))) stmt (foprab2 ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z) (B w) (B x) (B y) (B z) (C w) (C z) (D w) (D x) (D y) (D z)) ((= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv z) C))))) (<-> (A.e. x A (A.e. y B (e. C D))) (:--> F (X. A B) D))) stmt (1on () () (e. (1o) (On))) stmt (2on () () (e. (2o) (On))) stmt (df1o2 () () (= (1o) ({} ({/})))) stmt (df2o2 () () (= (2o) ({,} ({/}) ({} ({/}))))) stmt (1ne0 () () (-. (= (1o) ({/})))) stmt (ordgt0ge1 () () (-> (Ord A) (<-> (e. ({/}) A) (C_ (1o) A)))) stmt (ordge1n0 () () (-> (Ord A) (<-> (C_ (1o) A) (-. (= A ({/})))))) stmt (el1o () () (<-> (e. A (1o)) (= A ({/})))) stmt (0lt1o () () (e. 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A (On)) (e. B (On)) (e. C (On))) (<-> (e. A B) (e. (opr C (+o) A) (opr C (+o) B))))) stmt (oacan () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (= (opr A (+o) B) (opr A (+o) C)) (= B C)))) stmt (oaword () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (C_ A B) (C_ (opr C (+o) A) (opr C (+o) B))))) stmt (oawordri ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (-> (C_ A B) (C_ (opr A (+o) C) (opr B (+o) C))))) stmt (oaord1 () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. ({/}) B) (e. A (opr A (+o) B))))) stmt (oaword1 () () (-> (/\ (e. A (On)) (e. B (On))) (C_ A (opr A (+o) B)))) stmt (oaword2 () () (-> (/\ (e. A (On)) (e. B (On))) (C_ A (opr B (+o) A)))) stmt (oawordeulem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (S x) (S z)) ((e. A (On)) (e. 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B (On)) (e. C (On))) (= (opr (opr A (+o) B) (+o) C) (opr A (+o) (opr B (+o) C))))) stmt (oarec ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z)) () (-> (/\ (e. A (On)) (e. B (On))) (= (opr A (+o) B) (u. A ({|} x (E.e. y B (= (cv x) (opr A (+o) (cv y))))))))) stmt (omordi ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (/\ (e. B (On)) (e. C (On))) (e. ({/}) C)) (-> (e. A B) (e. (opr C (.o) A) (opr C (.o) B))))) stmt (omord2 () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) C)) (<-> (e. A B) (e. (opr C (.o) A) (opr C (.o) B))))) stmt (omord () () (-> (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (<-> (/\ (e. A B) (e. ({/}) C)) (e. (opr C (.o) A) (opr C (.o) B))))) stmt (omcan () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. ({/}) A)) (<-> (= (opr A (.o) B) (opr A (.o) C)) (= B C)))) stmt (omword () () (-> (/\ (/\/\ (e. A (On)) (e. B (On)) (e. C (On))) (e. 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B (V))) (-> (br A R B) (br B R A))) stmt (ecopoprtrn ((f g) (f h) (f t) (f s) (f r) (A f) (g h) (g t) (g s) (g r) (A g) (h t) (h s) (h r) (A h) (s t) (r t) (A t) (r s) (A s) (A r) (B f) (B g) (B h) (B t) (B s) (B r) (C f) (C g) (C h) (C t) (C s) (C r) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (s x) (r x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (s y) (r y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (s z) (r z) (F z) (v w) (u w) (f w) (g w) (h w) (t w) (s w) (r w) (F w) (u v) (f v) (g v) (h v) (t v) (s v) (r v) (F v) (f u) (g u) (h u) (t u) (s u) (r u) (F u) (F f) (F g) (F h) (F t) (F s) (F r) (R f) (R g) (R h) (R t) (R s) (R r) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h) (S t) (S s) (S r)) ((= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v))))))))))) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z)))) (e. B (V)) (e. C (V))) (-> (/\ (br A R B) (br B R C)) (br A R C))) stmt (ecopoprer ((f g) (f h) (g h) (x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (F x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (F y) (w z) (v z) (u z) (f z) (g z) (h z) (F z) (v w) (u w) (f w) (g w) (h w) (F w) (u v) (f v) (g v) (h v) (F v) (f u) (g u) (h u) (F u) (F f) (F g) (F h) (R f) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S f) (S g) (S h)) ((= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) (= (opr (cv z) F (cv u)) (opr (cv w) F (cv v))))))))))) (= (opr (cv x) F (cv y)) (opr (cv y) F (cv x))) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)) (= (opr (opr (cv x) F (cv y)) F (cv z)) (opr (cv x) F (opr (cv y) F (cv z)))) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (-> (= (opr (cv x) F (cv y)) (opr (cv x) F (cv z))) (= (cv y) (cv z))))) (Er R)) stmt (eceqopreq ((x y) (F x) (F y) (S x) (S y) (A x) (A y) (B x) (B y) (C x) (C y) (D x) (D y)) ((e. B (V)) (e. C (V)) (e. D (V)) (Er R) (= (dom R) (X. S S)) (= (dom F) (X. S S)) (-. (e. ({/}) S)) (-> (/\ (e. (cv x) S) (e. (cv y) S)) (e. (opr (cv x) F (cv y)) S)) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (<-> (br (<,> A B) R (<,> C D)) (= (opr A F D) (opr B F C))))) (-> (/\ (e. A S) (e. C S)) (<-> (= ([] (<,> A B) R) ([] (<,> C D) R)) (= (opr A F D) (opr B F C))))) stmt (th3qlem1 ((x y) (x z) (w x) (v x) (u x) (F x) (y z) (w y) (v y) (u y) (F y) (w z) (v z) (u z) (F z) (v w) (u w) (F w) (u v) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (A x) (A y) (A z) (A w) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) ((Er R) (= (dom R) S) (-> (/\ (/\ (e. (cv y) S) (e. (cv w) S)) (/\ (e. (cv z) S) (e. (cv v) S))) (-> (/\ (br (cv y) R (cv w)) (br (cv z) R (cv v))) (br (opr (cv y) F (cv z)) R (opr (cv w) F (cv v)))))) (-> (/\ (e. A (/. S R)) (e. B (/. S R))) (E* x (E. y (E. z (/\ (/\ (= A ([] (cv y) R)) (= B ([] (cv z) R))) (= (cv x) ([] (opr (cv y) F (cv z)) R)))))))) stmt (th3qlem2 ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (A x) (A y) (A z) (A w) (A v) (A u) (A t) (A s) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h)) ((e. R (V)) (Er R) (= (dom R) (X. S S)) (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h))))))) (-> (/\ (e. A (/. (X. S S) R)) (e. B (/. (X. S S) R))) (E* z (E. w (E. v (E. u (E. t (/\ (/\ (= A ([] (<,> (cv w) (cv v)) R)) (= B ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R)))))))))) stmt (th3qcor ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h) (G x) (G y) (G z) (G w) (G v) (G u) (G t) (G s) (G f) (G g) (G h)) ((e. R (V)) (Er R) (= (dom R) (X. S S)) (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h)))))) (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))))))) (Fun G)) stmt (th3q ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (R x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (R y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (R z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (R w) (u v) (t v) (s v) (f v) (g v) (h v) (R v) (t u) (s u) (f u) (g u) (h u) (R u) (s t) (f t) (g t) (h t) (R t) (f s) (g s) (h s) (R s) (f g) (f h) (R f) (g h) (R g) (R h) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (A x) (A y) (A z) (A w) (A v) (A u) (A t) (A s) (A f) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (F x) (F y) (F z) (F w) (F v) (F u) (F t) (F s) (F f) (F g) (F h) (G x) (G y) (G z) (G w) (G v) (G u) (G t) (G s) (G f) (G g) (G h)) ((e. R (V)) (Er R) (= (dom R) (X. S S)) (-> (/\ (/\ (/\ (e. (cv w) S) (e. (cv v) S)) (/\ (e. (cv u) S) (e. (cv t) S))) (/\ (/\ (e. (cv s) S) (e. (cv f) S)) (/\ (e. (cv g) S) (e. (cv h) S)))) (-> (/\ (br (<,> (cv w) (cv v)) R (<,> (cv u) (cv t))) (br (<,> (cv s) (cv f)) R (<,> (cv g) (cv h)))) (br (opr (<,> (cv w) (cv v)) F (<,> (cv s) (cv f))) R (opr (<,> (cv u) (cv t)) F (<,> (cv g) (cv h)))))) (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (/. (X. S S) R)) (e. (cv y) (/. (X. S S) R))) (E. w (E. v (E. u (E. t (/\ (/\ (= (cv x) ([] (<,> (cv w) (cv v)) R)) (= (cv y) ([] (<,> (cv u) (cv t)) R))) (= (cv z) ([] (opr (<,> (cv w) (cv v)) F (<,> (cv u) (cv t))) R))))))))))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (= (opr ([] (<,> A B) R) G ([] (<,> C D) R)) ([] (opr (<,> A B) F (<,> C D)) R)))) var (set a) var (set b) var (set c) var (set d) stmt (oprec ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d) (F x) (F y) (F z) (F t) (F s) (F g) (F h) (F a) (F b) (F c) (F d) (G x) (G y) (G z) (G t) (G s) (G g) (G h) (G a) (G b) (G c) (G d) (H x) (H y) (H z) (H w) (H v) (H u) (H f) (J x) (J y) (J z) (K x) (K y) (K z) (K w) (K v) (K u) (K f) (L x) (L y) (L z) (L w) (L v) (L u) (L f) (Q x) (Q y) (Q z) (Q a) (Q b) (Q c) (Q d) (R x) (R y) (R z) (R t) (R s) (R g) (R h) (R a) (R b) (R c) (R d) (S x) (S y) (S z) (S w) (S v) (S u) (S t) (S s) (S f) (S g) (S h) (S a) (S b) (S c) (S d) (ph x) (ph y) (ps z) (ps w) (ps v) (ps u) (ch z) (ch w) (ch v) (ch u)) ((e. H (V)) (e. K (V)) (e. L (V)) (e. R (V)) (Er R) (= (dom R) (X. S S)) (= R ({<,>|} x y (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. z (E. w (E. v (E. u (/\ (/\ (= (cv x) (<,> (cv z) (cv w))) (= (cv y) (<,> (cv v) (cv u)))) ph)))))))) (-> (/\ (/\ (= (cv z) (cv a)) (= (cv w) (cv b))) (/\ (= (cv v) (cv c)) (= (cv u) (cv d)))) (<-> ph ps)) (-> (/\ (/\ (= (cv z) (cv g)) (= (cv w) (cv h))) (/\ (= (cv v) (cv t)) (= (cv u) (cv s)))) (<-> ph ch)) (= G ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. S S)) (e. (cv y) (X. S S))) (E. w (E. v (E. u (E. f (/\ (/\ (= (cv x) (<,> (cv w) (cv v))) (= (cv y) (<,> (cv u) (cv f)))) (= (cv z) J))))))))) (-> (/\ (/\ (= (cv w) (cv a)) (= (cv v) (cv b))) (/\ (= (cv u) (cv g)) (= (cv f) (cv h)))) (= J K)) (-> (/\ (/\ (= (cv w) (cv c)) (= (cv v) (cv d))) (/\ (= (cv u) (cv t)) (= (cv f) (cv s)))) (= J L)) (-> (/\ (/\ (= (cv w) A) (= (cv v) B)) (/\ (= (cv u) C) (= (cv f) D))) (= J H)) (= F ({<<,>,>|} x y z (/\ (/\ (e. (cv x) Q) (e. (cv y) Q)) (E. a (E. b (E. c (E. d (/\ (/\ (= (cv x) ([] (<,> (cv a) (cv b)) R)) (= (cv y) ([] (<,> (cv c) (cv d)) R))) (= (cv z) ([] (opr (<,> (cv a) (cv b)) G (<,> (cv c) (cv d))) R)))))))))) (= Q (/. (X. S S) R)) (-> (/\ (/\ (/\ (e. (cv a) S) (e. (cv b) S)) (/\ (e. (cv c) S) (e. (cv d) S))) (/\ (/\ (e. (cv g) S) (e. (cv h) S)) (/\ (e. (cv t) S) (e. (cv s) S)))) (-> (/\ ps ch) (br K R L)))) (-> (/\ (/\ (e. A S) (e. B S)) (/\ (e. C S) (e. D S))) (= (opr ([] (<,> A B) R) F ([] (<,> C D) R)) ([] H R)))) stmt (ecoprcom ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F x) (F y) (F z) (F w) (R x) (R y) (R z) (R w) (S x) (S y) (S z) (S w) (C z) (C w)) ((= C (/. (X. S S) R)) (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (= (opr ([] (<,> (cv x) (cv y)) R) F ([] (<,> (cv z) (cv w)) R)) ([] (<,> D G) R))) (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv x) S) (e. (cv y) S))) (= (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv x) (cv y)) R)) ([] (<,> H J) R))) (= D H) (= G J)) (-> (/\ (e. A C) (e. B C)) (= (opr A F B) (opr B F A)))) stmt (ecoprass ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (F x) (F y) (F z) (F w) (F v) (F u) (R x) (R y) (R z) (R w) (R v) (R u) (S x) (S y) (S z) (S w) (S v) (S u) (D z) (D w) (D v) (D u)) ((= D (/. (X. S S) R)) (-> (/\ (/\ (e. (cv x) S) (e. (cv y) S)) (/\ (e. (cv z) S) (e. (cv w) S))) (= (opr ([] (<,> (cv x) (cv y)) R) F ([] (<,> (cv z) (cv w)) R)) ([] (<,> G H) R))) (-> (/\ (/\ (e. (cv z) S) (e. (cv w) S)) (/\ (e. (cv v) S) (e. (cv u) S))) (= (opr ([] (<,> (cv z) (cv w)) R) F ([] (<,> (cv v) (cv u)) R)) ([] (<,> N Q) R))) (-> (/\ (/\ (e. G S) (e. H S)) (/\ (e. (cv v) S) (e. 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D (V))) (-> (/\ (/\ (br A (~<_) B) (br C (~<_) D)) (= (i^i B D) ({/}))) (br (u. A C) (~<_) (u. B D)))) stmt (xpcomen ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (V)) (e. B (V))) (br (X. A B) (~~) (X. B A))) stmt (xpcomeng ((x y) (A x) (A y) (B y)) () (-> (/\ (e. A C) (e. B D)) (br (X. A B) (~~) (X. B A)))) stmt (xpassen ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((e. A (V)) (e. B (V)) (e. C (V))) (br (X. (X. A B) C) (~~) (X. A (X. B C)))) stmt (xpdom2 ((A t) (t x) (t y) (t z) (t w) (t v) (t u) (f t) (B t) (x y) (x z) (w x) (v x) (u x) (f x) (B x) (y z) (w y) (v y) (u y) (f y) (B y) (w z) (v z) (u z) (f z) (B z) (v w) (u w) (f w) (B w) (u v) (f v) (B v) (f u) (B u) (B f) (C t) (C x) (C y) (C z) (C w) (C v) (C u) (C f)) ((e. B (V)) (e. C (V))) (-> (br A (~<_) B) (br (X. C A) (~<_) (X. 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A (V)) (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (-> (C_ (ran (cv g)) A) (= (" (cv g) (\ B (" (cv f) (U. D)))) (\ A (U. D))))) stmt (sbthlem4 ((A x) (B x) (D x) (f x) (g x)) ((e. A (V)) (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x))))))) (-> (/\ (/\ (= (dom (cv g)) B) (C_ (ran (cv g)) A)) (Fun (`' (cv g)))) (= (" (`' (cv g)) (\ A (U. D))) (\ B (" (cv f) (U. D)))))) stmt (sbthlem5 ((H x) (A x) (B x) (D x) (f x) (g x)) ((e. A (V)) (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))) (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))))) (-> (/\ (= (dom (cv f)) A) (C_ (ran (cv g)) A)) (= (dom H) A))) stmt (sbthlem6 ((H x) (A x) (B x) (D x) (f x) (g x)) ((e. A (V)) (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))) (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))))) (-> (/\ (C_ (ran (cv f)) B) (/\ (/\ (= (dom (cv g)) B) (C_ (ran (cv g)) A)) (Fun (`' (cv g))))) (= (ran H) B))) stmt (sbthlem7 ((H x) (A x) (B x) (D x) (f x) (g x)) ((e. A (V)) (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))) (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))))) (-> (/\ (Fun (cv f)) (Fun (`' (cv g)))) (Fun H))) stmt (sbthlem8 ((H x) (A x) (B x) (D x) (f x) (g x)) ((e. A (V)) (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))) (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))))) (-> (/\ (Fun (`' (cv f))) (/\ (/\ (/\ (Fun (cv g)) (= (dom (cv g)) B)) (C_ (ran (cv g)) A)) (Fun (`' (cv g))))) (Fun (`' H)))) stmt (sbthlem9 ((H x) (A x) (B x) (D x) (f x) (g x)) ((e. A (V)) (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))) (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D)))))) (-> (/\ (:-1-1-> (cv f) A B) (:-1-1-> (cv g) B A)) (:-1-1-onto-> H A B))) stmt (sbthlem10 ((f g) (A f) (A g) (B f) (B g) (H x) (A x) (B x) (D x) (f x) (g x)) ((e. A (V)) (= D ({|} x (/\ (C_ (cv x) A) (C_ (" (cv g) (\ B (" (cv f) (cv x)))) (\ A (cv x)))))) (= H (u. (|` (cv f) (U. D)) (|` (`' (cv g)) (\ A (U. D))))) (e. B (V))) (-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B))) stmt (sbth ((x y) (x z) (w x) (f x) (g x) (A x) (y z) (w y) (f y) (g y) (A y) (w z) (f z) (g z) (A z) (f w) (g w) (A w) (f g) (A f) (A g) (B x) (B y) (B z) (B w) (B f) (B g)) () (-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B))) stmt (sbthbg () () (-> (e. B C) (<-> (/\ (br A (~<_) B) (br B (~<_) A)) (br A (~~) B)))) stmt (sbthcl ((x y)) () (= (~~) (i^i (~<_) (`' (~<_))))) stmt (dfsdom2 () () (= (~<) (\ (~<_) (`' (~<_))))) stmt (brsdom2 () ((e. A (V)) (e. B (V))) (<-> (br A (~<) B) (/\ (br A (~<_) B) (-. (br B (~<_) A))))) stmt (sdomnsym () () (-> (br A (~<) B) (-. (br B (~<) A)))) stmt (domnsym () () (-> (br A (~<_) B) (-. (br B (~<) A)))) stmt (0dom () () (br ({/}) (~<_) A)) stmt (dom0 () () (<-> (br A (~<_) ({/})) (= A ({/})))) stmt (0sdomg () () (-> (e. A B) (<-> (br ({/}) (~<) A) (-. (= A ({/})))))) stmt (0sdom () ((e. A (V))) (<-> (br ({/}) (~<) A) (-. (= A ({/}))))) stmt (sdom0 () () (-. (br A (~<) ({/})))) stmt (sdomdomtr () () (-> (e. C D) (-> (/\ (br A (~<) B) (br B (~<_) C)) (br A (~<) C)))) stmt (sdomentr () () (-> (e. C D) (-> (/\ (br A (~<) B) (br B (~~) C)) (br A (~<) C)))) stmt (ensdomtr () () (-> (/\ (br A (~~) B) (br B (~<) C)) (br A (~<) C))) stmt (sdomtr () () (-> (/\ (br A (~<) B) (br B (~<) C)) (br A (~<) C))) stmt (domsdomtr () () (-> (/\ (br A (~<_) B) (br B (~<) C)) (br A (~<) C))) stmt (enen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~~) C) (br B (~~) C)))) stmt (enen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~~) A) (br C (~~) B)))) stmt (domen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~<_) C) (br B (~<_) C)))) stmt (domen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~<_) A) (br C (~<_) B)))) stmt (sdomen1 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br A (~<) C) (br B (~<) C)))) stmt (sdomen2 () () (-> (/\ (e. B D) (br A (~~) B)) (<-> (br C (~<) A) (br C (~<) B)))) stmt (fodomr ((f g) (f z) (A f) (g z) (A g) (A z) (B f) (B g) (B z)) () (-> (/\/\ (e. A C) (br ({/}) (~<) B) (br B (~<_) A)) (E. f (:-onto-> (cv f) A B)))) stmt (canth2 ((x y) (f x) (A x) (f y) (A y) (A f)) ((e. A (V))) (br A (~<) (P~ A))) stmt (canth2g ((A x)) () (-> (e. A B) (br A (~<) (P~ A)))) stmt (xpen () ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V))) (-> (/\ (br A (~~) B) (br C (~~) D)) (br (X. A C) (~~) (X. B D)))) stmt (mapenlem1 ((f g) (f x) (f y) (f z) (f v) (A f) (g x) (g y) (g z) (g v) (A g) (x y) (x z) (v x) (A x) (y z) (v y) (A y) (v z) (A z) (A v) (B f) (B g) (B x) (B y) (B z) (B v) (C f) (C g) (C x) (C y) (C z) (C v) (D f) (D g) (D x) (D y) (D z) (D v) (H z) (H v)) ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V)) (= H ({<,>|} x y (/\ (e. (cv x) (opr A (^m) C)) (= (cv y) (o. (o. (cv f) (cv x)) (`' (cv g)))))))) (-> (/\ (/\ (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (:--> (cv z) C A)) (e. (cv v) C)) (= (` (` H (cv z)) (` (cv g) (cv v))) (` (cv f) (` (cv z) (cv v)))))) stmt (mapenlem2 ((f g) (f x) (f y) (f z) (f w) (f v) (A f) (g x) (g y) (g z) (g w) (g v) (A g) (x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B f) (B g) (B x) (B y) (B z) (B w) (B v) (C f) (C g) (C x) (C y) (C z) (C w) (C v) (D f) (D g) (D x) (D y) (D z) (D w) (D v) (H z) (H w) (H v)) ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V)) (= H ({<,>|} x y (/\ (e. (cv x) (opr A (^m) C)) (= (cv y) (o. (o. (cv f) (cv x)) (`' (cv g)))))))) (-> (/\ (:-1-1-onto-> (cv f) A B) (:-1-1-onto-> (cv g) C D)) (:-1-1-onto-> H (opr A (^m) C) (opr B (^m) D)))) stmt (mapen ((f g) (f x) (f y) (A f) (g x) (g y) (A g) (x y) (A x) (A y) (B f) (B g) (B x) (B y) (C f) (C g) (C x) (C y) (D f) (D g) (D x) (D y)) ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V))) (-> (/\ (br A (~~) B) (br C (~~) D)) (br (opr A (^m) C) (~~) (opr B (^m) D)))) stmt (mapdom1 ((A x) (B x) (C x)) ((e. A (V)) (e. B (V)) (e. C (V))) (-> (br A (~<_) B) (br (opr A (^m) C) (~<_) (opr B (^m) C)))) stmt (mapdom2lem ((x z) (w x) (A x) (w z) (A z) (A w) (B x) (B z) (B w) (C x) (C z) (C w)) ((e. A (V)) (e. B (V)) (e. C (V))) (-> (e. (cv x) (opr C (^m) (cv z))) (= (i^i (cv x) (X. (\ B (cv z)) ({} (cv w)))) ({/})))) stmt (mapdom2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (V)) (e. B (V)) (e. C (V))) (-> (/\ (br A (~<_) B) (-. (/\ (= A ({/})) (= C ({/}))))) (br (opr C (^m) A) (~<_) (opr C (^m) B)))) stmt (mapxpen ((x y) (x z) (w x) (v x) (f x) (g x) (A x) (y z) (w y) (v y) (f y) (g y) (A y) (w z) (v z) (f z) (g z) (A z) (v w) (f w) (g w) (A w) (f v) (g v) (A v) (f g) (A f) (A g) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (C x) (C y) (C z) (C w) (C v) (C f) (C g)) ((e. A (V)) (e. B (V)) (e. C (V))) (br (opr (opr A (^m) B) (^m) C) (~~) (opr A (^m) (X. B C)))) stmt (xpmapenlem1 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (V)) (e. B (V)) (e. C (V)) (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z))))))))) (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z))))))))) (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))))) (/\ (-> (= (cv y) (<,> D R)) (A. z (= (cv y) (<,> D R)))) (-> (= (cv y) (<,> D R)) (A. w (= (cv y) (<,> D R)))))) stmt (xpmapenlem2 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (V)) (e. B (V)) (e. C (V)) (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z))))))))) (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z))))))))) (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))))) (-> (/\ (= (cv y) (<,> D R)) (e. (cv z) C)) (/\ (= (` (U. (dom ({} (cv y)))) (cv z)) (U. (dom ({} (` (cv x) (cv z)))))) (= (` (U. (ran ({} (cv y)))) (cv z)) (U. (ran ({} (` (cv x) (cv z))))))))) stmt (xpmapenlem3 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (V)) (e. B (V)) (e. C (V)) (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z))))))))) (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z))))))))) (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))))) (-> (/\ (:--> (cv x) C (X. A B)) (= (cv y) (<,> D R))) (= (cv x) S))) stmt (xpmapenlem4 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (V)) (e. B (V)) (e. C (V)) (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z))))))))) (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z))))))))) (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))))) (-> (/\ (/\ (= (cv y) (<,> (U. (dom ({} (cv y)))) (U. (ran ({} (cv y)))))) (/\ (:--> (U. (dom ({} (cv y)))) C A) (:--> (U. (ran ({} (cv y)))) C B))) (= (cv x) S)) (/\ (:--> (cv x) C (X. A B)) (= (cv y) (<,> D R))))) stmt (xpmapenlem5 ((D y) (R y) (S x) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (V)) (e. B (V)) (e. C (V)) (= D ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (dom ({} (` (cv x) (cv z))))))))) (= R ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (U. (ran ({} (` (cv x) (cv z))))))))) (= S ({<,>|} z w (/\ (e. (cv z) C) (= (cv w) (<,> (` (U. (dom ({} (cv y)))) (cv z)) (` (U. (ran ({} (cv y)))) (cv z)))))))) (br (opr (X. A B) (^m) C) (~~) (X. (opr A (^m) C) (opr B (^m) C)))) stmt (xpmapen ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (V)) (e. B (V)) (e. C (V))) (br (opr (X. A B) (^m) C) (~~) (X. (opr A (^m) C) (opr B (^m) C)))) stmt (mapunen ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) ((e. A (V)) (e. B (V)) (e. C (V))) (-> (= (i^i A B) ({/})) (br (opr C (^m) (u. A B)) (~~) (X. (opr C (^m) A) (opr C (^m) B))))) stmt (pwen () ((e. A (V)) (e. B (V))) (-> (br A (~~) B) (br (P~ A) (~~) (P~ B)))) stmt (ssenen ((x y) (x z) (f x) (A x) (y z) (f y) (A y) (f z) (A z) (A f) (B x) (B y) (B z) (B f) (C x) (C y) (C z) (C f)) ((e. A (V)) (e. B (V)) (e. C (V))) (-> (br A (~~) B) (br ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) C))) (~~) ({|} x (/\ (C_ (cv x) B) (br (cv x) (~~) C)))))) stmt (limenpsi ((x y) (A x) (A y)) ((Lim A)) (-> (e. A B) (br A (~~) (\ A ({} ({/})))))) stmt (limensuci () ((Lim A)) (-> (e. A B) (br A (~~) (suc A)))) stmt (limensuc () () (-> (/\ (e. A B) (Lim A)) (br A (~~) (suc A)))) stmt (phplem1 () () (-> (/\ (e. A (om)) (e. B A)) (= (u. ({} A) (\ A ({} B))) (\ (suc A) ({} B))))) stmt (phplem2 () ((e. A (V)) (e. B (V))) (-> (/\ (e. A (om)) (e. B A)) (br A (~~) (\ (suc A) ({} B))))) stmt (phplem3 () ((e. A (V)) (e. B (V))) (-> (/\ (e. A (om)) (e. B (suc A))) (br A (~~) (\ (suc A) ({} B))))) stmt (phplem4 ((A f) (B f)) ((e. A (V)) (e. B (V))) (-> (/\ (e. A (om)) (e. B (om))) (-> (br (suc A) (~~) (suc B)) (br A (~~) B)))) stmt (nneneq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~~) B) (= A B)))) stmt (php ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (om)) (C: B A)) (-. (br A (~~) B)))) stmt (php2 ((A x) (B x)) () (-> (/\ (e. A (om)) (C: B A)) (br B (~<) A))) stmt (php3 ((A x) (x y) (x z) (f x) (A y) (A z) (A f) (y z) (f y) (f z) (B y) (B z) (B f)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (C: B A)) (br B (~<) A))) stmt (php4 () () (-> (e. A (om)) (br A (~<) (suc A)))) stmt (php5 () () (-> (e. A (om)) (-. (br A (~~) (suc A))))) stmt (onomeneq () () (-> (/\ (e. A (On)) (e. B (om))) (<-> (br A (~~) B) (= A B)))) stmt (onfin ((A x)) () (-> (e. A (On)) (<-> (E.e. x (om) (br A (~~) (cv x))) (e. A (om))))) stmt (nndomo () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<_) B) (C_ A B)))) stmt (nnsdomo () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<) B) (C: A B)))) stmt (omsucdom () () (-> (/\ (e. A (om)) (e. B (om))) (<-> (br A (~<) B) (br (suc A) (~<_) B)))) stmt (sucdomi () () (-> (/\ (e. A (om)) (e. B C)) (-> (br (suc A) (~<_) B) (br A (~<) B)))) stmt (0sdom1dom ((A x)) ((e. A (V))) (<-> (br ({/}) (~<) A) (br (1o) (~<_) A))) stmt (finsucdom ((A x) (B x)) () (-> (/\ (e. A (om)) (E.e. x (om) (br B (~~) (cv x)))) (<-> (br A (~<) B) (br (suc A) (~<_) B)))) stmt (pssinf ((B x)) () (-> (/\ (C: A B) (br A (~~) B)) (-. (E.e. x (om) (br B (~~) (cv x)))))) stmt (ominf ((x y)) () (-. (E.e. x (om) (br (om) (~~) (cv x))))) stmt (omsdomnn ((A x)) () (-> (e. A (om)) (/\ (br A (~<_) (om)) (-. (br (om) (~~) A))))) stmt (isfinite1 ((A x)) () (-> (E.e. x (om) (br A (~~) (cv x))) (/\ (br A (~<_) (om)) (-. (br (om) (~~) A))))) stmt (infsdomnn () ((e. A (V))) (-> (/\ (br (om) (~<_) A) (e. B (om))) (br B (~<) A))) stmt (infn0 () ((e. A (V))) (-> (br (om) (~<_) A) (-. (= A ({/}))))) stmt (pssnn ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (om)) (C: B A)) (E.e. x A (br B (~~) (cv x))))) stmt (ssnn ((A x) (B x)) () (-> (/\ (e. A (om)) (C_ B A)) (E.e. x (om) (br B (~~) (cv x))))) stmt (ssfi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (C_ B A)) (E.e. x (om) (br B (~~) (cv x))))) stmt (domfi ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (br B (~<_) A)) (E.e. x (om) (br B (~~) (cv x))))) stmt (unblem1 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (C_ B (om)) (A.e. x (om) (E.e. y B (e. (cv x) (cv y))))) (e. A B)) (e. (|^| (\ B (suc A))) B))) stmt (unblem2 ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (F z) (F w) (F v) (F u)) ((-> (e. (cv w) F) (A. x (e. (cv w) F))) (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om)))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (-> (e. (cv z) (om)) (e. (` F (cv z)) A)))) stmt (unblem3 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F z) (F w) (F v)) ((-> (e. (cv w) F) (A. x (e. (cv w) F))) (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om)))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (-> (e. (cv z) (om)) (e. (` F (cv z)) (` F (suc (cv z))))))) stmt (unblem4 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F z) (F w) (F v)) ((-> (e. (cv w) F) (A. x (e. (cv w) F))) (= F (|` (rec ({<,>|} x y (= (cv y) (|^| (\ A (suc (cv x)))))) (|^| A)) (om)))) (-> (/\ (C_ A (om)) (A.e. w (om) (E.e. v A (e. (cv w) (cv v))))) (:-1-1-> F (om) A))) stmt (unbnn ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) ((e. A (V))) (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (e. (cv x) (cv y))))) (br A (~~) (om)))) stmt (unbnn2 ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (C_ (cv x) (cv y))))) (br A (~~) (om)))) stmt (isfinite2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (br A (~<) (om)) (E.e. x (om) (br A (~~) (cv x))))) stmt (unfilem1 ((x y) (A x) (A y) (B x) (B y)) ((e. A (om)) (e. B (om)) (= F ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x))))))) (= (ran F) (\ (opr A (+o) B) A))) stmt (unfilem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (F z) (F w)) ((e. A (om)) (e. B (om)) (= F ({<,>|} x y (/\ (e. (cv x) B) (= (cv y) (opr A (+o) (cv x))))))) (:-1-1-onto-> F B (\ (opr A (+o) B) A))) stmt (unfilem3 ((x y) (f x) (A x) (f y) (A y) (A f) (B x) (B y) (B f)) () (-> (/\ (e. A (om)) (e. B (om))) (br B (~~) (\ (opr A (+o) B) A)))) stmt (fin2inf () () (-> (br A (~<) (om)) (e. (om) (V)))) stmt (unfi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (E.e. x (om) (br A (~~) (cv x))) (E.e. x (om) (br B (~~) (cv x)))) (E.e. x (om) (br (u. A B) (~~) (cv x))))) stmt (unfi2 ((A x) (B x)) () (-> (/\ (br A (~<) (om)) (br B (~<) (om))) (br (u. A B) (~<) (om)))) stmt (infcntss ((A x)) ((e. A (V))) (-> (br (om) (~<_) A) (E. x (/\ (C_ (cv x) A) (br (cv x) (~~) (om)))))) stmt (prfi ((A x) (B x)) () (E.e. x (om) (br ({,} A B) (~~) (cv x)))) stmt (unifi ((f k) (f m) (f n) (f x) (f y) (f z) (A f) (k m) (k n) (k x) (k y) (k z) (A k) (m n) (m x) (m y) (m z) (A m) (n x) (n y) (n z) (A n) (x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (E.e. n (om) (br A (~~) (cv n))) (A.e. x A (E.e. n (om) (br (cv x) (~~) (cv n))))) (E.e. n (om) (br (U. A) (~~) (cv n))))) stmt (unifi2 ((n x) (A n) (A x)) () (-> (/\ (br A (~<) (om)) (A.e. x A (br (cv x) (~<) (om)))) (br (U. A) (~<) (om)))) stmt (fiint ((x y) (x z) (w x) (v x) (f x) (A x) (y z) (w y) (v y) (f y) (A y) (w z) (v z) (f z) (A z) (v w) (f w) (A w) (f v) (A v) (A f)) () (<-> (A.e. x A (A.e. y A (e. (i^i (cv x) (cv y)) A))) (A. x (-> (/\/\ (C_ (cv x) A) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) A))))) stmt (abfii1 ((x y) (x z) (y z)) () (= (|^| ({|} x (/\ (C_ A (cv x)) (A.e. y (cv x) (A.e. z (cv x) (e. (i^i (cv y) (cv z)) (cv x))))))) (|^| ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) (cv x)) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))))))) stmt (abfii2 ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (= ({|} x (E. y (/\/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y)))))) (|^| ({|} x (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x)))))))) stmt (abfii3 ((w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (= (|^| ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))))) (|^| ({|} x (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x)))))))) stmt (abfii4 ((f t) (f u) (f v) (f w) (f x) (f y) (f z) (A f) (t u) (t v) (t w) (t x) (t y) (t z) (A t) (u v) (u w) (u x) (u y) (u z) (A u) (v w) (v x) (v y) (v z) (A v) (w x) (w y) (w z) (A w) (x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (= (|^| ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) (cv x)) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))))) (|^| ({|} x (/\ (C_ A (cv x)) (A. y (-> (/\/\ (C_ (cv y) A) (-. (= (cv y) ({/}))) (E.e. z (om) (br (cv y) (~~) (cv z)))) (e. (|^| (cv y)) (cv x))))))))) stmt (abfii5 ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (= (|^| ({|} x (/\ (C_ A (cv x)) (A.e. y (cv x) (A.e. z (cv x) (e. (i^i (cv y) (cv z)) (cv x))))))) ({|} x (E. y (/\/\ (C_ (cv y) A) (E.e. z (om) (br (cv y) (~~) (cv z))) (= (cv x) (|^| (cv y)))))))) stmt (fodomfi ((f g) (f m) (f n) (f k) (f y) (f x) (A f) (g m) (g n) (g k) (g y) (g x) (A g) (m n) (k m) (m y) (m x) (A m) (k n) (n y) (n x) (A n) (k y) (k x) (A k) (x y) (A y) (A x) (B f) (B g) (B m) (B x) (F f) (F g) (F m)) () (-> (/\ (E.e. n (om) (br A (~~) (cv n))) (:-onto-> F A B)) (br B (~<_) A))) stmt (fodomfib ((f n) (A f) (A n) (B f)) () (-> (E.e. n (om) (br A (~~) (cv n))) (<-> (/\ (-. (= A ({/}))) (E. f (:-onto-> (cv f) A B))) (/\ (br ({/}) (~<) B) (br B (~<_) A))))) stmt (iunfi ((w x) (w y) (w z) (n w) (A w) (x y) (x z) (n x) (A x) (y z) (n y) (A y) (n z) (A z) (A n) (B n) (B w) (B y) (B z)) () (-> (/\ (E.e. n (om) (br A (~~) (cv n))) (A.e. x A (E.e. n (om) (br B (~~) (cv n))))) (E.e. n (om) (br (U_ x A B) (~~) (cv n))))) stmt (axreg ((x y) (x z) (y z)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y))))))))) stmt (zfregcl ((x y) (A x) (x z) (A y) (y z) (A z)) ((e. A (V))) (-> (E. x (e. (cv x) A)) (E.e. x A (A.e. y (cv x) (-. (e. (cv y) A)))))) stmt (zfreg ((A x) (x y) (A y)) ((e. A (V))) (-> (-. (= A ({/}))) (E.e. x A (= (i^i (cv x) A) ({/}))))) stmt (zfreg2 ((A x) (x y) (A y)) ((e. A (V))) (-> (-. (= A ({/}))) (E.e. x A (= (i^i A (cv x)) ({/}))))) stmt (eirrv ((x y) (x z) (y z)) () (-. (e. (cv x) (cv x)))) stmt (eirr ((A x)) () (-. (e. A A))) stmt (sucprcreg ((A x)) () (<-> (-. (e. A (V))) (= (suc A) A))) stmt (zfregfr ((x y) (A x) (A y)) () (Fr (E) A)) stmt (en2lp ((A x) (A y) (x y) (B x) (B y)) () (-. (/\ (e. A B) (e. B A)))) stmt (preleq () ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V))) (-> (/\ (/\ (e. A B) (e. C D)) (= ({,} A B) ({,} C D))) (/\ (= A C) (= B D)))) stmt (opthreg () ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V))) (-> (= ({,} A ({,} A B)) ({,} C ({,} C D))) (/\ (= A C) (= B D)))) stmt (suc11reg () () (<-> (= (suc A) (suc B)) (= A B))) stmt (inf0 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) ((e. (om) (V))) (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) stmt (inf1 ((x y) (x z) (y z)) ((E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) (E. x (/\ (-. (= (cv x) ({/}))) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) stmt (inf2 ((x y) (x z) (y z)) ((E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) (E. x (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))))) stmt (inf3lema ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (G v) (G u) (G f) (F v) (F u) (F f) (A v) (A u) (A f) (B v) (B u) (B f)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (<-> (e. A (` G B)) (/\ (e. A (cv x)) (C_ (i^i A (cv x)) B)))) stmt (inf3lemb ((x y) (x z) (w x) (y z) (w y) (w z)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (= (` F ({/})) ({/}))) stmt (inf3lemc ((x y) (x z) (w x) (y z) (w y) (w z)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (e. A (om)) (= (` F (suc A)) (` G (` F A))))) stmt (inf3lemd ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (e. A (om)) (C_ (` F A) (cv x)))) stmt (inf3lem1 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (e. A (om)) (C_ (` F A) (` F (suc A))))) stmt (inf3lem2 ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (G v) (G u) (G f) (F v) (F u) (F f) (A v) (A u) (A f) (B v) (B u) (B f)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (-. (= (` F A) (cv x)))))) stmt (inf3lem3 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G v) (F v) (A v) (B v)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (-. (= (` F A) (` F (suc A))))))) stmt (inf3lem4 ((x y) (x z) (w x) (y z) (w y) (w z)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (e. A (om)) (C: (` F A) (` F (suc A)))))) stmt (inf3lem5 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (-> (/\ (e. A (om)) (e. B A)) (C: (` F B) (` F A))))) stmt (inf3lem6 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G v) (G u) (F v) (F u) (A v) (A u) (B v) (B u)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (:-1-1-> F (om) (P~ (cv x))))) stmt (inf3lem7 ((x y) (x z) (w x) (y z) (w y) (w z)) ((= G ({<,>|} y z (= (cv z) ({e.|} w (cv x) (C_ (i^i (cv w) (cv x)) (cv y)))))) (= F (|` (rec G ({/})) (om))) (e. A (V)) (e. B (V))) (-> (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))) (e. (om) (V)))) stmt (inf3 ((x y) (x z) (w x) (y z) (w y) (w z)) ((E. x (/\ (-. (= (cv x) ({/}))) (C_ (cv x) (U. (cv x)))))) (e. (om) (V))) stmt (infeq5 ((x y) (x z) (w x) (y z) (w y) (w z)) () (<-> (E. x (C: (cv x) (U. (cv x)))) (e. (om) (V)))) stmt (axinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x))))))))) stmt (inf4 ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (E. y (/\ (e. (cv y) (cv x)) (A. z (-. (e. (cv z) (cv y)))))) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv z) (cv x)) (A. w (<-> (e. (cv w) (cv z)) (\/ (e. (cv w) (cv y)) (= (cv w) (cv y)))))))))))) stmt (zfinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x)))))) stmt (omex ((x y)) () (e. (om) (V))) stmt (inf5 () () (E. x (C: (cv x) (U. (cv x))))) stmt (omelon () () (e. (om) (On))) stmt (dfom3 ((x y) (x z) (y z)) () (= (om) (|^| ({|} x (/\ (e. ({/}) (cv x)) (A.e. y (cv x) (e. (suc (cv y)) (cv x)))))))) stmt (elom3 ((A x)) () (<-> (e. A (om)) (A. x (-> (Lim (cv x)) (e. A (cv x)))))) stmt (dfom4 ((x y)) () (= (om) ({|} x (A. y (-> (Lim (cv y)) (e. (cv x) (cv y))))))) stmt (oancom () () (=/= (opr (1o) (+o) (om)) (opr (om) (+o) (1o)))) stmt (isfinite ((A x)) () (<-> (br A (~<) (om)) (E.e. x (om) (br A (~~) (cv x))))) stmt (nnsdom () () (-> (e. A (om)) (br A (~<) (om)))) stmt (omenps () () (br (om) (~~) (\ (om) ({} ({/}))))) stmt (omensuc () () (br (om) (~~) (suc (om)))) stmt (infensuc ((x y) (A x) (A y)) () (-> (/\ (e. A (On)) (C_ (om) A)) (br A (~~) (suc A)))) stmt (unbnnt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (om)) (A.e. x (om) (E.e. y A (e. (cv x) (cv y))))) (br A (~~) (om)))) stmt (noinfep ((x y) (x z) (F x) (y z) (F y) (F z)) () (-> (Fn F (om)) (E.e. x (om) (-. (e. (` F (suc (cv x))) (` F (cv x))))))) stmt (trcl ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A y) (A z) (A v) (F y) (F v) (F u)) ((e. A (V)) (= F (|` (rec ({<,>|} z w (= (cv w) (u. (cv z) (U. (cv z))))) A) (om))) (= C (U_ y (om) (` F (cv y))))) (/\/\ (C_ A C) (Tr C) (A. x (-> (/\ (C_ A (cv x)) (Tr (cv x))) (C_ C (cv x)))))) stmt (tz9.1 ((A x) (A y) (A z) (A w) (A v) (x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) ((e. A (V))) (E. x (/\/\ (C_ A (cv x)) (Tr (cv x)) (A. y (-> (/\ (C_ A (cv y)) (Tr (cv y))) (C_ (cv x) (cv y))))))) stmt (zfregs ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (-. (= A ({/}))) (E.e. x A (= (i^i (cv x) A) ({/}))))) stmt (setind ((x y) (A x) (A y)) () (-> (A. x (-> (C_ (cv x) A) (e. (cv x) A))) (= A (V)))) stmt (setind2 ((A x)) () (-> (C_ (P~ A) A) (= A (V)))) stmt (r1fnon ((x y)) () (Fn (R1) (On))) stmt (r10 ((x y)) () (= (` (R1) ({/})) ({/}))) stmt (r1suc ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (R1) (suc A)) (P~ (` (R1) A))))) stmt (r1lim ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (e. A B) (Lim A)) (= (` (R1) A) (U_ x A (` (R1) (cv x)))))) stmt (r1tr ((x y) (x z) (A x) (y z) (A y) (A z)) () (Tr (` (R1) A))) stmt (r1ord ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B (On)) (-> (e. A B) (e. (` (R1) A) (` (R1) B))))) stmt (r1ord2 () () (-> (e. B (On)) (-> (e. A B) (C_ (` (R1) A) (` (R1) B))))) stmt (r1ord3 () () (-> (/\ (e. A (On)) (e. B (On))) (-> (C_ A B) (C_ (` (R1) A) (` (R1) B))))) stmt (r1val1 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (` (R1) A) (U_ x A (P~ (` (R1) (cv x))))))) stmt (tz9.12lem1 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F x) (F y)) ((e. A (V)) (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))))))) (C_ (" F A) (On))) stmt (tz9.12lem2 ((w z) (v z) (A z) (v w) (A w) (A v)) ((e. A (V)) (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))))))) (e. (suc (U. (" F A))) (On))) stmt (tz9.12lem3 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (F x) (F y)) ((e. A (V)) (= F ({<,>|} z w (= (cv w) (|^| ({e.|} v (On) (e. (cv z) (` (R1) (cv v))))))))) (-> (A.e. x A (E.e. y (On) (e. (cv x) (` (R1) (cv y))))) (e. A (` (R1) (suc (suc (U. (" F A)))))))) stmt (tz9.12 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) ((e. A (V))) (-> (A.e. x A (E.e. y (On) (e. (cv x) (` (R1) (cv y))))) (E.e. y (On) (e. A (` (R1) (cv y)))))) stmt (tz9.13 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) ((e. A (V))) (E.e. x (On) (e. A (` (R1) (cv x))))) stmt (tz9.13g ((x y) (A x) (A y)) () (-> (e. A B) (E.e. x (On) (e. A (` (R1) (cv x)))))) stmt (rankwflem ((A x)) () (-> (e. A B) (E.e. x (On) (e. A (` (R1) (suc (cv x))))))) stmt (jech9.3 ((x y)) () (= (U_ x (On) (` (R1) (cv x))) (V))) stmt (unir1 () () (= (U. (" (R1) (On))) (V))) stmt (rankval ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (= (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x)))))))) stmt (rankvalg ((x y) (A x) (A y)) () (-> (e. A B) (= (` (rank) A) (|^| ({e.|} x (On) (e. A (` (R1) (suc (cv x))))))))) stmt (rankval2 ((A x)) () (-> (e. A B) (= (` (rank) A) (|^| ({e.|} x (On) (C_ A (` (R1) (cv x)))))))) stmt (rankon ((A x)) () (e. (` (rank) A) (On))) stmt (rankid ((x y) (A x) (A y)) ((e. A (V))) (e. A (` (R1) (suc (` (rank) A))))) stmt (rankr1lem () ((e. A (V))) (-> (e. B (On)) (-> (-. (e. A (` (R1) B))) (C_ B (` (rank) A))))) stmt (rankr1 ((x y) (A x) (A y) (B x) (B y)) ((e. A (V))) (<-> (= B (` (rank) A)) (/\ (-. (e. A (` (R1) B))) (e. A (` (R1) (suc B)))))) stmt (rankr1g ((A x) (B x)) () (-> (e. A C) (<-> (= B (` (rank) A)) (/\ (-. (e. A (` (R1) B))) (e. A (` (R1) (suc B))))))) stmt (ssrankr1 () ((e. A (V))) (-> (e. B (On)) (<-> (C_ B (` (rank) A)) (-. (e. A (` (R1) B)))))) stmt (rankr1a () ((e. A (V))) (-> (e. B (On)) (<-> (e. A (` (R1) B)) (e. (` (rank) A) B)))) stmt (r1val2 ((A x)) () (-> (e. A (On)) (= (` (R1) A) ({|} x (e. (` (rank) (cv x)) A))))) stmt (r1val3 ((x y) (A x) (A y)) () (-> (e. A (On)) (= (` (R1) A) (U_ x A (P~ ({|} y (e. (` (rank) (cv y)) (cv x)))))))) stmt (rankel () ((e. B (V))) (-> (e. A B) (e. (` (rank) A) (` (rank) B)))) stmt (rankval3 ((x y) (A x) (A y)) ((e. A (V))) (= (` (rank) A) (|^| ({e.|} x (On) (A.e. y A (e. (` (rank) (cv y)) (cv x))))))) stmt (bndrank ((x y) (A x) (A y)) () (-> (E.e. x (On) (A.e. y A (C_ (` (rank) (cv y)) (cv x)))) (e. A (V)))) stmt (unbndrank ((x y) (A x) (A y)) () (-> (-. (e. A (V))) (A.e. x (On) (E.e. y A (e. (cv x) (` (rank) (cv y))))))) stmt (rankpw () ((e. A (V))) (= (` (rank) (P~ A)) (suc (` (rank) A)))) stmt (ranklim ((A x) (B x)) () (-> (Lim B) (<-> (e. (` (rank) A) B) (e. (` (rank) (P~ A)) B)))) stmt (r1pw ((A x) (B x)) () (-> (e. B (On)) (<-> (e. A (` (R1) B)) (e. (P~ A) (` (R1) (suc B)))))) stmt (r1pwcl ((A x) (A y) (x y) (B x) (B y)) () (-> (Lim B) (<-> (e. A (` (R1) B)) (e. (P~ A) (` (R1) B))))) stmt (rankss () ((e. B (V))) (-> (C_ A B) (C_ (` (rank) A) (` (rank) B)))) stmt (ranksn ((x y) (A x) (A y)) ((e. A (V))) (= (` (rank) ({} A)) (suc (` (rank) A)))) stmt (rankuni2 ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (= (` (rank) (U. A)) (U_ x A (` (rank) (cv x))))) stmt (rankun ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((e. A (V)) (e. B (V))) (= (` (rank) (u. A B)) (u. (` (rank) A) (` (rank) B)))) stmt (rankpr () ((e. A (V)) (e. B (V))) (= (` (rank) ({,} A B)) (suc (u. (` (rank) A) (` (rank) B))))) stmt (rankop () ((e. A (V)) (e. B (V))) (= (` (rank) (<,> A B)) (suc (suc (u. (` (rank) A) (` (rank) B)))))) stmt (r1rankid () () (-> (e. A B) (C_ A (` (R1) (` (rank) A))))) stmt (rankonid ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (On)) (= (` (rank) A) A))) stmt (rankeq0 ((A x)) ((e. A (V))) (<-> (= A ({/})) (= (` (rank) A) ({/})))) stmt (rankr1id ((x y) (A x) (A y)) () (<-> (e. A (On)) (= (` (rank) (` (R1) A)) A))) stmt (rankuni ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (` (rank) (U. A)) (U. (` (rank) A)))) stmt (rankr1b () ((e. A (V))) (-> (e. B (On)) (<-> (C_ A (` (R1) B)) (C_ (` (rank) A) B)))) stmt (rankuniOLD ((x y) (A x) (A y)) ((e. A (V))) (= (` (rank) (U. A)) (U. (` (rank) A)))) stmt (rankuniss () ((e. A (V))) (C_ (` (rank) (U. A)) (` (rank) A))) stmt (rankval4 ((x y) (A x) (A y)) ((e. A (V))) (= (` (rank) A) (U_ x A (suc (` (rank) (cv x)))))) stmt (rankbnd ((A x) (B x)) ((e. A (V))) (<-> (A.e. x A (C_ (suc (` (rank) (cv x))) B)) (C_ (` (rank) A) B))) stmt (rankelun () ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V))) (-> (/\ (e. (` (rank) A) (` (rank) C)) (e. (` (rank) B) (` (rank) D))) (e. (` (rank) (u. A B)) (` (rank) (u. C D))))) stmt (rankxpl () ((e. A (V)) (e. B (V))) (-> (=/= (X. A B) ({/})) (C_ (` (rank) (u. A B)) (` (rank) (X. A B))))) stmt (rankxpu () ((e. A (V)) (e. B (V))) (C_ (` (rank) (X. A B)) (suc (suc (` (rank) (u. A B)))))) stmt (rankxplim ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((e. A (V)) (e. B (V))) (-> (/\ (Lim (` (rank) (u. A B))) (=/= (X. A B) ({/}))) (= (` (rank) (X. A B)) (` (rank) (u. A B))))) stmt (rankxplim2 () ((e. A (V)) (e. B (V))) (-> (Lim (` (rank) (X. A B))) (Lim (` (rank) (u. A B))))) stmt (rankxplim3 ((x y) (A x) (A y) (B x) (B y)) ((e. A (V)) (e. B (V))) (<-> (Lim (` (rank) (X. A B))) (Lim (U. (` (rank) (X. A B)))))) stmt (rankxpsuc ((A x) (B x)) ((e. A (V)) (e. B (V))) (-> (/\ (= (` (rank) (u. A B)) (suc C)) (=/= (X. A B) ({/}))) (= (` (rank) (X. A B)) (suc (suc (` (rank) (u. A B))))))) stmt (scottex ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (e. ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) (V))) stmt (scott0 ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (= A ({/})) (= ({e.|} x A (A.e. y A (C_ (` (rank) (cv x)) (` (rank) (cv y))))) ({/})))) stmt (scottexs ((x y) (x z) (y z) (ph y) (ph z)) () (e. ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (V))) stmt (scott0s ((x y) (x z) (y z) (ph y) (ph z)) () (<-> (E. x ph) (-. (= ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) ({/}))))) stmt (cplem1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C w) (D w)) ((= C ({e.|} y B (A.e. z B (C_ (` (rank) (cv y)) (` (rank) (cv z)))))) (= D (U_ x A C))) (A.e. x A (-> (-. (= B ({/}))) (-. (= (i^i B D) ({/})))))) stmt (cplem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w)) ((e. A (V))) (E. y (A.e. x A (-> (-. (= B ({/}))) (-. (= (i^i B (cv y)) ({/}))))))) stmt (cp ((ph z) (ph w) (w z) (x y) (x z) (w x) (y z) (w y)) () (E. w (A.e. x (cv z) (-> (E. y ph) (E.e. y (cv w) ph))))) stmt (bnd ((ph z) (ph w) (w z) (x y) (x z) (w x) (y z) (w y)) () (-> (A.e. x (cv z) (E. y ph)) (E. w (A.e. x (cv z) (E.e. y (cv w) ph))))) stmt (bnd2 ((ph z) (ph w) (ph v) (w z) (v z) (v w) (x z) (w x) (v x) (A x) (A z) (A w) (A v) (x y) (B x) (y z) (w y) (v y) (B y) (B z) (B w) (B v)) ((e. A (V))) (-> (A.e. x A (E.e. y B ph)) (E. z (/\ (C_ (cv z) B) (A.e. x A (E.e. y (cv z) ph)))))) stmt (kardex ((x y) (x z) (A x) (y z) (A y) (A z)) ((e. A (V))) (e. ({|} x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))) (V))) stmt (karden ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C z) (D z)) ((e. A (V)) (e. B (V)) (= C ({|} x (/\ (br (cv x) (~~) A) (A. y (-> (br (cv y) (~~) A) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))) (= D ({|} x (/\ (br (cv x) (~~) B) (A. y (-> (br (cv y) (~~) B) (C_ (` (rank) (cv x)) (` (rank) (cv y))))))))) (<-> (= C D) (br A (~~) B))) stmt (htalem ((x y) (A x) (A y) (R x) (R y)) ((e. A (V)) (= B (U. ({e.|} x A (A.e. y A (-. (br (cv y) R (cv x)))))))) (-> (/\ (We R A) (-. (= A ({/})))) (e. B A))) stmt (hta ((v z) (A z) (A v) (x y) (R x) (x z) (v x) (R y) (y z) (v y) (R z) (R v) (ph y) (ph z) (ph v)) ((= A ({|} x (/\ ph (A. y (-> ([/] (cv y) x ph) (C_ (` (rank) (cv x)) (` (rank) (cv y)))))))) (= B (U. ({e.|} x A (A.e. y A (-. (br (cv y) R (cv x)))))))) (-> (We R A) (-> ph ([/] B x ph)))) stmt (aceq1 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (y z) (w y) (v y) (u y) (t y) (h y) (w z) (v z) (u z) (t z) (h z) (v w) (u w) (t w) (h w) (u v) (t v) (h v) (t u) (h u) (h t)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. x (A. z (<-> (E. x (/\ (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (/\ (e. (cv z) (cv x)) (e. (cv x) (cv y))))) (= (cv z) (cv x))))))))))) stmt (aceq0 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. v (A. u (<-> (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) (= (cv u) (cv v))))))))))) stmt (aceq2 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (<-> (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u)))))))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v)))))))))) stmt (aceq3lem ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (f x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (f y) (w z) (v z) (u z) (t z) (h z) (g z) (f z) (v w) (u w) (t w) (h w) (g w) (f w) (u v) (t v) (h v) (g v) (f v) (t u) (h u) (g u) (f u) (h t) (g t) (f t) (g h) (f h) (f g) (F t) (F h) (F g)) ((= F ({<,>|} w v (/\ (e. (cv w) (dom (cv y))) (= (cv v) (` (cv f) ({|} u (br (cv w) (cv y) (cv u))))))))) (-> (A. x (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))) (E. f (/\ (C_ (cv f) (cv y)) (Fn (cv f) (dom (cv y))))))) stmt (aceq3 ((f x) (f y) (f z) (f w) (f v) (f u) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))))) stmt (aceq4 ((f x) (f z) (f y) (f w) (f v) (x z) (x y) (w x) (v x) (y z) (w z) (v z) (w y) (v y) (v w)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. f (/\ (Fn (cv f) (cv x)) (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z))))))))) stmt (aceq5lem1 ((v w) (v y) (g v) (t v) (w y) (g w) (t w) (g y) (t y) (g t)) () (<-> (E! v (e. (cv v) (i^i (X. ({} (cv w)) (cv w)) (cv y)))) (E! g (/\ (e. (cv g) (cv w)) (e. (<,> (cv w) (cv g)) (cv y)))))) stmt (aceq5lem2 ((u w) (t w) (h w) (g w) (t u) (h u) (g u) (h t) (g t) (g h) (A w) (A g)) ((= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))))))) (<-> (e. (<,> (cv w) (cv g)) (U. A)) (/\ (e. (cv w) (cv h)) (e. (cv g) (cv w))))) stmt (aceq5lem3 ((u w) (t w) (h w) (t u) (h u) (h t) (A w)) ((= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t)))))))) (<-> (e. (X. ({} (cv w)) (cv w)) A) (/\ (-. (= (cv w) ({/}))) (e. (cv w) (cv h))))) stmt (aceq5lem4 ((x z) (x y) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w z) (v z) (u z) (t z) (h z) (g z) (w y) (v y) (u y) (t y) (h y) (g y) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (B z) (B w) (B g) (A x) (A y) (A z) (A w) (A g)) ((= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t))))))) (= B (i^i (U. A) (cv y))) (<-> ph (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))))) (-> ph (E. y (A.e. z A (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) stmt (aceq5lem5 ((f x) (f z) (f y) (f w) (f v) (f u) (f t) (f h) (f g) (x z) (x y) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w z) (v z) (u z) (t z) (h z) (g z) (w y) (v y) (u y) (t y) (h y) (g y) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (B z) (B w) (B f) (B g) (A x) (A y) (A z) (A w) (A g)) ((= A ({|} u (/\ (-. (= (cv u) ({/}))) (E.e. t (cv h) (= (cv u) (X. ({} (cv t)) (cv t))))))) (= B (i^i (U. A) (cv y))) (<-> ph (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))))) (-> ph (E. f (A.e. w (cv h) (-> (-. (= (cv w) ({/}))) (e. (` (cv f) (cv w)) (cv w))))))) stmt (aceq5 ((f x) (f z) (f y) (f w) (f v) (f h) (f u) (f t) (x z) (x y) (w x) (v x) (h x) (u x) (t x) (y z) (w z) (v z) (h z) (u z) (t z) (w y) (v y) (h y) (u y) (t y) (v w) (h w) (u w) (t w) (h v) (u v) (t v) (h u) (h t) (t u)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))))) stmt (aceq6a ((x z) (f x) (x y) (w x) (v x) (u x) (g x) (f z) (y z) (w z) (v z) (u z) (g z) (f y) (f w) (f v) (f u) (f g) (w y) (v y) (u y) (g y) (v w) (u w) (g w) (u v) (g v) (g u)) () (-> (A. x (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))))))) (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))))) stmt (aceq6b ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (y z) (w y) (v y) (u y) (f y) (g y) (w z) (v z) (u z) (f z) (g z) (v w) (u w) (f w) (g w) (u v) (f v) (g v) (f u) (g u) (f g)) () (-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))))))))) stmt (aceq7 ((x z) (f x) (x y) (w x) (v x) (u x) (f z) (y z) (w z) (v z) (u z) (f y) (f w) (f v) (f u) (w y) (v y) (u y) (v w) (u w) (u v)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u))))))))))) stmt (axac ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. x (A. y (A. z (-> (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) stmt (ac2 ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. y (A.e. z (cv x) (A.e. w (cv z) (E!e. v (cv z) (E.e. u (cv y) (/\ (e. (cv z) (cv u)) (e. (cv v) (cv u))))))))) stmt (ac3 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E!e. w (cv z) (E.e. v (cv y) (/\ (e. (cv z) (cv v)) (e. (cv w) (cv v))))))))) stmt (ac7 ((f x) (f y) (f z) (f w) (f v) (f u) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) stmt (ac7g ((f x) (R f) (R x)) () (-> (e. R A) (E. f (/\ (C_ (cv f) R) (Fn (cv f) (dom R)))))) stmt (ac4 ((x z) (f x) (f z)) () (E. f (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (e. (` (cv f) (cv z)) (cv z)))))) stmt (ac4c ((x y) (f x) (A x) (f y) (A y) (A f)) ((e. A (V))) (E. f (A.e. x A (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x)))))) stmt (ac5 ((f x) (f y) (A f) (x y) (A x) (A y)) ((e. A (V))) (E. f (/\ (Fn (cv f) A) (A.e. x A (-> (-. (= (cv x) ({/}))) (e. (` (cv f) (cv x)) (cv x))))))) stmt (ac5b ((f x) (A x) (A f)) ((e. A (V))) (-> (A.e. x A (-. (= (cv x) ({/})))) (E. f (/\ (:--> (cv f) A (U. A)) (A.e. x A (e. (` (cv f) (cv x)) (cv x))))))) stmt (ac6lem ((f z) (g z) (C z) (f g) (C f) (C g) (H z) (H f) (H g) (f g) (f x) (f y) (f z) (A f) (g x) (g y) (g z) (A g) (x y) (x z) (A x) (y z) (A y) (A z) (B f) (B g) (B x) (B y) (B z) (ph z) (f ph)) ((e. A (V)) (e. B (V)) (= C ({e.|} y B ph)) (= H ({<,>|} x z (/\ (e. (cv x) A) (= (cv z) C))))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A (e. (` (cv f) (cv x)) C)))))) stmt (ac6 ((f x) (f y) (f z) (f w) (A f) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B f) (B x) (B y) (B z) (B w) (ph z) (ph w) (f ph) (ps y)) ((e. A (V)) (e. B (V)) (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A ps))))) stmt (ac6s ((x y) (f x) (x z) (A x) (f y) (y z) (A y) (f z) (A f) (A z) (B x) (B y) (B f) (B z) (f ph) (ph z) (ps y) (ps z)) ((e. A (V)) (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps))) (-> (A.e. x A (E.e. y B ph)) (E. f (/\ (:--> (cv f) A B) (A.e. x A ps))))) stmt (ac6s2 ((x y) (f x) (A x) (f y) (A y) (A f) (f ph) (ps y)) ((e. A (V)) (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps))) (-> (A.e. x A (E. y ph)) (E. f (/\ (Fn (cv f) A) (A.e. x A ps))))) stmt (ac6s3 ((x y) (f x) (A x) (f y) (A y) (A f) (f ph) (ps y)) ((e. A (V)) (-> (= (cv y) (` (cv f) (cv x))) (<-> ph ps))) (-> (A.e. x A (E. y ph)) (E. f (A.e. x A ps)))) stmt (ac6s4 ((f x) (f y) (A f) (x y) (A x) (A y) (B f) (B y)) ((e. A (V))) (-> (A.e. x A (=/= B ({/}))) (E. f (/\ (Fn (cv f) A) (A.e. x A (e. (` (cv f) (cv x)) B)))))) stmt (ac8 ((x z) (x y) (w x) (v x) (f x) (y z) (w z) (v z) (f z) (w y) (v y) (f y) (v w) (f w) (f v)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) stmt (ac9s ((f x) (A f) (A x) (B f)) ((e. A (V))) (<-> (A.e. x A (=/= B ({/}))) (=/= (X_ x A B) ({/})))) stmt (kmlem1 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph x) (ph y) (ph v) (ps x) (ps v)) () (-> (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) ph))) (E. y (A.e. z (cv x) ps)))) (A. x (-> (A.e. z (cv x) (A.e. w (cv x) ph)) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) ps))))))) stmt (kmlem2 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph y) (ph w) (ph u)) () (<-> (E. y (A.e. z (cv x) (-> ph (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (E. y (/\ (-. (e. (cv y) (cv x))) (A.e. z (cv x) (-> ph (E! v (e. (cv v) (i^i (cv z) (cv y)))))))))) stmt (kmlem3 ((x z) (w x) (v x) (w z) (v z) (v w)) () (<-> (-. (= (\ (cv z) (U. (\ (cv x) ({} (cv z))))) ({/}))) (E.e. v (cv z) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (-. (e. (cv v) (i^i (cv z) (cv w))))))))) stmt (kmlem4 ((v y) (x y) (v x) (y z) (v z) (w y) (v w)) () (-> (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) (= (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (cv w)) ({/})))) stmt (kmlem5 () () (-> (/\ (e. (cv w) (cv x)) (-. (= (cv z) (cv w)))) (= (i^i (\ (cv z) (U. (\ (cv x) ({} (cv z))))) (\ (cv w) (U. (\ (cv x) ({} (cv w)))))) ({/})))) stmt (kmlem6 ((x z) (w x) (v x) (w z) (v z) (v w) (ph v) (A v)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> ph (= A ({/})))))) (A.e. z (cv x) (E.e. v (cv z) (A.e. w (cv x) (-> ph (-. (e. (cv v) A)))))))) stmt (kmlem7 ((x z) (w x) (v x) (w z) (v z) (v w)) () (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (-. (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w)))))))))) stmt (kmlem8 ((x z) (w x) (u x) (t x) (h x) (w z) (u z) (t z) (h z) (u w) (t w) (h w) (t u) (h u) (h t) (A z) (A w) (A h)) ((= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))))))) (A.e. z A (A.e. w A (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) stmt (kmlem9 ((h ph) (x y) (x z) (w x) (u x) (t x) (h x) (y z) (w y) (u y) (t y) (h y) (w z) (u z) (t z) (h z) (u w) (t w) (h w) (t u) (h u) (h t) (A y) (A z) (A w) (A h)) ((= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))))))) (-> (A. h (-> (A.e. z (cv h) (A.e. w (cv h) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/}))))) (E. y (A.e. z (cv h) ph)))) (E. y (A.e. z A ph)))) stmt (kmlem10 ((x z) (u x) (t x) (u z) (t z) (t u) (A z)) ((= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))))))) (-> (e. (cv z) (cv x)) (= (i^i (cv z) (U. A)) (\ (cv z) (U. (\ (cv x) ({} (cv z)))))))) stmt (kmlem11 ((x y) (x z) (v x) (u x) (t x) (y z) (v y) (u y) (t y) (v z) (u z) (t z) (u v) (t v) (t u) (A y) (A z) (A v)) ((= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))))))) (-> (A.e. z (cv x) (-. (= (\ (cv z) (U. (\ (cv x) ({} (cv z))))) ({/})))) (-> (A.e. z A (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))) (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (i^i (cv y) (U. A)))))))))) stmt (kmlem12 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (w z) (v z) (u z) (t z) (h z) (g z) (v w) (u w) (t w) (h w) (g w) (u v) (t v) (h v) (g v) (t u) (h u) (g u) (h t) (g t) (g h) (A y) (A z) (A w) (A v) (A h) (A g)) ((= A ({|} u (E.e. t (cv x) (= (cv u) (\ (cv t) (U. (\ (cv x) ({} (cv t)))))))))) (<-> (A. x (-> (/\ (A.e. z (cv x) (-. (= (cv z) ({/})))) (A.e. z (cv x) (A.e. w (cv x) (-> (-. (= (cv z) (cv w))) (= (i^i (cv z) (cv w)) ({/})))))) (E. y (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (A. x (-> (-. (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w)))))))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y))))))))))) stmt (kmlem13 ((x y) (x z) (v x) (y z) (v y) (v z)) () (<-> (-> (-. (E.e. z (cv x) (A.e. v (cv z) ph))) (E. y (A.e. z (cv x) (-> (-. (= (cv z) ({/}))) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))) (\/ (E.e. z (cv x) (A.e. v (cv z) ph)) (E. y (/\ (-. (e. (cv y) (cv x))) (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y)))))))))) stmt (kmlem14 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u)) ((<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))) (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (<-> (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))) (E. y (A. z (E. v (A. u (/\ (e. (cv y) (cv x)) ph))))))) stmt (kmlem15 ((x y) (x z) (v x) (u x) (y z) (v y) (u y) (v z) (u z) (u v) (ph u)) ((<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))) (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (<-> (/\ (-. (e. (cv y) (cv x))) ch) (A. z (E. v (A. u (/\ (-. (e. (cv y) (cv x))) ps)))))) stmt (kmlem16 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (ph u)) ((<-> ph (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (<-> ps (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))) (<-> ch (A.e. z (cv x) (E! v (e. (cv v) (i^i (cv z) (cv y))))))) (<-> (\/ (E.e. z (cv x) (A.e. v (cv z) (E.e. w (cv x) (/\ (-. (= (cv z) (cv w))) (e. (cv v) (i^i (cv z) (cv w))))))) (E. y (/\ (-. (e. (cv y) (cv x))) ch))) (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) ph) (/\ (-. (e. (cv y) (cv x))) ps)))))))) stmt (aceqkm ((x y) (x z) (w x) (v x) (u x) (f x) (t x) (h x) (y z) (w y) (v y) (u y) (f y) (t y) (h y) (w z) (v z) (u z) (f z) (t z) (h z) (v w) (u w) (f w) (t w) (h w) (u v) (f v) (t v) (h v) (f u) (t u) (h u) (f t) (f h) (h t)) () (<-> (A. x (E. f (/\ (C_ (cv f) (cv x)) (Fn (cv f) (dom (cv x)))))) (A. x (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (/\ (-. (e. (cv y) (cv x))) (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v)))))))))))))) stmt (ackm ((x y) (x z) (v x) (u x) (f x) (y z) (v y) (u y) (f y) (v z) (u z) (f z) (u v) (f v) (f u)) () (A. x (E. y (A. z (E. v (A. u (\/ (/\ (e. (cv y) (cv x)) (-> (e. (cv z) (cv y)) (/\ (/\ (e. (cv v) (cv x)) (-. (= (cv y) (cv v)))) (e. (cv z) (cv v))))) (/\ (-. (e. (cv y) (cv x))) (-> (e. (cv z) (cv x)) (/\ (/\ (e. (cv v) (cv z)) (e. (cv v) (cv y))) (-> (/\ (e. (cv u) (cv z)) (e. (cv u) (cv y))) (= (cv u) (cv v))))))))))))) stmt (numthlem ((x y) (f x) (g x) (A x) (f y) (g y) (A y) (f g) (A f) (A g) (B x) (B y) (B f) (F x) (F y) (F f) (G x) (G y) (G f)) ((e. A (V)) (= B ({|} f (E.e. x (On) (/\ (Fn (cv f) (cv x)) (A.e. y (cv x) (= (` (cv f) (cv y)) (` G (|` (cv f) (cv y))))))))) (= F (U. B)) (= G ({<,>|} f y (= (cv y) (` (cv g) (\ A (ran (cv f)))))))) (E.e. x (On) (E. f (:-1-1-onto-> (cv f) (cv x) A)))) stmt (numth ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h)) ((e. A (V))) (E.e. x (On) (E. f (:-1-1-onto-> (cv f) (cv x) A)))) stmt (numth2 ((x y) (x z) (A x) (y z) (A y) (A z)) () (E.e. x (On) (br (cv x) (~~) A))) stmt (numthcor ((x y) (A x) (A y)) () (-> (e. A B) (E.e. x (On) (br A (~<) (cv x))))) stmt (weth ((x y) (x z) (w x) (f x) (A x) (y z) (w y) (f y) (A y) (w z) (f z) (A z) (f w) (A w) (A f)) ((e. A (V))) (E. x (We (cv x) A))) stmt (zornlem1 ((w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D u) (D v) (D f) (D t) (R x) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (-> (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/}))))) (e. (` F (cv x)) D))) stmt (zornlem2 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (-> (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/}))))) (-> (e. (cv y) (cv x)) (br (` F (cv y)) R (` F (cv x)))))) stmt (zornlem3 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (-> (/\ (Po R A) (/\ (e. (cv x) (On)) (/\ (We (cv w) A) (-. (= D ({/})))))) (-> (e. (cv y) (cv x)) (-. (= (` F (cv x)) (` F (cv y))))))) stmt (zornlem4 ((x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (A h) (t z) (f t) (g t) (t u) (t v) (A t) (f z) (g z) (u z) (v z) (A z) (f g) (f u) (f v) (A f) (g u) (g v) (A g) (u v) (A u) (A v) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v)))))))))) (-> (/\ (Po R A) (We (cv w) A)) (E.e. x (On) (= D ({/}))))) stmt (zornlem5 ((u x) (v x) (f x) (t x) (s x) (H x) (u v) (f u) (t u) (s u) (H u) (f v) (t v) (s v) (H v) (f t) (f s) (H f) (s t) (H t) (H s) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (s x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (s y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (s w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h s) (A h) (t z) (f t) (g t) (t u) (t v) (s t) (A t) (f z) (g z) (u z) (v z) (s z) (A z) (f g) (f u) (f v) (f s) (A f) (g u) (g v) (g s) (A g) (u v) (s u) (A u) (s v) (A v) (A s) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F s) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))) (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z)))))) (-> (/\ (/\ (We (cv w) A) (e. (cv x) (On))) (A.e. y (cv x) (-. (= H ({/}))))) (C_ (" F (cv x)) A))) stmt (zornlem6 ((u x) (v x) (f x) (t x) (s x) (r x) (a x) (b x) (H x) (u v) (f u) (t u) (s u) (r u) (a u) (b u) (H u) (f v) (t v) (s v) (r v) (a v) (b v) (H v) (f t) (f s) (f r) (a f) (b f) (H f) (s t) (r t) (a t) (b t) (H t) (r s) (a s) (b s) (H s) (a r) (b r) (H r) (a b) (H a) (H b) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (r x) (s x) (a x) (b x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (r y) (s y) (a y) (b y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (r w) (s w) (a w) (b w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h r) (h s) (a h) (b h) (A h) (t z) (f t) (g t) (t u) (t v) (r t) (s t) (a t) (b t) (A t) (f z) (g z) (u z) (v z) (r z) (s z) (a z) (b z) (A z) (f g) (f u) (f v) (f r) (f s) (a f) (b f) (A f) (g u) (g v) (g r) (g s) (a g) (b g) (A g) (u v) (r u) (s u) (a u) (b u) (A u) (r v) (s v) (a v) (b v) (A v) (r s) (a r) (b r) (A r) (a s) (b s) (A s) (a b) (A a) (A b) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F r) (F s) (F a) (F b) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (D a) (D b) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s) (R r) (R a) (R b)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))) (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z)))))) (-> (Po R A) (-> (/\ (/\ (We (cv w) A) (e. (cv x) (On))) (A.e. y (cv x) (-. (= H ({/}))))) (Or R (" F (cv x)))))) stmt (zornlem7 ((u x) (v x) (f x) (t x) (s x) (r x) (a x) (b x) (H x) (u v) (f u) (t u) (s u) (r u) (a u) (b u) (H u) (f v) (t v) (s v) (r v) (a v) (b v) (H v) (f t) (f s) (f r) (a f) (b f) (H f) (s t) (r t) (a t) (b t) (H t) (r s) (a s) (b s) (H s) (a r) (b r) (H r) (a b) (H a) (H b) (x y) (w x) (h x) (t x) (x z) (f x) (g x) (u x) (v x) (r x) (s x) (a x) (b x) (A x) (w y) (h y) (t y) (y z) (f y) (g y) (u y) (v y) (r y) (s y) (a y) (b y) (A y) (h w) (t w) (w z) (f w) (g w) (u w) (v w) (r w) (s w) (a w) (b w) (A w) (h t) (h z) (f h) (g h) (h u) (h v) (h r) (h s) (a h) (b h) (A h) (t z) (f t) (g t) (t u) (t v) (r t) (s t) (a t) (b t) (A t) (f z) (g z) (u z) (v z) (r z) (s z) (a z) (b z) (A z) (f g) (f u) (f v) (f r) (f s) (a f) (b f) (A f) (g u) (g v) (g r) (g s) (a g) (b g) (A g) (u v) (r u) (s u) (a u) (b u) (A u) (r v) (s v) (a v) (b v) (A v) (r s) (a r) (b r) (A r) (a s) (b s) (A s) (a b) (A a) (A b) (B h) (B t) (B f) (F x) (F y) (F z) (F v) (F u) (F f) (F g) (F h) (F t) (F r) (F s) (F a) (F b) (G h) (G t) (G f) (C t) (D y) (D u) (D v) (D f) (D t) (D a) (D b) (R x) (R y) (R z) (R w) (R g) (R u) (R v) (R f) (R t) (R s) (R r) (R a) (R b)) ((e. A (V)) (= B ({|} f (E.e. h (On) (/\ (Fn (cv f) (cv h)) (A.e. t (cv h) (= (` (cv f) (cv t)) (` G (|` (cv f) (cv t))))))))) (= F (U. B)) (= C ({e.|} z A (A.e. g (ran (cv f)) (br (cv g) R (cv z))))) (= D ({e.|} z A (A.e. g (" F (cv x)) (br (cv g) R (cv z))))) (= G ({<,>|} f t (= (cv t) (U. ({e.|} v C (A.e. u C (-. (br (cv u) (cv w) (cv v))))))))) (= H ({e.|} z A (A.e. g (" F (cv y)) (br (cv g) R (cv z)))))) (-> (/\ (Po R A) (A. s (-> (/\ (C_ (cv s) A) (Or R (cv s))) (E.e. a A (A.e. r (cv s) (\/ (br (cv r) R (cv a)) (= (cv r) (cv a)))))))) (E.e. a A (A.e. b A (-. (br (cv a) R (cv b))))))) stmt (zorn2lem ((x y) (x z) (y z) (w x) (w y)) () (<-> (br (cv z) ({<,>|} x y (C: (cv x) (cv y))) (cv w)) (C: (cv z) (cv w)))) var (set q) stmt (zorn ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (s x) (r x) (q x) (a x) (b x) (c x) (d x) (j x) (k x) (m x) (n x) (R x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (s y) (r y) (q y) (a y) (b y) (c y) (d y) (j y) (k y) (m y) (n y) (R y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (s z) (r z) (q z) (a z) (b z) (c z) (d z) (j z) (k z) (m z) (n z) (R z) (v w) (u w) (f w) (g w) (h w) (t w) (s w) (r w) (q w) (a w) (b w) (c w) (d w) (j w) (k w) (m w) (n w) (R w) (u v) (f v) (g v) (h v) (t v) (s v) (r v) (q v) (a v) (b v) (c v) (d v) (j v) (k v) (m v) (n v) (R v) (f u) (g u) (h u) (t u) (s u) (r u) (q u) (a u) (b u) (c u) (d u) (j u) (k u) (m u) (n u) (R u) (f g) (f h) (f t) (f s) (f r) (f q) (a f) (b f) (c f) (d f) (f j) (f k) (f m) (f n) (R f) (g h) (g t) (g s) (g r) (g q) (a g) (b g) (c g) (d g) (g j) (g k) (g m) (g n) (R g) (h t) (h s) (h r) (h q) (a h) (b h) (c h) (d h) (h j) (h k) (h m) (h n) (R h) (s t) (r t) (q t) (a t) (b t) (c t) (d t) (j t) (k t) (m t) (n t) (R t) (r s) (q s) (a s) (b s) (c s) (d s) (j s) (k s) (m s) (n s) (R s) (q r) (a r) (b r) (c r) (d r) (j r) (k r) (m r) (n r) (R r) (a q) (b q) (c q) (d q) (j q) (k q) (m q) (n q) (R q) (a b) (a c) (a d) (a j) (a k) (a m) (a n) (R a) (b c) (b d) (b j) (b k) (b m) (b n) (R b) (c d) (c j) (c k) (c m) (c n) (R c) (d j) (d k) (d m) (d n) (R d) (j k) (j m) (j n) (R j) (k m) (k n) (R k) (m n) (R m) (R n) (A x) (A y) (A z) (A w) (A v) (A u) (A f) (A g) (A h) (A t) (A s) (A r) (A q) (A a) (A b) (A c) (A d) (A j) (A k) (A m) (A n)) ((e. A (V))) (-> (/\ (Po R A) (A. w (-> (/\ (C_ (cv w) A) (Or R (cv w))) (E.e. x A (A.e. z (cv w) (\/ (br (cv z) R (cv x)) (= (cv z) (cv x)))))))) (E.e. x A (A.e. y A (-. (br (cv x) R (cv y))))))) stmt (zorn2 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A x) (A y) (A z) (A w) (A v) (A u)) ((e. A (V))) (-> (A. z (-> (/\ (C_ (cv z) A) (A.e. x (cv z) (A.e. y (cv z) (\/ (C_ (cv x) (cv y)) (C_ (cv y) (cv x)))))) (e. (U. (cv z)) A))) (E.e. x A (A.e. y A (-. (C: (cv x) (cv y))))))) stmt (fodom ((F f) (A f) (B f)) ((e. A (V))) (-> (:-onto-> F A B) (br B (~<_) A))) stmt (fodomg ((A x) (B x) (F x)) () (-> (e. A C) (-> (:-onto-> F A B) (br B (~<_) A)))) stmt (fodomb ((A f) (B f)) ((e. A (V))) (<-> (/\ (-. (= A ({/}))) (E. f (:-onto-> (cv f) A B))) (/\ (br ({/}) (~<) B) (br B (~<_) A)))) stmt (imadomg () () (-> (e. A B) (-> (Fun F) (br (" F A) (~<_) A)))) stmt (fnrndomg () () (-> (e. A B) (-> (Fn F A) (br (ran F) (~<_) A)))) stmt (unidom ((f g) (f h) (f x) (f y) (f z) (A f) (g h) (g x) (g y) (g z) (A g) (h x) (h y) (h z) (A h) (x y) (x z) (A x) (y z) (A y) (A z) (B f) (B g) (B h) (B x) (B y) (B z)) ((e. A (V)) (e. B (V))) (-> (A.e. x A (br (cv x) (~<_) B)) (br (U. A) (~<_) (X. A B)))) stmt (unidomg ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B z)) () (-> (/\/\ (e. A C) (e. B D) (A.e. x A (br (cv x) (~<_) B))) (br (U. A) (~<_) (X. A B)))) stmt (uniimadom ((x y) (A x) (A y) (B x) (B y) (F x) (F y)) ((e. A (V)) (e. B (V))) (-> (/\ (Fun F) (A.e. x A (br (` F (cv x)) (~<_) B))) (br (U. (" F A)) (~<_) (X. A B)))) stmt (uniimadomf ((x z) (A x) (A z) (x y) (B x) (y z) (B y) (B z) (F y) (F z)) ((-> (e. (cv y) F) (A. x (e. (cv y) F))) (e. A (V)) (e. B (V))) (-> (/\ (Fun F) (A.e. x A (br (` F (cv x)) (~<_) B))) (br (U. (" F A)) (~<_) (X. A B)))) stmt (iundom ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B z) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V))) (-> (A.e. x A (br C (~<_) B)) (br (U_ x A C) (~<_) (X. A B)))) stmt (oncardval ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (On)) (= (` (card) A) (|^| ({e.|} x (On) (br (cv x) (~~) A)))))) stmt (oncardon ((A x)) () (-> (e. A (On)) (e. (` (card) A) (On)))) stmt (oncardid ((A x)) () (-> (e. A (On)) (br (` (card) A) (~~) A))) stmt (cardonle ((A x)) () (-> (e. A (On)) (C_ (` (card) A) A))) stmt (card0 () () (= (` (card) ({/})) ({/}))) stmt (cardnn () () (-> (e. A (om)) (= (` (card) A) A))) stmt (cardom () () (= (` (card) (om)) (om))) stmt (cardval ((x y) (x z) (A x) (y z) (A y) (A z)) () (= (` (card) A) (|^| ({e.|} x (On) (br (cv x) (~~) A))))) stmt (cardon ((A x)) () (e. (` (card) A) (On))) stmt (cardid ((A x)) () (br (` (card) A) (~~) A)) stmt (oncard ((x y) (A x) (A y)) () (<-> (E. x (= A (` (card) (cv x)))) (= A (` (card) A)))) stmt (cardne ((A x) (B x)) () (-> (e. A (` (card) B)) (-. (br A (~~) B)))) stmt (carden () () (-> (/\ (e. A C) (e. B D)) (<-> (= (` (card) A) (` (card) B)) (br A (~~) B)))) stmt (cardeq0 () () (-> (e. A B) (<-> (= (` (card) A) ({/})) (= A ({/}))))) stmt (cardsn () () (-> (e. A B) (= (` (card) ({} A)) (1o)))) stmt (carddomi () () (-> (e. A C) (-> (C_ (` (card) A) (` (card) B)) (br A (~<_) B)))) stmt (carddom () () (-> (/\ (e. A C) (e. B D)) (<-> (C_ (` (card) A) (` (card) B)) (br A (~<_) B)))) stmt (cardsdom () () (-> (/\ (e. A C) (e. B D)) (<-> (e. (` (card) A) (` (card) B)) (br A (~<) B)))) stmt (domtri () () (-> (/\ (e. A C) (e. B D)) (<-> (br A (~<_) B) (-. (br B (~<) A))))) stmt (entri () () (-> (/\ (e. A C) (e. B D)) (\/\/ (br A (~<) B) (br A (~~) B) (br B (~<) A)))) stmt (entri2 () () (-> (/\ (e. A C) (e. B D)) (\/ (br A (~<_) B) (br B (~<) A)))) stmt (entri3 () () (-> (/\ (e. A C) (e. B D)) (\/ (br A (~<_) B) (br B (~<_) A)))) stmt (sucdom ((A x) (B x)) () (-> (/\ (e. A (om)) (e. B C)) (<-> (br A (~<) B) (br (suc A) (~<_) B)))) stmt (unxpdomlem ((x y) (x z) (w x) (v x) (u x) (t x) (f x) (A x) (y z) (w y) (v y) (u y) (t y) (f y) (A y) (w z) (v z) (u z) (t z) (f z) (A z) (v w) (u w) (t w) (f w) (A w) (u v) (t v) (f v) (A v) (t u) (f u) (A u) (f t) (A t) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B f)) ((e. A (V)) (e. B (V))) (-> (/\ (br (1o) (~<) A) (br (1o) (~<) B)) (br (u. A B) (~<_) (X. A B)))) stmt (unxpdom ((x y) (A x) (A y) (B y)) () (-> (/\ (br (1o) (~<) A) (br (1o) (~<) B)) (br (u. A B) (~<_) (X. A B)))) stmt (unxpdom2 () ((e. A (V)) (e. B (V))) (-> (/\ (br (1o) (~<) A) (br B (~<_) A)) (br (u. A B) (~<_) (X. A A)))) stmt (sucxpdom ((A x)) () (-> (br (1o) (~<) A) (br (suc A) (~<_) (X. A A)))) stmt (sdomel () () (-> (/\ (e. A (On)) (e. B (On))) (-> (br A (~<) B) (e. A B)))) stmt (sdomsdomcard () () (<-> (br A (~<) B) (br A (~<) (` (card) B)))) stmt (cardidm () () (= (` (card) (` (card) A)) (` (card) A))) stmt (canth3 () () (-> (e. A B) (e. (` (card) A) (` (card) (P~ A))))) stmt (cardlim ((A x)) () (<-> (C_ (om) (` (card) A)) (Lim (` (card) A)))) stmt (cardsdomel () () (-> (e. A (On)) (<-> (br A (~<) B) (e. A (` (card) B))))) stmt (iscard ((A x)) () (<-> (= (` (card) A) A) (/\ (e. A (On)) (A.e. x A (br (cv x) (~<) A))))) stmt (iscard2 ((x y) (A x) (A y)) () (<-> (= (` (card) A) A) (/\ (e. A (On)) (A.e. x (On) (-> (br A (~~) (cv x)) (C_ A (cv x))))))) stmt (cardval2 ((A x)) () (= (` (card) A) ({e.|} x (On) (br (cv x) (~<) A)))) stmt (ondomon ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A B) (e. ({e.|} x (On) (br (cv x) (~<_) A)) (On)))) stmt (ondomcard ((x y) (A x) (A y)) () (-> (e. A B) (= (` (card) ({e.|} x (On) (br (cv x) (~<_) A))) ({e.|} x (On) (br (cv x) (~<_) A))))) stmt (carduni ((x y) (A x) (A y) (B y)) () (-> (e. A B) (-> (A.e. x A (= (` (card) (cv x)) (cv x))) (= (` (card) (U. A)) (U. A))))) stmt (cardiun ((x y) (x z) (A x) (y z) (A y) (A z) (B y) (B z)) () (-> (e. A C) (-> (A.e. x A (= (` (card) B) B)) (= (` (card) (U_ x A B)) (U_ x A B))))) stmt (cardmin ((x y) (A x) (A y) (B y)) () (-> (e. A B) (= (` (card) (|^| ({e.|} x (On) (br A (~<) (cv x))))) (|^| ({e.|} x (On) (br A (~<) (cv x))))))) stmt (cardprc ((x y)) () (-. (e. ({|} x (= (` (card) (cv x)) (cv x))) (V)))) stmt (alephfnon ((x y) (x z) (y z)) () (Fn (aleph) (On))) stmt (aleph0 ((x y) (x z) (y z)) () (= (` (aleph) ({/})) (om))) stmt (alephlim ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A B) (Lim A)) (= (` (aleph) A) (U_ x A (` (aleph) (cv x)))))) stmt (alephon ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (e. (` (aleph) A) (On))) stmt (alephsuc ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (On)) (= (` (aleph) (suc A)) (|^| ({e.|} x (On) (br (` (aleph) A) (~<) (cv x))))))) stmt (alephcard ((x y) (A x) (A y)) () (= (` (card) (` (aleph) A)) (` (aleph) A))) stmt (alephnbtwn ((A x) (B x)) () (-> (= (` (card) B) B) (-. (/\ (e. (` (aleph) A) B) (e. B (` (aleph) (suc A))))))) stmt (alephnbtwn2 () () (-. (/\ (br (` (aleph) A) (~<) B) (br B (~<) (` (aleph) (suc A)))))) stmt (alephsucpw () () (br (` (aleph) (suc A)) (~<_) (P~ (` (aleph) A)))) stmt (aleph1 () () (br (` (aleph) (1o)) (~<_) (opr (2o) (^m) (` (aleph) ({/}))))) stmt (alephordlem1 ((x y) (A x) (A y)) () (-> (e. A (On)) (br (` (aleph) A) (~<) (` (aleph) (suc A))))) stmt (alephordlem2 ((A x) (B x)) () (-> (/\ (e. B (V)) (Lim B)) (-> (e. A B) (br (` (aleph) A) (~<_) (` (aleph) B))))) stmt (alephordi ((x y) (A x) (A y) (B x) (B y)) () (-> (e. B (On)) (-> (e. A B) (br (` (aleph) A) (~<) (` (aleph) B))))) stmt (alephord () () (-> (/\ (e. A (On)) (e. B (On))) (<-> (e. 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(cv y))))))))))) stmt (cf0 ((x y)) () (= (` (cf) ({/})) ({/}))) stmt (cardcf ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (= (` (card) (` (cf) A)) (` (cf) A))) stmt (cflecard ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (C_ (` (cf) A) (` (card) A))) stmt (cfle () () (C_ (` (cf) A) A)) stmt (cfeq0 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (e. A (On)) (<-> (= (` (cf) A) ({/})) (= A ({/}))))) stmt (cfsuc ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v)) () (-> (e. A (On)) (= (` (cf) (suc A)) (1o)))) stmt (cfom ((x y) (x z) (w x) (y z) (w y) (w z)) () (= (` (cf) (om)) (om))) stmt (cdavalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A C) (e. B D)) (= (opr A (+c) B) (u. (X. A ({} ({/}))) (X. B ({} (1o))))))) stmt (cdaval () ((e. A (V)) (e. B (V))) (= (opr A (+c) B) (u. (X. A ({} ({/}))) (X. 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(cv y) (cv x)))))) stmt (axpowndlem1 () () (-> (A. x (= (cv x) (cv y))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x)))))))) stmt (axpowndlem2 ((w x) (y z) (w z) (w y)) () (-> (-. (A. x (= (cv x) (cv y)))) (-> (-. (A. x (= (cv x) (cv z)))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x))))))))) stmt (axpowndlem3 ((w x) (y z) (w y) (w z)) () (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x))))))) stmt (axpowndlem4 ((w x) (w y) (w z)) () (-> (-. (A. y (= (cv y) (cv x)))) (-> (-. (A. y (= (cv y) (cv z)))) (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x))))))))) stmt (axpownd ((w x) (w y)) () (-> (-. (= (cv x) (cv y))) (E. x (A. y (-> (A. x (-> (E. z (e. (cv x) (cv y))) (A. y (e. (cv x) (cv z))))) (e. (cv y) (cv x))))))) stmt (axregndlem1 () () (-> (A. x (= (cv x) (cv z))) (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y)))))))))) stmt (axregndlem2 ((w x) (y z) (w z) (w y)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y))))))))) stmt (axregnd ((w x) (w y) (w z)) () (-> (e. (cv x) (cv y)) (E. x (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv x)) (-. (e. (cv z) (cv y))))))))) stmt (axinfndlem1 ((w x) (y z) (w z) (w y)) () (-> (A. x (e. (cv y) (cv z))) (E. x (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))))) stmt (axinfnd ((w x) (w y) (w z)) () (E. x (-> (e. (cv y) (cv z)) (/\ (e. (cv y) (cv x)) (A. y (-> (e. (cv y) (cv x)) (E. z (/\ (e. (cv y) (cv z)) (e. (cv z) (cv x)))))))))) stmt (axacndlem1 () () (-> (A. x (= (cv x) (cv y))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))))) stmt (axacndlem2 () () (-> (A. x (= (cv x) (cv z))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))))) stmt (axacndlem3 () () (-> (A. y (= (cv y) (cv z))) (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w))))))))))) stmt (axacndlem4 ((v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) stmt (axacndlem5 ((v x) (v y) (w z) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) stmt (axacnd ((v x) (v y) (v z) (v w)) () (E. x (A. y (A. z (-> (A. x (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w)))) (E. w (A. y (<-> (E. w (/\ (/\ (e. (cv y) (cv z)) (e. (cv z) (cv w))) (/\ (e. (cv y) (cv w)) (e. (cv w) (cv x))))) (= (cv y) (cv w)))))))))) stmt (zfcndext ((x y) (x z) (y z)) () (-> (A. z (<-> (e. (cv z) (cv x)) (e. (cv z) (cv y)))) (= (cv x) (cv y)))) stmt (zfcndrep ((x y) (x z) (w x) (y z) (w y) (w z)) () (-> (A. w (E. y (A. z (-> (A. y ph) (= (cv z) (cv y)))))) (E. y (A. z (<-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv w) (cv x)) (A. y ph)))))))) stmt (zfcndun ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (-> (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x)))) (e. (cv z) (cv y)))))) stmt (zfcndpow ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (A. z (-> (A. w (-> (e. (cv w) (cv z)) (e. (cv w) (cv x)))) (e. (cv z) (cv y)))))) stmt (zfcndreg ((x y) (x z) (y z)) () (-> (E. y (e. (cv y) (cv x))) (E. y (/\ (e. (cv y) (cv x)) (A. z (-> (e. (cv z) (cv y)) (-. (e. (cv z) (cv x))))))))) stmt (zfcndinf ((x y) (x z) (w x) (y z) (w y) (w z)) () (E. y (/\ (e. (cv x) (cv y)) (A. z (-> (e. (cv z) (cv y)) (E. w (/\ (e. (cv z) (cv w)) (e. (cv w) (cv y))))))))) stmt (zfcndac ((x y) (x z) (w x) (v x) (u x) (t x) (y z) (w y) (v y) (u y) (t y) (w z) (v z) (u z) (t z) (v w) (u w) (t w) (u v) (t v) (t u)) () (E. y (A. z (A. w (-> (/\ (e. (cv z) (cv w)) (e. (cv w) (cv x))) (E. v (A. u (<-> (E. t (/\ (/\ (e. (cv u) (cv w)) (e. (cv w) (cv t))) (/\ (e. (cv u) (cv t)) (e. (cv t) (cv y))))) (= (cv u) (cv v)))))))))) stmt (grothlem ((v w) (u w) (h w) (g w) (f w) (u v) (h v) (g v) (f v) (h u) (g u) (f u) (g h) (f h) (f g)) () (<-> (br (cv u) (cv w) (cv v)) (E. g (/\ (e. 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(/\ (e. A (om)) (e. ({/}) A)))) stmt (pinn () () (-> (e. A (N.)) (e. A (om)))) stmt (pion () () (-> (e. A (N.)) (e. A (On)))) stmt (piord () () (-> (e. A (N.)) (Ord A))) stmt (niex () () (e. (N.) (V))) stmt (0npi () () (-. (e. ({/}) (N.)))) stmt (1pi () () (e. (1o) (N.))) stmt (addpiord () () (-> (/\ (e. A (N.)) (e. B (N.))) (= (opr A (+N) B) (opr A (+o) B)))) stmt (mulpiord () () (-> (/\ (e. A (N.)) (e. B (N.))) (= (opr A (.N) B) (opr A (.o) B)))) stmt (mulidpi () () (-> (e. A (N.)) (= (opr A (.N) (1o)) A))) stmt (ltpiord ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (N.)) (e. B (N.))) (<-> (br A ( (/\ (e. A (N.)) (e. B (N.))) (e. (opr A (+N) B) (N.)))) stmt (mulclpi () () (-> (/\ (e. A (N.)) (e. B (N.))) (e. (opr A (.N) B) (N.)))) stmt (addcompi () ((e. A (V)) (e. B (V))) (= (opr A (+N) B) (opr B (+N) A))) stmt (addasspi () ((e. B (V)) (e. C (V))) (= (opr (opr A (+N) B) (+N) C) (opr A (+N) (opr B (+N) C)))) stmt (mulcompi () ((e. A (V)) (e. 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(V))) stmt (0npq () () (-. (e. ({/}) (Q.)))) stmt (ltrelpq ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (C_ ( (/\ (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (/\ (/\ (e. F (N.)) (e. G (N.))) (/\ (e. R (N.)) (e. S (N.))))) (-> (/\ (= (opr A (.N) D) (opr B (.N) C)) (= (opr F (.N) S) (opr G (.N) R))) (br (<,> (opr (opr A (.N) G) (+N) (opr B (.N) F)) (opr B (.N) G)) (~Q) (<,> (opr (opr C (.N) S) (+N) (opr D (.N) R)) (opr D (.N) S)))))) stmt (mulcmpblnq ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (F x) (F y) (F z) (G x) (G y) (G z) (R x) (R y) (R z) (S x) (S y) (S z)) ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V)) (e. F (V)) (e. G (V)) (e. R (V)) (e. S (V))) (-> (/\ (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (/\ (/\ (e. F (N.)) (e. G (N.))) (/\ (e. R (N.)) (e. S (N.))))) (-> (/\ (= (opr A (.N) D) (opr B (.N) C)) (= (opr F (.N) S) (opr G (.N) R))) (br (<,> (opr A (.N) F) (opr B (.N) G)) (~Q) (<,> (opr C (.N) R) (opr D (.N) S)))))) stmt (addpipq ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (= (opr ([] (<,> A B) (~Q)) (+Q) ([] (<,> C D) (~Q))) ([] (<,> (opr (opr A (.N) D) (+N) (opr B (.N) C)) (opr B (.N) D)) (~Q))))) stmt (mulpipq ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (N.)) (e. B (N.))) (/\ (e. C (N.)) (e. D (N.)))) (= (opr ([] (<,> A B) (~Q)) (.Q) ([] (<,> C D) (~Q))) ([] (<,> (opr A (.N) C) (opr B (.N) D)) (~Q))))) stmt (ordpipq ((A x) (A y) (A z) (A w) (A v) (A u) (A f) (x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V))) (<-> (br ([] (<,> A B) (~Q)) ( C D) (~Q))) (br (opr A (.N) D) ( (/\ (e. A (Q.)) (e. B (Q.))) (e. (opr A (+Q) B) (Q.)))) stmt (dmaddpq ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (+Q)) (X. (Q.) (Q.)))) stmt (mulclpq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (Q.)) (e. B (Q.))) (e. (opr A (.Q) B) (Q.)))) stmt (dmmulpq ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (.Q)) (X. (Q.) (Q.)))) stmt (addcompq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (V)) (e. B (V))) (= (opr A (+Q) B) (opr B (+Q) A))) stmt (addasspq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((e. B (V)) (e. C (V))) (= (opr (opr A (+Q) B) (+Q) C) (opr A (+Q) (opr B (+Q) C)))) stmt (mulcompq ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (V)) (e. B (V))) (= (opr A (.Q) B) (opr B (.Q) A))) stmt (mulasspq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((e. B (V)) (e. C (V))) (= (opr (opr A (.Q) B) (.Q) C) (opr A (.Q) (opr B (.Q) C)))) stmt (distrpqlem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. A (V)) (e. B (V)) (e. C (V))) (-> (/\/\ (e. A (N.)) (e. B (N.)) (e. C (N.))) (= ([] (<,> (opr A (.N) B) (opr A (.N) C)) (~Q)) ([] (<,> B C) (~Q))))) stmt (distrpq ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((e. B (V)) (e. C (V))) (= (opr A (.Q) (opr B (+Q) C)) (opr (opr A (.Q) B) (+Q) (opr A (.Q) C)))) stmt (1qec () ((e. A (V))) (-> (e. A (N.)) (= (1Q) ([] (<,> A A) (~Q))))) stmt (mulidpq ((x y) (A x) (A y)) () (-> (e. A (Q.)) (= (opr A (.Q) (1Q)) A))) stmt (recmulpq ((x y) (A x) (A y) (B x) (B y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) ((e. B (V))) (-> (e. A (Q.)) (<-> (= (` (*Q) A) B) (= (opr A (.Q) B) (1Q))))) stmt (recidpq () () (-> (e. A (Q.)) (= (opr A (.Q) (` (*Q) A)) (1Q)))) stmt (recclpq () () (-> (e. A (Q.)) (e. (` (*Q) A) (Q.)))) stmt (recrecpq () ((e. A (V))) (-> (e. A (Q.)) (= (` (*Q) (` (*Q) A)) A))) stmt (dmrecpq ((x y)) () (= (dom (*Q)) (Q.))) stmt (ltsopq ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (w z) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (Or ( (e. C (Q.)) (<-> (br A ( (e. C (Q.)) (<-> (br A ( (/\ (e. A (Q.)) (e. B (Q.))) (br A ( (/\ (e. A (Q.)) (e. B (Q.))) (<-> (br A ( (/\ (e. A (Q.)) (e. B (Q.))) (<-> (br A ( (e. A (Q.)) (E. x (= (opr (cv x) (+Q) (cv x)) A)))) stmt (nsmallpq ((A x)) () (-> (e. A (Q.)) (E. x (br (cv x) ( (br A ( (br A ( (e. A (P.)) (/\ (/\ (C: ({/}) A) (C: A (Q.))) (A.e. x A (/\ (A. y (-> (br (cv y) ( (e. A (P.)) (-. (= A ({/}))))) stmt (prpssnq ((x y) (A x) (A y)) () (-> (e. A (P.)) (C: A (Q.)))) stmt (elprpq () () (-> (/\ (e. A (P.)) (e. B A)) (e. B (Q.)))) stmt (0npr () () (-. (e. ({/}) (P.)))) stmt (prcdpq ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\ (e. A (P.)) (e. B A)) (-> (br C ( (/\ (/\ (e. A (P.)) (e. B A)) (e. C (Q.))) (-> (-. (e. C A)) (br B ( (/\ (e. A (P.)) (e. B A)) (E. x (/\ (e. (cv x) A) (br B ( (/\ (e. A (P.)) (e. B A)) (E. x (e. (opr B (+Q) (cv x)) A)))) stmt (ltrelpr ((x y)) () (C_ (,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A F B) ({|} f (E. g (E. h (/\ (/\ (e. (cv g) A) (e. (cv h) B)) (= (cv f) (opr (cv g) G (cv h)))))))))) stmt (genpelv ((f g) (f h) (C f) (g h) (C g) (C h) (F h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))) (e. C (V))) (-> (/\ (e. A (P.)) (e. B (P.))) (<-> (e. C (opr A F B)) (E. f (E. g (/\ (/\ (e. (cv f) A) (e. (cv g) B)) (= C (opr (cv f) G (cv g))))))))) stmt (genpprecl ((f g) (f h) (C f) (g h) (C g) (C h) (D f) (D g) (D h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (-> (/\ (e. C A) (e. D B)) (e. (opr C G D) (opr A F B))))) stmt (genpdm ((x y) (x z) (y z) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u)) ((= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (= (dom F) (X. (P.) (P.)))) stmt (genpn0 ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z))))))))))) (-> (/\ (e. A (P.)) (e. B (P.))) (C: ({/}) (opr A F B)))) stmt (genpss ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))) (-> (/\ (e. (cv g) (Q.)) (e. (cv h) (Q.))) (e. (opr (cv g) G (cv h)) (Q.)))) (-> (/\ (e. A (P.)) (e. B (P.))) (C_ (opr A F B) (Q.)))) stmt (genpnnp ((v w) (A w) (A v) (B w) (B v) (F w) (F v) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))) (-> (/\ (e. (cv w) (Q.)) (e. (cv v) (Q.))) (e. (opr (cv w) G (cv v)) (Q.))) (-> (e. (cv z) (Q.)) (<-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (-. (= (opr A F B) (Q.))))) stmt (genpcd ((F h) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))) (-> (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (-> (e. (cv f) (opr A F B)) (-> (br (cv x) (,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))) (-> (e. (cv v) (Q.)) (<-> (br (cv z) ( (/\ (e. A (P.)) (e. B (P.))) (-> (e. (cv f) (opr A F B)) (E. x (/\ (e. (cv x) (opr A F B)) (br (cv f) (,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))) (-> (/\ (e. (cv x) (Q.)) (e. (cv y) (Q.))) (e. (opr (cv x) G (cv y)) (Q.))) (-> (e. (cv h) (Q.)) (<-> (br (cv f) ( (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (e. (opr A F B) (P.)))) stmt (genpass ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (t x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (t y) (w z) (v z) (u z) (f z) (g z) (h z) (t z) (v w) (u w) (f w) (g w) (h w) (t w) (u v) (f v) (g v) (h v) (t v) (f u) (g u) (h u) (t u) (f g) (f h) (f t) (g h) (g t) (h t) (A x) (A f) (A g) (A h) (A t) (B x) (B f) (B g) (B h) (B t) (C x) (C y) (C z) (C f) (C g) (C h) (C t) (F x) (F y) (F z) (F h) (F t) (G t) (x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (G x) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (f w) (g w) (h w) (G w) (u v) (f v) (g v) (h v) (G v) (f u) (g u) (h u) (G u) (G f) (G g) (G h) (F f) (F g)) ((= F ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (P.)) (e. (cv v) (P.))) (= (cv u) ({|} x (E.e. y (cv w) (E.e. z (cv v) (= (cv x) (opr (cv y) G (cv z)))))))))) (e. B (V)) (e. C (V)) (= (dom F) (X. (P.) (P.))) (-> (/\ (e. (cv f) (P.)) (e. (cv g) (P.))) (e. (opr (cv f) F (cv g)) (P.))) (= (opr (opr (cv f) G (cv g)) G (cv h)) (opr (cv f) G (opr (cv g) G (cv h))))) (= (opr (opr A F B) F C) (opr A F (opr B F C)))) stmt (plpv ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A (+P.) B) ({|} x (E. y (E. z (/\ (/\ (e. (cv y) A) (e. (cv z) B)) (= (cv x) (opr (cv y) (+Q) (cv z)))))))))) stmt (mpv ((x y) (x z) (f x) (g x) (h x) (A x) (y z) (f y) (g y) (h y) (A y) (f z) (g z) (h z) (A z) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B f) (B g) (B h) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (= (opr A (.P.) B) ({|} x (E. y (E. z (/\ (/\ (e. (cv y) A) (e. (cv z) B)) (= (cv x) (opr (cv y) (.Q) (cv z)))))))))) stmt (dmplp ((x y) (x z) (y z) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (+P.)) (X. (P.) (P.)))) stmt (dmmp ((x y) (x z) (y z) (w x) (v x) (u x) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (.P.)) (X. (P.) (P.)))) stmt (1pr ((x y)) () (e. (1P) (P.))) stmt (addclprlem1 ((x y) (x z) (w x) (y z) (w y) (w z) (g y) (g z) (g w) (h y) (h z) (h w)) () (-> (/\ (/\ (e. A (P.)) (e. (cv g) A)) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (e. (opr A (+P.) B) (P.)))) stmt (mulclprlem ((x y) (x z) (w x) (v x) (u x) (g x) (h x) (y z) (w y) (v y) (u y) (g y) (h y) (w z) (v z) (u z) (g z) (h z) (v w) (u w) (g w) (h w) (u v) (g v) (h v) (g u) (h u) (g h) (A x) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (/\ (e. A (P.)) (e. (cv g) A)) (/\ (e. B (P.)) (e. (cv h) B))) (e. (cv x) (Q.))) (-> (br (cv x) ( (/\ (e. A (P.)) (e. B (P.))) (e. (opr A (.P.) B) (P.)))) stmt (addcompr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((e. A (V)) (e. B (V))) (= (opr A (+P.) B) (opr B (+P.) A))) stmt (addasspr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) ((e. B (V)) (e. C (V))) (= (opr (opr A (+P.) B) (+P.) C) (opr A (+P.) (opr B (+P.) C)))) stmt (mulcompr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((e. A (V)) (e. B (V))) (= (opr A (.P.) B) (opr B (.P.) A))) stmt (mulasspr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) ((e. B (V)) (e. C (V))) (= (opr (opr A (.P.) B) (.P.) C) (opr A (.P.) (opr B (.P.) C)))) stmt (distrlem1pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) () (-> (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (C_ (opr A (.P.) (opr B (+P.) C)) (opr (opr A (.P.) B) (+P.) (opr A (.P.) C))))) stmt (distrlem2pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) () (-> (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (-> (/\ (e. (cv x) A) (/\ (e. (cv y) B) (e. (cv z) C))) (e. (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv x) (.Q) (cv z))) (opr A (.P.) (opr B (+P.) C)))))) stmt (distrlem3pr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) () (-> (/\ (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (/\ (e. (cv x) A) (/\ (e. (cv y) B) (e. (cv z) C)))) (/\ (e. (cv x) (Q.)) (/\ (e. (cv y) (Q.)) (e. (cv z) (Q.)))))) stmt (distrlem4pr ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f)) () (-> (/\ (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (/\ (/\ (e. (cv x) A) (e. (cv f) A)) (/\ (e. (cv y) B) (e. (cv z) C)))) (e. (opr (opr (cv x) (.Q) (cv y)) (+Q) (opr (cv f) (.Q) (cv z))) (opr A (.P.) (opr B (+P.) C))))) stmt (distrlem5pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (A y) (w z) (v z) (u z) (f z) (g z) (h z) (A z) (v w) (u w) (f w) (g w) (h w) (A w) (u v) (f v) (g v) (h v) (A v) (f u) (g u) (h u) (A u) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (B g) (B h) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (C g) (C h)) () (-> (/\ (e. A (P.)) (/\ (e. B (P.)) (e. C (P.)))) (C_ (opr (opr A (.P.) B) (+P.) (opr A (.P.) C)) (opr A (.P.) (opr B (+P.) C))))) stmt (distrpr () ((e. B (V)) (e. C (V))) (= (opr A (.P.) (opr B (+P.) C)) (opr (opr A (.P.) B) (+P.) (opr A (.P.) C)))) stmt (1idpr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (A y) (w z) (v z) (u z) (f z) (g z) (A z) (v w) (u w) (f w) (g w) (A w) (u v) (f v) (g v) (A v) (f u) (g u) (A u) (f g) (A f) (A g)) () (-> (e. A (P.)) (= (opr A (.P.) (1P)) A))) stmt (ltprord ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (P.)) (e. B (P.))) (<-> (br A ( (/\ (e. A (P.)) (e. B (P.))) (\/\/ (C: A B) (= A B) (C: B A)))) stmt (ltsopr ((x y) (x z) (y z)) () (Or ( (e. C (N.)) (-> (/\ (/\ (e. B (Q.)) (A. x (-> (e. (cv x) A) (e. (opr (cv x) (+Q) B) A)))) (e. (cv y) A)) (e. (opr (cv y) (+Q) (opr ([] (<,> C (1o)) (~Q)) (.Q) B)) A)))) stmt (prlem934b ((w z) (v z) (u z) (v w) (u w) (u v) (f z) (f w) (f v) (f u) (g z) (h z) (g w) (h w) (g v) (h v) (g u) (h u) (f g) (f h) (g h)) () (-> (/\ (/\ (e. (cv u) (N.)) (e. (cv w) (N.))) (/\ (e. (cv v) (N.)) (e. (cv z) (N.)))) (\/ (= (opr ([] (<,> (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))) ([] (<,> (cv v) (cv u)) (~Q))) (br ([] (<,> (cv v) (cv u)) (~Q)) ( (opr (cv w) (.N) (cv v)) (1o)) (~Q)) (.Q) ([] (<,> (cv z) (cv w)) (~Q))))))) stmt (prlem934 ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (P.)) (e. B (Q.))) (E. x (/\ (e. (cv x) A) (-. (e. (opr (cv x) (+Q) B) A)))))) stmt (ltaddpr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (P.)) (e. B (P.))) (br A ( (e. C (P.)) (-> (= (opr A (+P.) B) C) (br A ( (e. B (P.)) (-> (C: A B) (-. (= C ({/})))))) stmt (ltexprlem2 ((x y) (A x) (A y) (B x) (B y) (C x)) ((= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B)))))) (-> (e. B (P.)) (C: C (Q.)))) stmt (ltexprlem3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C z)) ((= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B)))))) (-> (e. B (P.)) (-> (e. (cv x) C) (A. z (-> (br (cv z) ( (e. B (P.)) (-> (e. (cv x) C) (E. z (/\ (e. (cv z) C) (br (cv x) ( (/\ (e. B (P.)) (C: A B)) (e. C (P.)))) stmt (ltexprlem6 ((x y) (x z) (w x) (v x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (f y) (g y) (h y) (A y) (w z) (v z) (f z) (g z) (h z) (A z) (v w) (f w) (g w) (h w) (A w) (f v) (g v) (h v) (A v) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (C x) (C z) (C w) (C v) (C f) (C g) (C h) (u x) (u y) (u z) (u w) (u v) (f u) (g u) (h u)) ((= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B)))))) (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (C: A B)) (C_ (opr A (+P.) C) B))) stmt (ltexprlem7 ((x y) (x z) (w x) (v x) (f x) (g x) (h x) (A x) (y z) (w y) (v y) (f y) (g y) (h y) (A y) (w z) (v z) (f z) (g z) (h z) (A z) (v w) (f w) (g w) (h w) (A w) (f v) (g v) (h v) (A v) (f g) (f h) (A f) (g h) (A g) (A h) (B x) (B y) (B z) (B w) (B v) (C x) (C z) (C w) (C v) (C f) (C g) (C h) (u x) (u y) (u z) (u w) (u v) (f u) (g u) (h u)) ((= C ({|} x (E. y (/\ (-. (e. (cv y) A)) (e. (opr (cv y) (+Q) (cv x)) B)))))) (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (C: A B)) (C_ B (opr A (+P.) C)))) stmt (ltexpri ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. B (V))) (-> (br A ( (e. C (P.)) (-> (br A ( (e. C (P.)) (<-> (br A ( (/\ (e. A (P.)) (e. B (P.))) (-> (= (opr A (+P.) B) (opr A (+P.) C)) (= B C)))) stmt (prlem936a ((w x) (v x) (u x) (v w) (u w) (u v) (w y) (v y) (u y) (w z) (v z) (u z)) () (-> (/\ (e. (cv x) (Q.)) (/\ (e. (cv z) (Q.)) (e. (cv y) (Q.)))) (<-> (br (opr (cv y) (+Q) (cv z)) ( (/\ (e. (opr (cv y) (.Q) B) A) ph) (e. (opr (cv y) (+Q) (cv z)) A)) (-> (/\ (/\ (e. A (P.)) (e. (opr (cv y) (+Q) (cv z)) A)) (/\ (e. (cv x) (Q.)) (e. (cv z) (Q.)))) (-> ps ch)) (-> (/\ (e. (cv x) (Q.)) (/\ (e. (cv z) (Q.)) (e. (cv y) (Q.)))) (<-> ch th)) (-> (/\ (/\ (/\ (br (1Q) ( th ta)) (-> (/\ (e. A (P.)) ta) (-> ps (-. (e. (opr (cv x) (.Q) B) A))))) (-> (/\ (/\ (e. A (P.)) (e. (cv z) (Q.))) (/\ (/\ ph (e. (cv y) (Q.))) (/\ (br (1Q) ( (/\ (e. (cv x) A) ps) (/\ (e. (cv x) A) (-. (e. (opr (cv x) (.Q) B) A)))))) stmt (prlem936 ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (v x) (B x) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v)) ((e. B (V))) (-> (/\ (e. A (P.)) (br (1Q) ( (e. A (P.)) (C: ({/}) B))) stmt (reclem2pr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B z)) ((= B ({|} x (E. y (/\ (br (cv x) ( (e. A (P.)) (e. B (P.)))) stmt (reclem3pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (A y) (w z) (v z) (u z) (f z) (g z) (A z) (v w) (u w) (f w) (g w) (A w) (u v) (f v) (g v) (A v) (f u) (g u) (A u) (f g) (A f) (A g) (B x) (B z) (B w) (B v) (B u) (B f) (B g)) ((= B ({|} x (E. y (/\ (br (cv x) ( (e. A (P.)) (C_ (1P) (opr A (.P.) B)))) stmt (reclem4pr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (f y) (g y) (A y) (w z) (v z) (u z) (f z) (g z) (A z) (v w) (u w) (f w) (g w) (A w) (u v) (f v) (g v) (A v) (f u) (g u) (A u) (f g) (A f) (A g) (B x) (B z) (B w) (B v) (B u) (B f) (B g)) ((= B ({|} x (E. y (/\ (br (cv x) ( (e. A (P.)) (= (opr A (.P.) B) (1P)))) stmt (recexpr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (P.)) (E. x (/\ (e. (cv x) (P.)) (= (opr A (.P.) (cv x)) (1P)))))) stmt (suplem1pr ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (/\ (C_ A (P.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (P.)) (A. y (-> (e. (cv y) (P.)) (-> (e. (cv y) A) (br (cv y) ( (C_ A (P.)) (/\ (-> (e. (cv y) A) (-. (br (U. A) ( (br (cv y) ( (/\ (/\ (C_ A (P.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (P.)) (A. y (-> (e. (cv y) (P.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (P.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (<-> (br (<,> A B) (~R) (<,> C D)) (= (opr A (+P.) D) (opr B (+P.) C))))) stmt (dmenr ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (= (dom (~R)) (X. (P.) (P.)))) stmt (enrer ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (Er (~R))) stmt (enreceq () () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (<-> (= ([] (<,> A B) (~R)) ([] (<,> C D) (~R))) (= (opr A (+P.) D) (opr B (+P.) C))))) stmt (enrex ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (e. (~R) (V))) stmt (srex () () (e. (R.) (V))) stmt (ltrelsr ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (C_ ( (/\ (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (/\ (/\ (e. F (P.)) (e. G (P.))) (/\ (e. R (P.)) (e. S (P.))))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (br (<,> (opr A (+P.) F) (opr B (+P.) G)) (~R) (<,> (opr C (+P.) R) (opr D (+P.) S)))))) stmt (mulcmpblnrlem ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z) (D x) (D y) (D z) (F x) (F y) (F z) (G x) (G y) (G z) (R x) (R y) (R z) (S x) (S y) (S z)) ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V)) (e. F (V)) (e. G (V)) (e. R (V)) (e. S (V))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (= (opr (opr D (.P.) F) (+P.) (opr (opr (opr A (.P.) F) (+P.) (opr B (.P.) G)) (+P.) (opr (opr C (.P.) S) (+P.) (opr D (.P.) R)))) (opr (opr D (.P.) F) (+P.) (opr (opr (opr A (.P.) G) (+P.) (opr B (.P.) F)) (+P.) (opr (opr C (.P.) R) (+P.) (opr D (.P.) S))))))) stmt (mulcmpblnr () ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V)) (e. F (V)) (e. G (V)) (e. R (V)) (e. S (V))) (-> (/\ (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (/\ (/\ (e. F (P.)) (e. G (P.))) (/\ (e. R (P.)) (e. S (P.))))) (-> (/\ (= (opr A (+P.) D) (opr B (+P.) C)) (= (opr F (+P.) S) (opr G (+P.) R))) (br (<,> (opr (opr A (.P.) F) (+P.) (opr B (.P.) G)) (opr (opr A (.P.) G) (+P.) (opr B (.P.) F))) (~R) (<,> (opr (opr C (.P.) R) (+P.) (opr D (.P.) S)) (opr (opr C (.P.) S) (+P.) (opr D (.P.) R))))))) stmt (addsrpr ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (= (opr ([] (<,> A B) (~R)) (+R) ([] (<,> C D) (~R))) ([] (<,> (opr A (+P.) C) (opr B (+P.) D)) (~R))))) stmt (mulsrpr ((x y) (x z) (w x) (v x) (u x) (t x) (s x) (f x) (g x) (h x) (a x) (b x) (c x) (d x) (A x) (y z) (w y) (v y) (u y) (t y) (s y) (f y) (g y) (h y) (a y) (b y) (c y) (d y) (A y) (w z) (v z) (u z) (t z) (s z) (f z) (g z) (h z) (a z) (b z) (c z) (d z) (A z) (v w) (u w) (t w) (s w) (f w) (g w) (h w) (a w) (b w) (c w) (d w) (A w) (u v) (t v) (s v) (f v) (g v) (h v) (a v) (b v) (c v) (d v) (A v) (t u) (s u) (f u) (g u) (h u) (a u) (b u) (c u) (d u) (A u) (s t) (f t) (g t) (h t) (a t) (b t) (c t) (d t) (A t) (f s) (g s) (h s) (a s) (b s) (c s) (d s) (A s) (f g) (f h) (a f) (b f) (c f) (d f) (A f) (g h) (a g) (b g) (c g) (d g) (A g) (a h) (b h) (c h) (d h) (A h) (a b) (a c) (a d) (A a) (b c) (b d) (A b) (c d) (A c) (A d) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B s) (B f) (B g) (B h) (B a) (B b) (B c) (B d) (C x) (C y) (C z) (C w) (C v) (C u) (C t) (C s) (C f) (C g) (C h) (C a) (C b) (C c) (C d) (D x) (D y) (D z) (D w) (D v) (D u) (D t) (D s) (D f) (D g) (D h) (D a) (D b) (D c) (D d)) () (-> (/\ (/\ (e. A (P.)) (e. B (P.))) (/\ (e. C (P.)) (e. D (P.)))) (= (opr ([] (<,> A B) (~R)) (.R) ([] (<,> C D) (~R))) ([] (<,> (opr (opr A (.P.) C) (+P.) (opr B (.P.) D)) (opr (opr A (.P.) D) (+P.) (opr B (.P.) C))) (~R))))) stmt (ltsrpr ((A x) (A y) (A z) (A w) (A v) (A u) (A f) (x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) ((e. A (V)) (e. B (V)) (e. C (V)) (e. D (V))) (<-> (br ([] (<,> A B) (~R)) ( C D) (~R))) (br (opr A (+P.) D) ( (br (0R) ( A B) (~R))) (br B ( (/\ (e. A (R.)) (e. B (R.))) (e. (opr A (+R) B) (R.)))) stmt (mulclsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) () (-> (/\ (e. A (R.)) (e. B (R.))) (e. (opr A (.R) B) (R.)))) stmt (dmaddsr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (+R)) (X. (R.) (R.)))) stmt (dmmulsr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (= (dom (.R)) (X. (R.) (R.)))) stmt (addcomsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (V)) (e. B (V))) (= (opr A (+R) B) (opr B (+R) A))) stmt (addasssr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((e. B (V)) (e. C (V))) (= (opr (opr A (+R) B) (+R) C) (opr A (+R) (opr B (+R) C)))) stmt (mulcomsr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (V)) (e. B (V))) (= (opr A (.R) B) (opr B (.R) A))) stmt (mulasssr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((e. B (V)) (e. C (V))) (= (opr (opr A (.R) B) (.R) C) (opr A (.R) (opr B (.R) C)))) stmt (distrsr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (g x) (h x) (f y) (g y) (h y) (f z) (g z) (h z) (f w) (g w) (h w) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) ((e. B (V)) (e. C (V))) (= (opr A (.R) (opr B (+R) C)) (opr (opr A (.R) B) (+R) (opr A (.R) C)))) stmt (m1p1sr () () (= (opr (-1R) (+R) (1R)) (0R))) stmt (m1m1sr () () (= (opr (-1R) (.R) (-1R)) (1R))) stmt (ltsosr ((x y) (x z) (w x) (v x) (u x) (f x) (g x) (h x) (y z) (w y) (v y) (u y) (f y) (g y) (h y) (w z) (v z) (u z) (f z) (g z) (h z) (v w) (u w) (f w) (g w) (h w) (u v) (f v) (g v) (h v) (f u) (g u) (h u) (f g) (f h) (g h)) () (Or ( (e. A (R.)) (= (opr A (+R) (0R)) A))) stmt (1idsr ((x y) (A x) (A y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (-> (e. A (R.)) (= (opr A (.R) (1R)) A))) stmt (00sr ((x y) (A x) (A y)) () (-> (e. A (R.)) (= (opr A (.R) (0R)) (0R)))) stmt (ltasr ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u) (f x) (f y) (f z) (f w) (f v) (f u)) ((e. A (V)) (e. B (V))) (-> (e. C (R.)) (<-> (br A ( (e. A (R.)) (= (opr A (+R) (opr A (.R) (-1R))) (0R)))) stmt (negexsr ((A x)) () (-> (e. A (R.)) (E. x (/\ (e. (cv x) (R.)) (= (opr A (+R) (cv x)) (0R)))))) stmt (recexsrlem ((x y) (x z) (A x) (y z) (A y) (A z) (w x) (v x) (u x) (f x) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) ((e. A (V))) (-> (br (0R) ( (/\ (br (0R) ( (/\ (br (0R) ( (e. A (R.)) (-> (-. (= A (0R))) (br (0R) ( (e. A (R.)) (-> (-. (= A (0R))) (E. x (/\ (e. (cv x) (R.)) (= (opr A (.R) (cv x)) (1R))))))) stmt (ssgt0sr () ((e. A (V)) (e. B (V))) (-> (/\ (e. A (R.)) (e. B (R.))) (-> (-. (/\ (= A (0R)) (= B (0R)))) (br (0R) ( (br (0R) ( (opr A (+P.) (1P)) (1P)) (~R))) (e. A (P.)))) stmt (ltpsrpr () ((e. A (V)) (e. B (V))) (<-> (br ([] (<,> (opr A (+P.) (1P)) (1P)) (~R)) ( (opr B (+P.) (1P)) (1P)) (~R))) (br A ( (br (0R) ( (opr (cv x) (+P.) (1P)) (1P)) (~R)) A))))) stmt (suppsrlem ((w x) (A x) (A w) (B x) (B w)) ((= B ({|} w (e. ([] (<,> (opr (cv w) (+P.) (1P)) (1P)) (~R)) A)))) (-> (/\ (A. x (-> (e. (cv x) A) (br (0R) ( (opr (cv w) (+P.) (1P)) (1P)) (~R)) A)))) (-> (/\ (/\ (A. x (-> (e. (cv x) A) (br (0R) ( (br (0R) ( (e. (cv y) A) (br (cv y) ( (br (0R) ( (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (/\ (/\ (A. x (-> (e. (cv x) A) (br (0R) ( (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (/\ (E. y (/\ (e. (cv y) A) (br (0R) ( (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (opr C (+R) (opr A (+R) (-1R))) (R.)) (e. A (R.)))) stmt (supsrlem2 ((A x) (C x)) ((e. C (R.))) (<-> (e. A (R.)) (E. x (/\ (e. (cv x) (R.)) (= (opr C (+R) (opr (cv x) (+R) (-1R))) A))))) stmt (supsrlem3 ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z) (C x) (C y) (C z)) ((e. C (R.)) (e. A (V)) (e. B (V))) (<-> (br (opr C (+R) (opr A (+R) (-1R))) ( (e. D B) (e. (opr C (+R) (opr D (+R) (-1R))) A))) stmt (supsrlem5 ((w y) (A y) (A w) (B y) (B w) (C y) (C w)) ((e. C (R.)) (= B ({|} w (e. (opr C (+R) (opr (cv w) (+R) (-1R))) A)))) (-> (e. C A) (E. y (/\ (e. (cv y) B) (br (0R) ( (/\ (e. C A) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (/\ (/\ (C_ A (R.)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (R.)) (A. y (-> (e. (cv y) (R.)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (R.)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (e. (<,> A B) (CC)) (/\ (e. A (R.)) (e. B (R.))))) stmt (opelreal () () (<-> (e. (<,> A (0R)) (RR)) (e. A (R.)))) stmt (elreal ((x y) (A x) (A y)) () (<-> (e. A (RR)) (E. x (/\ (e. (cv x) (R.)) (= (<,> (cv x) (0R)) A))))) stmt (0ncn () () (-. (e. ({/}) (CC)))) stmt (ltrelre ((x y) (x z) (w x) (y z) (w y) (w z)) () (C_ ( (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr (<,> A B) (+) (<,> C D)) (<,> (opr A (+R) C) (opr B (+R) D))))) stmt (mulcnsr ((x y) (x z) (w x) (v x) (u x) (f x) (A x) (y z) (w y) (v y) (u y) (f y) (A y) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B y) (B z) (B w) (B v) (B u) (B f) (C x) (C y) (C z) (C w) (C v) (C u) (C f) (D x) (D y) (D z) (D w) (D v) (D u) (D f)) () (-> (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr (<,> A B) (x.) (<,> C D)) (<,> (opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) C) (+R) (opr A (.R) D)))))) stmt (eqresr () ((e. A (V))) (<-> (= (<,> A (0R)) (<,> B (0R))) (= A B))) stmt (addresr () () (-> (/\ (e. A (R.)) (e. B (R.))) (= (opr (<,> A (0R)) (+) (<,> B (0R))) (<,> (opr A (+R) B) (0R))))) stmt (mulresr () ((e. B (V))) (-> (/\ (e. A (R.)) (e. B (R.))) (= (opr (<,> A (0R)) (x.) (<,> B (0R))) (<,> (opr A (.R) B) (0R))))) stmt (ltresr ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (V)) (e. B (V))) (<-> (br (<,> A (0R)) ( B (0R))) (br A ( (cv w) (0R)) A)))) (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (/\ (C_ B (R.)) (-. (= B ({/})))))) stmt (supre ((w x) (v x) (u x) (A x) (v w) (u w) (A w) (u v) (A v) (A u) (x y) (x z) (B x) (y z) (w y) (v y) (u y) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) ((= B ({|} w (e. (<,> (cv w) (0R)) A)))) (-> (/\ (/\ (C_ A (RR)) (-. (= A ({/})))) (E. x (/\ (e. (cv x) (RR)) (A. y (-> (e. (cv y) (RR)) (-> (e. (cv y) A) (br (cv y) ( (e. (cv y) (RR)) (/\ (-> (e. (cv y) A) (-. (br (cv x) ( (br (cv y) ( (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr ([] (<,> A B) (`' (E))) (+) ([] (<,> C D) (`' (E)))) ([] (<,> (opr A (+R) C) (opr B (+R) D)) (`' (E)))))) stmt (mulcnsrec () () (-> (/\ (/\ (e. A (R.)) (e. B (R.))) (/\ (e. C (R.)) (e. D (R.)))) (= (opr ([] (<,> A B) (`' (E))) (x.) ([] (<,> C D) (`' (E)))) ([] (<,> (opr (opr A (.R) C) (+R) (opr (-1R) (.R) (opr B (.R) D))) (opr (opr B (.R) C) (+R) (opr A (.R) D))) (`' (E)))))) stmt (axaddopr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (:--> (+) (X. (CC) (CC)) (CC))) stmt (axmulopr ((x y) (x z) (w x) (v x) (u x) (f x) (y z) (w y) (v y) (u y) (f y) (w z) (v z) (u z) (f z) (v w) (u w) (f w) (u v) (f v) (f u)) () (:--> (x.) (X. (CC) (CC)) (CC))) stmt (axcnex () () (e. (CC) (V))) stmt (axresscn () () (C_ (RR) (CC))) stmt (ax1re () () (e. (1) (RR))) stmt (axicn () () (e. (i) (CC))) stmt (axaddcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (CC))) (e. (opr A (+) B) (CC)))) stmt (axaddrcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (RR)) (e. B (RR))) (e. (opr A (+) B) (RR)))) stmt (axmulcl ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CC)) (e. B (CC))) (e. (opr A (x.) 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(= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv x) (<_) (cv y))))) (e. (sup A (RR) (`' (<))) (RR)))) stmt (nnunb ((x y) (x z) (y z)) () (-. (E.e. x (RR) (A.e. y (NN) (\/ (br (cv y) (<) (cv x)) (= (cv y) (cv x))))))) stmt (arch ((x y) (A x) (A y)) () (-> (e. A (RR)) (E.e. x (NN) (br A (<) (cv x))))) stmt (nnreclt ((A n)) () (-> (/\ (e. A (RR)) (br (0) (<) A)) (E.e. n (NN) (br (opr (1) (/) (cv n)) (<) A)))) stmt (xrsupexmnf ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR*) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))))) (E.e. x (RR*) (/\ (A.e. y (u. A ({} (-oo))) (-. (br (cv x) (<) (cv y)))) (A.e. y (RR*) (-> (br (cv y) (<) (cv x)) (E.e. z (u. A ({} (-oo))) (br (cv y) (<) (cv z))))))))) stmt (xrinfmexpnf ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR*) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y))))))) (E.e. x (RR*) (/\ (A.e. y (u. A ({} (+oo))) (-. (br (cv y) (<) (cv x)))) (A.e. y (RR*) (-> (br (cv x) (<) (cv y)) (E.e. z (u. A ({} (+oo))) (br (cv z) (<) (cv y))))))))) stmt (xrsupsslem ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (RR*)) (\/ (C_ A (RR)) (e. (+oo) A))) (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR*) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))))))) stmt (xrinfmsslem ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\ (C_ A (RR*)) (\/ (C_ A (RR)) (e. (-oo) A))) (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR*) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y))))))))) stmt (xrsupss ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (RR*)) (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv x) (<) (cv y)))) (A.e. y (RR*) (-> (br (cv y) (<) (cv x)) (E.e. z A (br (cv y) (<) (cv z))))))))) stmt (xrinfmss ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (RR*)) (E.e. x (RR*) (/\ (A.e. y A (-. (br (cv y) (<) (cv x)))) (A.e. y (RR*) (-> (br (cv x) (<) (cv y)) (E.e. z A (br (cv z) (<) (cv y))))))))) stmt (xrub ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (C_ A (RR*)) (e. B (RR*))) (<-> (A.e. x (RR) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))) (A.e. x (RR*) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y)))))))) stmt (supxrOLD ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (/\ (A.e. x A (-. (br B (<) (cv x)))) (A.e. x (RR*) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) (= (sup A (RR*) (<)) B))) stmt (supxr ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (/\ (A.e. x A (-. (br B (<) (cv x)))) (A.e. x (RR) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) (= (sup A (RR*) (<)) B))) stmt (supxr2OLD ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (/\ (A.e. x A (br (cv x) (<_) B)) (A.e. x (RR*) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) (= (sup A (RR*) (<)) B))) stmt (supxr2 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\ (C_ A (RR*)) (e. B (RR*))) (/\ (A.e. x A (br (cv x) (<_) B)) (A.e. x (RR) (-> (br (cv x) (<) B) (E.e. y A (br (cv x) (<) (cv y))))))) (= (sup A (RR*) (<)) B))) stmt (supxrre ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (/\/\ (C_ A (RR)) (-. (= A ({/}))) (E.e. x (RR) (A.e. y A (br (cv y) (<_) (cv x))))) (= (sup A (RR*) (<)) (sup A (RR) (<))))) stmt (supxrcl ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (C_ A (RR*)) (e. 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(sup A (RR*) (<)) (RR)))) stmt (supxrgtmnf () () (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (br (-oo) (<) (sup A (RR*) (<))))) stmt (supxrre1 () () (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (<-> (e. (sup A (RR*) (<)) (RR)) (br (sup A (RR*) (<)) (<) (+oo))))) stmt (supxrre2 () () (-> (/\ (C_ A (RR)) (-. (= A ({/})))) (<-> (e. (sup A (RR*) (<)) (RR)) (-. (= (sup A (RR*) (<)) (+oo)))))) stmt (xrsup0 ((y z)) () (= (sup ({/}) (RR*) (<)) (-oo))) stmt (supxrub ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (C_ A (RR*)) (e. B A)) (br B (<_) (sup A (RR*) (<))))) stmt (supxrleub ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\/\ (C_ A (RR*)) (e. B (RR*)) (A.e. x A (br (cv x) (<_) B))) (br (sup A (RR*) (<)) (<_) B))) stmt (elnn0 () () (<-> (e. A (NN0)) (\/ (e. A (NN)) (= A (0))))) stmt (nnssnn0 () () (C_ (NN) (NN0))) stmt (nn0ssre () () (C_ (NN0) (RR))) stmt (nn0sscn () () (C_ (NN0) (CC))) stmt (nn0ex () () (e. (NN0) (V))) stmt (nnnn0t () () (-> (e. A (NN)) (e. A (NN0)))) stmt (nnnn0 () ((e. N (NN))) (e. N (NN0))) stmt (nn0ret () () (-> (e. A (NN0)) (e. A (RR)))) stmt (nn0cnt () () (-> (e. A (NN0)) (e. A (CC)))) stmt (nn0re () ((e. A (NN0))) (e. A (RR))) stmt (nn0cn () ((e. A (NN0))) (e. A (CC))) stmt (0nn0 () () (e. (0) (NN0))) stmt (1nn0 () () (e. (1) (NN0))) stmt (2nn0 () () (e. (2) (NN0))) stmt (lt0nnn0 () () (-> (/\ (e. A (RR)) (br A (<) (0))) (-. (e. A (NN0))))) stmt (nn0ge0t () () (-> (e. N (NN0)) (br (0) (<_) N))) stmt (nn0ge0 () ((e. N (NN0))) (br (0) (<_) N)) stmt (nn0le0eq0t () () (-> (e. N (NN0)) (<-> (br N (<_) (0)) (= N (0))))) stmt (nn0addclt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (e. (opr M (+) N) (NN0)))) stmt (nn0addcl () ((e. M (NN0)) (e. N (NN0))) (e. (opr M (+) N) (NN0))) stmt (nn0mulcl () ((e. M (NN0)) (e. N (NN0))) (e. (opr M (x.) N) (NN0))) stmt (nn0mulclt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (e. (opr M (x.) N) (NN0)))) stmt (peano2nn0 () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN0)))) stmt (nnnn0addclt () () (-> (/\ (e. M (NN)) (e. N (NN0))) (e. (opr M (+) N) (NN)))) stmt (nn0nnaddclt () () (-> (/\ (e. M (NN0)) (e. N (NN))) (e. (opr M (+) N) (NN)))) stmt (nn0ltp1let () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<) N) (br (opr M (+) (1)) (<_) N)))) stmt (nn0leltp1t () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (br M (<) (opr N (+) (1)))))) stmt (nn0ltlem1t () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1)))))) stmt (nn0addge1t () () (-> (/\ (e. A (RR)) (e. N (NN0))) (br A (<_) (opr A (+) N)))) stmt (nn0addge2t () () (-> (/\ (e. A (RR)) (e. N (NN0))) (br A (<_) (opr N (+) A)))) stmt (nn0addge1 () ((e. A (RR)) (e. N (NN0))) (br A (<_) (opr A (+) N))) stmt (nn0addge2 () ((e. A (RR)) (e. N (NN0))) (br A (<_) (opr N (+) A))) stmt (nn0le2x () ((e. N (NN0))) (br N (<_) (opr (2) (x.) N))) stmt (nn0lele2x () ((e. M (NN0)) (e. N (NN0))) (-> (br N (<_) M) (br N (<_) (opr (2) (x.) M)))) stmt (elz ((N x)) () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/\/ (= N (0)) (e. N (NN)) (e. (-u N) (NN)))))) stmt (nnnegz () () (-> (e. N (NN)) (e. (-u N) (ZZ)))) stmt (zret () () (-> (e. N (ZZ)) (e. N (RR)))) stmt (zcnt () () (-> (e. N (ZZ)) (e. N (CC)))) stmt (zre () ((e. A (ZZ))) (e. A (RR))) stmt (zssre () () (C_ (ZZ) (RR))) stmt (zsscn () () (C_ (ZZ) (CC))) stmt (zex () () (e. (ZZ) (V))) stmt (elnnz () () (<-> (e. N (NN)) (/\ (e. N (ZZ)) (br (0) (<) N)))) stmt (0z () () (e. (0) (ZZ))) stmt (elnn0z () () (<-> (e. N (NN0)) (/\ (e. N (ZZ)) (br (0) (<_) N)))) stmt (elznn0nn () () (<-> (e. N (ZZ)) (\/ (e. N (NN0)) (/\ (e. N (RR)) (e. (-u N) (NN)))))) stmt (elznn0 () () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/ (e. N (NN0)) (e. (-u N) (NN0)))))) stmt (elznn () () (<-> (e. N (ZZ)) (/\ (e. N (RR)) (\/ (e. N (NN)) (e. (-u N) (NN0)))))) stmt (nnssz () () (C_ (NN) (ZZ))) stmt (nn0ssz () () (C_ (NN0) (ZZ))) stmt (nnzt () () (-> (e. N (NN)) (e. N (ZZ)))) stmt (nn0zt () () (-> (e. N (NN0)) (e. N (ZZ)))) stmt (elnnz1 () () (<-> (e. N (NN)) (/\ (e. N (ZZ)) (br (1) (<_) N)))) stmt (znnnlt1t () () (-> (e. N (ZZ)) (<-> (-. (e. N (NN))) (br N (<) (1))))) stmt (nnzrab () () (= (NN) ({e.|} x (ZZ) (br (1) (<_) (cv x))))) stmt (nn0zrab () () (= (NN0) ({e.|} x (ZZ) (br (0) (<_) (cv x))))) stmt (1z () () (e. (1) (ZZ))) stmt (2z () () (e. (2) (ZZ))) stmt (nn0subt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (e. (opr N (-) M) (NN0))))) stmt (nn0sub2t () () (-> (/\/\ (e. M (NN0)) (e. N (NN0)) (br M (<_) N)) (e. (opr N (-) M) (NN0)))) stmt (znegclt () () (-> (e. N (ZZ)) (e. (-u N) (ZZ)))) stmt (nn0negz () () (-> (e. N (NN0)) (e. (-u N) (ZZ)))) stmt (zaddclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (+) N) (ZZ)))) stmt (peano2z () () (-> (e. N (ZZ)) (e. (opr N (+) (1)) (ZZ)))) stmt (peano2zd () ((-> ph (e. N (ZZ)))) (-> ph (e. (opr N (+) (1)) (ZZ)))) stmt (zsubclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (-) N) (ZZ)))) stmt (peano2zm () () (-> (e. N (ZZ)) (e. (opr N (-) (1)) (ZZ)))) stmt (zrevaddclt () () (-> (e. N (ZZ)) (<-> (/\ (e. M (CC)) (e. (opr M (+) N) (ZZ))) (e. M (ZZ))))) stmt (elnn0nn () () (<-> (e. N (NN0)) (/\ (e. N (CC)) (e. (opr N (+) (1)) (NN))))) stmt (nn0p1nnt () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN)))) stmt (elnnnn0 () () (<-> (e. N (NN)) (/\ (e. N (CC)) (e. (opr N (-) (1)) (NN0))))) stmt (elnnnn0b () () (<-> (e. N (NN)) (/\ (e. N (NN0)) (br (0) (<) N)))) stmt (elnnnn0c () () (<-> (e. N (NN)) (/\ (e. N (NN0)) (br (1) (<_) N)))) stmt (znnsubt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (e. (opr N (-) M) (NN))))) stmt (znn0subt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (e. (opr N (-) M) (NN0))))) stmt (znn0sub2t () () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (e. (opr N (-) M) (NN0)))) stmt (zmulclt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (e. (opr M (x.) N) (ZZ)))) stmt (zltp1let () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (br (opr M (+) (1)) (<_) N)))) stmt (zleltp1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (br M (<) (opr N (+) (1)))))) stmt (zlem1ltt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N)))) stmt (zltlem1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1)))))) stmt (nn0lem1ltt () () (-> (/\ (e. M (NN0)) (e. N (NN0))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N)))) stmt (nnlem1ltt () () (-> (/\ (e. M (NN)) (e. N (NN))) (<-> (br M (<_) N) (br (opr M (-) (1)) (<) N)))) stmt (nnltlem1t () () (-> (/\ (e. M (NN)) (e. N (NN))) (<-> (br M (<) N) (br M (<_) (opr N (-) (1)))))) stmt (z2get ((M k) (N k)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (E.e. k (ZZ) (/\ (br M (<_) (cv k)) (br N (<_) (cv k)))))) stmt (zextlet ((M k) (N k)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (A.e. k (ZZ) (<-> (br (cv k) (<_) M) (br (cv k) (<_) N)))) (= M N))) stmt (zextltt ((M k) (N k)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (A.e. k (ZZ) (<-> (br (cv k) (<) M) (br (cv k) (<) N)))) (= M N))) stmt (recnzt () () (-> (/\ (e. A (RR)) (br (1) (<) A)) (-. (e. (opr (1) (/) A) (ZZ))))) stmt (btwnnzt () () (-> (/\/\ (e. A (ZZ)) (br A (<) B) (br B (<) (opr A (+) (1)))) (-. (e. B (ZZ))))) stmt (gtndivt () () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br B (<) A)) (-. (e. (opr B (/) A) (ZZ))))) stmt (halfnz () () (-. (e. (opr (1) (/) (2)) (ZZ)))) stmt (primet ((A x)) () (-> (e. A (NN)) (<-> (A.e. x (NN) (-> (e. (opr A (/) (cv x)) (NN)) (\/ (= (cv x) (1)) (= (cv x) A)))) (A.e. x (NN) (-> (/\/\ (br (1) (<) (cv x)) (br (cv x) (<_) A) (e. (opr A (/) (cv x)) (NN))) (= (cv x) A)))))) stmt (msqznn () () (-> (/\ (e. A (ZZ)) (=/= A (0))) (e. (opr A (x.) A) (NN)))) stmt (nneo ((j k) (N j) (N k)) ((e. N (NN))) (<-> (e. (opr N (/) (2)) (NN)) (-. (e. (opr (opr N (+) (1)) (/) (2)) (NN))))) stmt (nneot () () (-> (e. N (NN)) (<-> (e. (opr N (/) (2)) (NN)) (-. (e. (opr (opr N (+) (1)) (/) (2)) (NN)))))) stmt (zeot () () (-> (e. N (ZZ)) (\/ (e. (opr N (/) (2)) (ZZ)) (e. (opr (opr N (+) (1)) (/) (2)) (ZZ))))) stmt (zneo () () (-> (/\ (e. A (ZZ)) (e. B (ZZ))) (-. (= (opr (2) (x.) A) (opr (opr (2) (x.) B) (+) (1)))))) stmt (peano2uz2 ((A x) (B x)) () (-> (/\ (e. A (ZZ)) (e. B ({e.|} x (ZZ) (br A (<_) (cv x))))) (e. (opr B (+) (1)) ({e.|} x (ZZ) (br A (<_) (cv x)))))) stmt (dfuz ((x y) (x z) (w x) (v x) (u x) (m x) (N x) (y z) (w y) (v y) (u y) (m y) (N y) (w z) (v z) (u z) (m z) (N z) (v w) (u w) (m w) (N w) (u v) (m v) (N v) (m u) (N u) (N m)) ((e. N (ZZ))) (= ({e.|} z (ZZ) (br N (<_) (cv z))) (|^| ({|} x (/\ (e. N (cv x)) (A.e. y (cv x) (e. (opr (cv y) (+) (1)) (cv x)))))))) stmt (peano5uz ((x y) (A x) (A y) (k x) (N x) (k y) (N y) (N k)) ((e. A (V)) (e. N (ZZ))) (-> (/\ (e. N A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ ({e.|} k (ZZ) (br N (<_) (cv k))) A))) stmt (peano5uzt ((A x) (k x) (N x) (N k)) ((e. A (V))) (-> (e. N (ZZ)) (-> (/\ (e. N A) (A.e. x A (e. (opr (cv x) (+) (1)) A))) (C_ ({e.|} k (ZZ) (br N (<_) (cv k))) A)))) stmt (uzind ((j w) (N j) (N w) (j ps) (ch j) (j th) (j ta) (k w) (k ph) (ph w) (j k) (M j) (M k) (M w)) ((-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) (-> (e. M (ZZ)) ps) (-> (/\/\ (e. M (ZZ)) (e. (cv k) (ZZ)) (br M (<_) (cv k))) (-> ch th))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) ta)) stmt (uzind2 ((N j) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (M k)) ((-> (= (cv j) (opr M (+) (1))) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) (-> (e. M (ZZ)) ps) (-> (/\/\ (e. M (ZZ)) (e. (cv k) (ZZ)) (br M (<) (cv k))) (-> ch th))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) ta)) stmt (uzind3 ((j k) (N j) (N k) (j ps) (ch j) (j th) (j ta) (m ph) (j m) (M j) (k m) (M m) (M k)) ((-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv m)) (<-> ph ch)) (-> (= (cv j) (opr (cv m) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) (-> (e. M (ZZ)) ps) (-> (/\ (e. M (ZZ)) (e. (cv m) ({e.|} k (ZZ) (br M (<_) (cv k))))) (-> ch th))) (-> (/\ (e. M (ZZ)) (e. N ({e.|} k (ZZ) (br M (<_) (cv k))))) ta)) stmt (uzindOLD ((x z) (w x) (A x) (w z) (A z) (A w) (ps x) (ps z) (ch x) (ch z) (th x) (th z) (ta x) (ta z) (w y) (ph y) (ph w) (x y) (B x) (y z) (B y) (B z) (B w)) ((-> (= (cv x) B) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (/\ (/\ (e. (cv y) (ZZ)) (e. B (ZZ))) (br B (<_) (cv y))) (-> ch th))) (-> (/\ (/\ (e. A (ZZ)) (e. B (ZZ))) (br B (<_) A)) ta)) stmt (uzind3OLD ((x z) (A x) (A z) (ps x) (ch x) (th x) (ta x) (ph y) (x y) (B x) (y z) (B y) (B z)) ((-> (= (cv x) B) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (/\ (e. B (ZZ)) (e. (cv y) ({e.|} z (ZZ) (br B (<_) (cv z))))) (-> ch th))) (-> (/\ (e. B (ZZ)) (e. A ({e.|} z (ZZ) (br B (<_) (cv z))))) ta)) stmt (uzwoOLD ((x y) (w x) (v x) (u x) (A x) (w y) (v y) (u y) (A y) (v w) (u w) (A w) (u v) (A v) (A u) (y z) (B y) (w z) (v z) (u z) (B z) (B w) (B v) (B u)) () (-> (/\ (e. B (ZZ)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) stmt (uzwo2OLD ((x y) (w x) (A x) (w y) (A y) (A w) (y z) (B y) (w z) (B z) (B w)) () (-> (/\ (e. B (ZZ)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) stmt (nn0ind ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((-> (= (cv x) (0)) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (e. (cv y) (NN0)) (-> ch th))) (-> (e. A (NN0)) ta)) stmt (nn0indALT ((x y) (A x) (ps x) (ch x) (th x) (ta x) (ph y)) ((-> (e. (cv y) (NN0)) (-> ch th)) ps (-> (= (cv x) (0)) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta))) (-> (e. A (NN0)) ta)) stmt (nn0ind-raph ((x y) (x z) (y z) (A x) (A z) (ps x) (ps z) (ch x) (ch z) (th x) (th z) (ta x) (ta z) (ph y) (ph z)) ((-> (= (cv x) (0)) (<-> ph ps)) (-> (= (cv x) (cv y)) (<-> ph ch)) (-> (= (cv x) (opr (cv y) (+) (1))) (<-> ph th)) (-> (= (cv x) A) (<-> ph ta)) ps (-> (e. (cv y) (NN0)) (-> ch th))) (-> (e. A (NN0)) ta)) stmt (btwnz ((x z) (A x) (A z) (A y)) () (-> (e. A (RR)) (/\ (E.e. x (ZZ) (br (cv x) (<) A)) (E.e. y (ZZ) (br A (<) (cv y)))))) stmt (uzwo3lem1 ((v z) (B z) (B v) (x y) (v x) (R x) (v y) (R y) (R v) (y z)) ((= R ({e.|} z (ZZ) (br B (<_) (cv z))))) (-> (e. B (RR)) (E!e. x R (A.e. y R (br (cv x) (<_) (cv y)))))) stmt (uzwo3lem2 ((x y) (x z) (w x) (v x) (t x) (y z) (w y) (v y) (t y) (w z) (v z) (t z) (v w) (t w) (t v) (A x) (A y) (B t) (S y) (S t) (R w) (v z) (B z) (B v) (x y) (v x) (R x) (v y) (R y) (R v) (y z)) ((= R ({e.|} z (ZZ) (br B (<_) (cv z)))) (= S (U. ({e.|} w R (A.e. v R (br (cv w) (<_) (cv v))))))) (-> (/\ (e. B (RR)) (/\ (C_ A R) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) stmt (uzwo3 ((x y) (u x) (t x) (A x) (u y) (t y) (A y) (t u) (A u) (A t) (w z) (v z) (u z) (t z) (B z) (v w) (u w) (t w) (B w) (u v) (t v) (B v) (B u) (B t)) () (-> (/\ (e. B (RR)) (/\ (C_ A ({e.|} z (ZZ) (br B (<_) (cv z)))) (=/= A ({/})))) (E!e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) stmt (zmin ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br A (<_) (cv x)) (A.e. y (ZZ) (-> (br A (<_) (cv y)) (br (cv x) (<_) (cv y)))))))) stmt (zmax ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br (cv x) (<_) A) (A.e. y (ZZ) (-> (br (cv y) (<_) A) (br (cv y) (<_) (cv x)))))))) stmt (zbtwnre ((x y) (A x) (A y)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br A (<_) (cv x)) (br (cv x) (<) (opr A (+) (1))))))) stmt (rebtwnz ((x y) (A x) (A y)) () (-> (e. A (RR)) (E!e. x (ZZ) (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1))))))) stmt (flvalt ((x y) (x z) (A x) (y z) (A y) (A z)) () (-> (e. A (RR)) (= (` (floor) A) (U. ({e.|} x (ZZ) (/\ (br (cv x) (<_) A) (br A (<) (opr (cv x) (+) (1))))))))) stmt (flclt ((A z)) () (-> (e. A (RR)) (e. (` (floor) A) (ZZ)))) stmt (flleltt ((x y) (A x) (A y)) () (-> (e. A (RR)) (/\ (br (` (floor) A) (<_) A) (br A (<) (opr (` (floor) A) (+) (1)))))) stmt (fllet () () (-> (e. A (RR)) (br (` (floor) A) (<_) A))) stmt (flltp1t () () (-> (e. A (RR)) (br A (<) (opr (` (floor) A) (+) (1))))) stmt (flget () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (br B (<_) A) (br B (<_) (` (floor) A))))) stmt (flltt () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (br A (<) B) (br (` (floor) A) (<) B)))) stmt (flidt () () (-> (e. A (ZZ)) (= (` (floor) A) A))) stmt (flval2t ((x y) (A x) (A y)) () (-> (e. A (RR)) (= (` (floor) A) (U. ({e.|} x (ZZ) (/\ (br (cv x) (<_) A) (A.e. y (ZZ) (-> (br (cv y) (<_) A) (br (cv y) (<_) (cv x)))))))))) stmt (flval3t ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (RR)) (= (` (floor) A) (sup ({e.|} x (ZZ) (br (cv x) (<_) A)) (RR) (<))))) stmt (flwordit () () (-> (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)) (br (` (floor) A) (<_) (` (floor) B)))) stmt (flbit ((A x) (B x)) () (-> (/\ (e. A (RR)) (e. B (ZZ))) (<-> (= (` (floor) A) B) (/\ (br B (<_) A) (br A (<) (opr B (+) (1))))))) stmt (flge0nn0t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (e. (` (floor) A) (NN0)))) stmt (flge1nnt () () (-> (/\ (e. A (RR)) (br (1) (<_) A)) (e. (` (floor) A) (NN)))) stmt (fladdzt () () (-> (/\ (e. A (RR)) (e. B (ZZ))) (= (` (floor) (opr A (+) B)) (opr (` (floor) A) (+) B)))) stmt (btwnzge0t () () (-> (/\ (/\ (e. A (RR)) (e. B (ZZ))) (/\ (br B (<_) A) (br A (<) (opr B (+) (1))))) (<-> (br (0) (<_) A) (br (0) (<_) B)))) stmt (flhalft () () (-> (e. N (ZZ)) (br N (<_) (opr (2) (x.) (` (floor) (opr (opr N (+) (1)) (/) (2))))))) stmt (elq ((x y) (x z) (A x) (y z) (A y) (A z)) () (<-> (e. A (QQ)) (E.e. x (ZZ) (E.e. y (NN) (= A (opr (cv x) (/) (cv y))))))) stmt (znq ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (ZZ)) (e. B (NN))) (e. (opr A (/) B) (QQ)))) stmt (qret ((x y) (A x) (A y)) () (-> (e. A (QQ)) (e. A (RR)))) stmt (zqt ((x y) (A x) (A y)) () (-> (e. A (ZZ)) (e. A (QQ)))) stmt (zssq () () (C_ (ZZ) (QQ))) stmt (nn0ssq () () (C_ (NN0) (QQ))) stmt (nnssq () () (C_ (NN) (QQ))) stmt (qssre () () (C_ (QQ) (RR))) stmt (qsscn () () (C_ (QQ) (CC))) stmt (nnqt () () (-> (e. A (NN)) (e. A (QQ)))) stmt (qcnt () () (-> (e. A (QQ)) (e. A (CC)))) stmt (qex () () (e. (QQ) (V))) stmt (qaddclt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (+) B) (QQ)))) stmt (qnegclt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (QQ)) (e. (-u A) (QQ)))) stmt (qmulclt ((x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u)) () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (x.) B) (QQ)))) stmt (qsubclt () () (-> (/\ (e. A (QQ)) (e. B (QQ))) (e. (opr A (-) B) (QQ)))) stmt (qrecclt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A (QQ)) (=/= A (0))) (e. (opr (1) (/) A) (QQ)))) stmt (qdivclt () () (-> (/\ (/\ (e. A (QQ)) (e. B (QQ))) (=/= B (0))) (e. (opr A (/) B) (QQ)))) stmt (qrevaddclt () () (-> (e. B (QQ)) (<-> (/\ (e. A (CC)) (e. (opr A (+) B) (QQ))) (e. A (QQ))))) stmt (nnrecqt () () (-> (e. A (NN)) (e. (opr (1) (/) A) (QQ)))) stmt (qbtwnre ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (br A (<) B)) (E.e. x (QQ) (/\ (br A (<) (cv x)) (br (cv x) (<) B))))) stmt (monoord ((y z) (A y) (A z) (B y) (x y) (x z) (F x) (F y) (F z)) ((:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (` F (cv x)) (<_) (` F (opr (cv x) (+) (1)))))) (-> (/\/\ (e. A (NN)) (e. B (NN)) (br A (<_) B)) (br (` F A) (<_) (` F B)))) stmt (om2uz0 ((x y) (C x) (C y)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (= (` G ({/})) C)) stmt (om2uzsuc ((x y) (w x) (v x) (w y) (v y) (v w) (G w) (G v) (A w) (A v) (C x) (C y) (C w) (C v)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. A (om)) (= (` G (suc A)) (opr (` G A) (+) (1))))) stmt (om2uzuz ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (A z) (A w) (A v) (C x) (C y) (C z) (C w) (C v)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. A (om)) (e. (` G A) ({e.|} z (ZZ) (br C (<_) (cv z)))))) stmt (om2uzlt ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (A z) (A w) (A v) (B z) (B w) (B v) (C x) (C y) (C z) (C w) (C v)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (/\ (e. A (om)) (e. B (om))) (-> (e. A B) (br (` G A) (<) (` G B))))) stmt (om2uzlt2 ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (/\ (e. A (om)) (e. B (om))) (<-> (e. A B) (br (` G A) (<) (` G B))))) stmt (om2uzran ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (G z) (G w) (G v) (G u) (C x) (C y) (C z) (C w) (C v) (C u)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (= (ran G) ({e.|} z (ZZ) (br C (<_) (cv z))))) stmt (om2uzf1o ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (G z) (G w) (G v) (C x) (C y) (C z) (C w) (C v)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (:-1-1-onto-> G (om) ({e.|} z (ZZ) (br C (<_) (cv z))))) stmt (uzrdgval ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. B ({e.|} z (ZZ) (br C (<_) (cv z)))) (= (` (o. (rec F A) (`' G)) B) (` (rec F A) (` (`' G) B))))) stmt (uzrdgini ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. A B) (= (` (o. (rec F A) (`' G)) C) A))) stmt (uzrdgsuc ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. B ({e.|} z (ZZ) (br C (<_) (cv z)))) (= (` (o. (rec F A) (`' G)) (opr B (+) (1))) (` F (` (o. (rec F A) (`' G)) B))))) stmt (uzrdginip1 ((x y) (x z) (y z) (G z) (A z) (B z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (-> (e. A B) (= (` (o. (rec F A) (`' G)) (opr C (+) (1))) (` F A)))) stmt (uzrdgfnuz ((x y) (x z) (y z) (G z) (A z) (C x) (C y) (C z)) ((e. C (ZZ)) (= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) C) (om)))) (Fn (o. (rec F A) (`' G)) ({e.|} z (ZZ) (br C (<_) (cv z))))) stmt (seq1lem1 ((x y) (v x) (v y) (w z) (v z) (u z) (t z) (f z) (v w) (u w) (t w) (f w) (u v) (t v) (f v) (t u) (f u) (f t) (A v) (B w) (B v) (B u) (B t) (B f) (u x) (t x) (f x) (C x) (C v) (C u) (C t) (C f) (G v) (G u) (G t) (G f)) ((= G (|` (rec ({<,>|} x y (= (cv y) (opr (cv x) (+) (1)))) (1)) (om)))) (-> (e. A (NN)) (= (` (1st) (` (o. (rec ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) B))) (<,> (1) C)) (`' G)) A)) A))) stmt (seq1lem2 () () (-> (e. A (NN)) (e. (opr A (+) (1)) (\ (NN) ({} (1)))))) stmt (peano3nn0 () () (-> (e. N (NN0)) (e. (opr N (+) (1)) (NN)))) stmt (seq1rval ((A z) (S z) (S w) (w z) (F z) (F w)) ((= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1)))))))) (e. A (V))) (= (` H A) (<,> (opr (` (1st) A) (+) (1)) (opr (` (2nd) A) S (` F (opr (` (1st) A) (+) (1))))))) stmt (seq1val ((x y) (f x) (g x) (h x) (G x) (f y) (g y) (h y) (G y) (f g) (f h) (G f) (g h) (G g) (G h) (H x) (H y) (H f) (H g) (H h) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (f z) (g z) (h z) (F z) (f w) (g w) (h w) (F w) (F f) (F g) (F h) (S x) (S y) (S z) (S w) (S f) (S g) (S h)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (= (opr S (seq1) F) ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) (cv x)))))))) stmt (seq1fnlem ((x y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (Fn (opr S (seq1) F) (NN))) stmt (seq1val2 ((x y) (A x) (A y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (-> (e. A (NN)) (= (` (opr S (seq1) F) A) (` (2nd) (` (o. (rec H (<,> (1) (` F (1)))) (`' G)) A))))) stmt (seq11lem ((w z) (F z) (F w) (S z) (S w)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (= (` (opr S (seq1) F) (1)) (` F (1)))) stmt (seq1suclem ((x y) (A x) (A y) (G x) (G y) (H x) (H y) (x z) (w x) (F x) (y z) (w y) (F y) (w z) (F z) (F w) (S x) (S y) (S z) (S w)) ((e. S (V)) (e. F (V)) (= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om))) (= H ({<,>|} z w (= (cv w) (<,> (opr (` (1st) (cv z)) (+) (1)) (opr (` (2nd) (cv z)) S (` F (opr (` (1st) (cv z)) (+) (1))))))))) (-> (e. A (NN)) (= (` (opr S (seq1) F) (opr A (+) (1))) (opr (` (opr S (seq1) F) A) S (` F (opr A (+) (1))))))) stmt (seq11 ((w z) (S z) (S w) (F z) (F w)) ((e. S (V)) (e. F (V))) (= (` (opr S (seq1) F) (1)) (` F (1)))) stmt (seq1p1 ((w z) (S z) (S w) (F z) (F w)) ((e. S (V)) (e. F (V))) (-> (e. A (NN)) (= (` (opr S (seq1) F) (opr A (+) (1))) (opr (` (opr S (seq1) F) A) S (` F (opr A (+) (1))))))) stmt (seq1m1 () ((e. S (V)) (e. F (V))) (-> (/\ (e. N (NN)) (br (1) (<) N)) (= (` (opr S (seq1) F) N) (opr (` (opr S (seq1) F) (opr N (-) (1))) S (` F N))))) stmt (seq1fn ((w z) (S z) (S w) (F z) (F w)) ((e. S (V)) (e. F (V))) (Fn (opr S (seq1) F) (NN))) stmt (seq1rn ((x y) (x z) (w x) (v x) (S x) (y z) (w y) (v y) (S y) (w z) (v z) (S z) (v w) (S w) (S v) (F x) (F y) (F z) (F w) (F v) (C x) (C y) (C z) (C w) (C v) (D x) (D y) (D z) (D w) (D v)) ((e. S (V)) (e. F (V))) (-> (/\/\ (e. (` F (1)) C) (:--> (|` F (\ (NN) ({} (1)))) (\ (NN) ({} (1))) D) (:--> S (X. C D) C)) (C_ (ran (opr S (seq1) F)) C))) stmt (seq1rn2 () ((e. S (V)) (e. F (V))) (-> (/\ (:--> F (NN) C) (:--> S (X. C C) C)) (C_ (ran (opr S (seq1) F)) C))) stmt (seq1f () ((e. S (V)) (e. F (V))) (-> (/\ (:--> F (NN) C) (:--> S (X. C C) C)) (:--> (opr S (seq1) F) (NN) C))) stmt (seq1cl () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. A (NN)) (:--> F (NN) C) (:--> S (X. C C) C)) (e. (` (opr S (seq1) F) A) C))) stmt (seq1res ((x y) (x z) (S x) (y z) (S y) (S z) (F x) (F y) (F z)) ((e. S (V)) (e. F (V))) (= (opr S (seq1) (|` F (NN))) (opr S (seq1) F))) stmt (ser1ft ((F f)) () (-> (:--> F (NN) (CC)) (:--> (opr (+) (seq1) F) (NN) (CC)))) stmt (ser1f () ((:--> F (NN) (CC))) (:--> (opr (+) (seq1) F) (NN) (CC))) stmt (ser1cl () ((:--> F (NN) (CC))) (-> (e. A (NN)) (e. (` (opr (+) (seq1) F) A) (CC)))) stmt (ser1recl ((A x) (x y) (F x) (F y)) ((:--> F (NN) (RR))) (-> (e. A (NN)) (e. (` (opr (+) (seq1) F) A) (RR)))) stmt (ser1re ((F x)) ((:--> F (NN) (RR))) (:--> (opr (+) (seq1) F) (NN) (RR))) stmt (ser1cl2 ((x y) (A y) (B x)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A)))) (A.e. x (NN) (e. A (CC)))) (-> (e. B (NN)) (e. (` (opr (+) (seq1) F) B) (CC)))) stmt (ser1f2 ((x y) (x z) (y z) (A y) (F z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A)))) (A.e. x (NN) (e. A (CC)))) (:--> (opr (+) (seq1) F) (NN) (CC))) stmt (ser11 ((B y) (x y) (A y) (B x)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A)))) (e. B (V)) (-> (= (cv x) (1)) (= A B))) (= (` (opr (+) (seq1) F) (1)) B)) stmt (ser1p1 ((B y) (x y) (C x) (C y) (x y) (A y) (B x)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) A)))) (e. C (V)) (-> (= (cv x) (opr B (+) (1))) (= A C))) (-> (e. B (NN)) (= (` (opr (+) (seq1) F) (opr B (+) (1))) (opr (` (opr (+) (seq1) F) B) (+) C)))) stmt (ser1mono ((A y) (x y) (F x) (F y)) ((:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (-> (e. A (NN)) (br (` (opr (+) (seq1) F) A) (<_) (` (opr (+) (seq1) F) (opr A (+) (1)))))) stmt (ser1add2 ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N k) (N m)) ((:--> F (NN) (CC)) (:--> G (NN) (CC)) (e. H (V)) (-> (/\/\ (e. (cv k) (NN)) (e. N (NN)) (br (cv k) (<_) N)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N))))) stmt (ser1add ((j k) (k m) (F k) (j m) (F j) (F m) (G k) (G j) (G m) (H k) (H j) (H m) (N j) (N k) (N m)) ((:--> F (NN) (CC)) (:--> G (NN) (CC)) (e. H (V)) (-> (/\ (e. (cv k) (NN)) (br (cv k) (<_) N)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))) (-> (e. N (NN)) (= (` (opr (+) (seq1) H) N) (opr (` (opr (+) (seq1) F) N) (+) (` (opr (+) (seq1) G) N))))) stmt (shftfval ((x y) (w x) (f x) (g x) (A x) (w y) (f y) (g y) (A y) (f w) (g w) (A w) (f g) (A f) (A g) (F x) (F y) (F w) (F f) (F g) (B x) (B y)) ((e. F (V))) (-> (e. A B) (= (opr F (shift) A) ({<,>|} x y (/\ (e. (cv x) (CC)) (= (cv y) (` F (opr (cv x) (-) A)))))))) stmt (shftfn ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((e. F (V))) (-> (e. A B) (Fn (opr F (shift) A) (CC)))) stmt (shftres () ((e. F (V))) (-> (/\ (e. A C) (C_ B (CC))) (Fn (|` (opr F (shift) A) B) B))) stmt (shftresvalt () ((e. F (V))) (-> (e. B C) (= (` (|` (opr F (shift) A) C) B) (` (opr F (shift) A) B)))) stmt (shftvalt ((x y) (A x) (A y) (F x) (F y) (B x) (B y) (C x) (C y)) ((e. F (V))) (-> (/\ (e. A C) (e. B (CC))) (= (` (opr F (shift) A) B) (` F (opr B (-) A))))) stmt (shftval2t () ((e. F (V))) (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. C (CC))) (= (` (opr F (shift) (opr A (-) B)) (opr A (+) C)) (` F (opr B (+) C))))) stmt (shftval3t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) (opr A (-) B)) A) (` F B)))) stmt (shftval4t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) (-u A)) B) (` F (opr A (+) B))))) stmt (shftval5t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr F (shift) A) (opr B (+) A)) (` F B)))) stmt (shftf ((A x) (F x) (B x) (C x)) ((e. F (V))) (-> (/\/\ (e. A D) (C_ B (CC)) (A.e. x B (e. (` F (opr (cv x) (-) A)) C))) (:--> (|` (opr F (shift) A) B) B C))) stmt (2shft ((x y) (A x) (A y) (F x) (F y) (B x) (B y)) ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (opr (opr F (shift) A) (shift) B) (opr F (shift) (opr A (+) B))))) stmt (shftcan2t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr (opr F (shift) (-u A)) (shift) A) B) (` F B)))) stmt (shftcan1t () ((e. F (V))) (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (opr (opr F (shift) A) (shift) (-u A)) B) (` F B)))) stmt (shftidt () ((e. F (V))) (-> (e. A (CC)) (= (` (opr F (shift) (0)) A) (` F A)))) stmt (seq1shftid ((j k) (j n) (S j) (k n) (S k) (S n) (F j) (F k) (F n)) ((e. S (V)) (e. F (V))) (= (opr S (seq1) (opr F (shift) (0))) (opr S (seq1) F))) stmt (uzvalt ((j k) (j y) (N j) (k y) (N k) (N y)) () (-> (e. N (ZZ)) (= (` (ZZ>) N) ({e.|} k (ZZ) (br N (<_) (cv k)))))) stmt (eluz1t ((N k) (M k)) () (-> (e. M (ZZ)) (<-> (e. N (` (ZZ>) M)) (/\ (e. N (ZZ)) (br M (<_) N))))) stmt (eluz2t ((j k) (j y) (N j) (k y) (N k) (N y) (M k)) () (<-> (e. N (` (ZZ>) M)) (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)))) stmt (eluz1 () ((e. M (ZZ))) (<-> (e. N (` (ZZ>) M)) (/\ (e. N (ZZ)) (br M (<_) N)))) stmt (eluzelz () () (-> (e. N (` (ZZ>) M)) (e. N (ZZ)))) stmt (eluzel2 () () (-> (e. N (` (ZZ>) M)) (e. M (ZZ)))) stmt (eluzle () () (-> (e. N (` (ZZ>) M)) (br M (<_) N))) stmt (eluzt () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (e. N (` (ZZ>) M)) (br M (<_) N)))) stmt (uzidt () () (-> (e. M (ZZ)) (e. M (` (ZZ>) M)))) stmt (uztrn () () (-> (/\ (e. M (` (ZZ>) K)) (e. K (` (ZZ>) N))) (e. M (` (ZZ>) N)))) stmt (uznegit () () (-> (e. N (` (ZZ>) M)) (e. (-u M) (` (ZZ>) (-u N))))) stmt (uzssz ((j k) (j y) (M j) (k y) (M k) (M y)) () (C_ (` (ZZ>) M) (ZZ))) stmt (uzss ((M k) (N k)) () (-> (e. N (` (ZZ>) M)) (C_ (` (ZZ>) N) (` (ZZ>) M)))) stmt (uz11t () () (-> (e. M (ZZ)) (<-> (= (` (ZZ>) M) (` (ZZ>) N)) (= M N)))) stmt (eluzp1m1t () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (e. (opr N (-) (1)) (` (ZZ>) M)))) stmt (eluzp1lt () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (br M (<) N))) stmt (eluzp1p1t () () (-> (e. N (` (ZZ>) M)) (e. (opr N (+) (1)) (` (ZZ>) (opr M (+) (1)))))) stmt (nn0uz () () (= (NN0) (` (ZZ>) (0)))) stmt (nnuz () () (= (NN) (` (ZZ>) (1)))) stmt (elnnuz () () (<-> (e. N (NN)) (e. N (` (ZZ>) (1))))) stmt (elnn0uz () () (<-> (e. N (NN0)) (e. N (` (ZZ>) (0))))) stmt (raluz ((M n)) () (-> (e. M (ZZ)) (<-> (A.e. n (` (ZZ>) M) ph) (A.e. n (ZZ) (-> (br M (<_) (cv n)) ph))))) stmt (raluz2 ((M n)) () (<-> (A.e. n (` (ZZ>) M) ph) (-> (e. M (ZZ)) (A.e. n (ZZ) (-> (br M (<_) (cv n)) ph))))) stmt (rexuz ((M n)) () (-> (e. M (ZZ)) (<-> (E.e. n (` (ZZ>) M) ph) (E.e. n (ZZ) (/\ (br M (<_) (cv n)) ph))))) stmt (rexuz2 ((M n)) () (<-> (E.e. n (` (ZZ>) M) ph) (/\ (e. M (ZZ)) (E.e. n (ZZ) (/\ (br M (<_) (cv n)) ph))))) stmt (2rexuz ((m n)) () (<-> (E. m (E.e. n (` (ZZ>) (cv m)) ph)) (E.e. m (ZZ) (E.e. n (ZZ) (/\ (br (cv m) (<_) (cv n)) ph))))) stmt (peano2uz () () (-> (e. N (` (ZZ>) M)) (e. (opr N (+) (1)) (` (ZZ>) M)))) stmt (peano2uzr () () (-> (/\ (e. M (ZZ)) (e. N (` (ZZ>) (opr M (+) (1))))) (e. N (` (ZZ>) M)))) stmt (uzaddclt ((K j) (j k) (M j) (M k) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (e. K (NN0))) (e. (opr N (+) K) (` (ZZ>) M)))) stmt (uzind4 ((j m) (N j) (N m) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (k m) (M k) (M m)) ((-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) (-> (e. M (ZZ)) ps) (-> (e. (cv k) (` (ZZ>) M)) (-> ch th))) (-> (e. N (` (ZZ>) M)) ta)) stmt (uzind4ALT ((N j) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (M k)) ((-> (e. M (ZZ)) ps) (-> (e. (cv k) (` (ZZ>) M)) (-> ch th)) (-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta))) (-> (e. N (` (ZZ>) M)) ta)) stmt (uzind4s ((k m) (j m) (M m) (j k) (M k) (M j) (N j) (j ph) (m ph)) ((-> (e. M (ZZ)) ([/] M k ph)) (-> (e. (cv k) (` (ZZ>) M)) (-> ph ([/] (opr (cv k) (+) (1)) k ph)))) (-> (e. N (` (ZZ>) M)) ([/] N k ph))) stmt (uzind4s2 ((k m) (k n) (M k) (m n) (M m) (M n) (N m) (k ph) (m ph) (n ph) (j k) (j m) (j n)) ((-> (e. M (ZZ)) ([/] M j ph)) (-> (e. (cv k) (` (ZZ>) M)) (-> ([/] (cv k) j ph) ([/] (opr (cv k) (+) (1)) j ph)))) (-> (e. N (` (ZZ>) M)) ([/] N j ph))) stmt (uzind4i ((j m) (N j) (N m) (j ps) (ch j) (j th) (j ta) (k ph) (j k) (M j) (k m) (M k) (M m)) ((e. M (ZZ)) (-> (= (cv j) M) (<-> ph ps)) (-> (= (cv j) (cv k)) (<-> ph ch)) (-> (= (cv j) (opr (cv k) (+) (1))) (<-> ph th)) (-> (= (cv j) N) (<-> ph ta)) ps (-> (e. (cv k) (` (ZZ>) M)) (-> ch th))) (-> (e. N (` (ZZ>) M)) ta)) stmt (uzwo ((h t) (h n) (h m) (M h) (n t) (m t) (M t) (m n) (M n) (M m) (h j) (h k) (S h) (j t) (j k) (j m) (j n) (S j) (k t) (S t) (k m) (k n) (S k) (S m) (S n)) () (-> (/\ (C_ S (` (ZZ>) M)) (-. (= S ({/})))) (E.e. j S (A.e. k S (br (cv j) (<_) (cv k)))))) stmt (uzwo2 ((j k) (S j) (S k)) () (-> (/\ (C_ S (` (ZZ>) M)) (-. (= S ({/})))) (E!e. j S (A.e. k S (br (cv j) (<_) (cv k)))))) stmt (nnwo ((x y) (A x) (A y)) () (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) stmt (nnwof ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (A z)) ((-> (e. (cv z) A) (A. x (e. (cv z) A))) (-> (e. (cv z) A) (A. y (e. (cv z) A)))) (-> (/\ (C_ A (NN)) (=/= A ({/}))) (E.e. x A (A.e. y A (br (cv x) (<_) (cv y)))))) stmt (nnwos ((x y) (x z) (y z) (ph y) (ph z) (ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps))) (-> (E.e. x (NN) ph) (E.e. x (NN) (/\ ph (A.e. y (NN) (-> ps (br (cv x) (<_) (cv y)))))))) stmt (indstr ((x y) (ph y) (ps x)) ((-> (= (cv x) (cv y)) (<-> ph ps)) (-> (e. (cv x) (NN)) (-> (A.e. y (NN) (-> (br (cv y) (<) (cv x)) ps)) ph))) (-> (e. (cv x) (NN)) ph)) stmt (uzinfm ((j k) (M j) (M k)) ((e. M (ZZ))) (= (sup (` (ZZ>) M) (RR) (`' (<))) M)) stmt (nninfm () () (= (sup (NN) (RR) (`' (<))) (1))) stmt (nn0infm () () (= (sup (NN0) (RR) (`' (<))) (0))) stmt (infmssuzcl ((j k) (S j) (S k)) () (-> (/\ (C_ S (` (ZZ>) M)) (-. (= S ({/})))) (e. (sup S (RR) (`' (<))) S))) stmt (fzvalt ((k m) (k n) (k z) (M k) (m n) (m z) (M m) (n z) (M n) (M z) (N k) (N m) (N n) (N z)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (= (opr M (...) N) ({e.|} k (ZZ) (/\ (br M (<_) (cv k)) (br (cv k) (<_) N)))))) stmt (elfz1t ((K j) (M j) (N j)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (e. K (opr M (...) N)) (/\/\ (e. K (ZZ)) (br M (<_) K) (br K (<_) N))))) stmt (elfzt () () (-> (/\/\ (e. K (ZZ)) (e. M (ZZ)) (e. N (ZZ))) (<-> (e. K (opr M (...) N)) (/\ (br M (<_) K) (br K (<_) N))))) stmt (elfz2t ((k n) (k m) (k z) (M k) (m n) (n z) (M n) (m z) (M m) (M z) (N k) (N n) (N m) (N z)) () (-> (e. N A) (<-> (e. K (opr M (...) N)) (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N)))))) stmt (elfzlem ((k n) (k m) (k z) (M k) (m n) (n z) (M n) (m z) (M m) (M z) (N k) (N n) (N m) (N z)) () (-> (e. K (opr M (...) N)) (/\ (/\ (e. M (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N))))) stmt (elfz5t () () (-> (/\ (e. K (` (ZZ>) M)) (e. N (ZZ))) (<-> (e. K (opr M (...) N)) (br K (<_) N)))) stmt (elfz4t () () (-> (/\ (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (/\ (br M (<_) K) (br K (<_) N))) (e. K (opr M (...) N)))) stmt (elfzuzb () () (-> (e. N A) (<-> (e. K (opr M (...) N)) (/\ (e. K (` (ZZ>) M)) (e. N (` (ZZ>) K)))))) stmt (eluzfzt () () (-> (/\ (e. K (` (ZZ>) M)) (e. N (` (ZZ>) K))) (e. K (opr M (...) N)))) stmt (elfzuz3t () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. N (` (ZZ>) K)))) stmt (elfzel2 () ((e. N (V))) (-> (e. K (opr M (...) N)) (e. N (ZZ)))) stmt (elfzel2g () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. N (ZZ)))) stmt (elfzel1 () () (-> (e. K (opr M (...) N)) (e. M (ZZ)))) stmt (elfzelz () () (-> (e. K (opr M (...) N)) (e. K (ZZ)))) stmt (elfzle1 () () (-> (e. K (opr M (...) N)) (br M (<_) K))) stmt (elfzle2 () () (-> (e. K (opr M (...) N)) (br K (<_) N))) stmt (elfzle3 () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (br M (<_) N))) stmt (elfzuz2t () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. N (` (ZZ>) M)))) stmt (eluzfz1t () () (-> (e. N (` (ZZ>) M)) (e. M (opr M (...) N)))) stmt (elfzuzt () () (-> (e. K (opr M (...) N)) (e. K (` (ZZ>) M)))) stmt (eluzfz2t () () (-> (e. N (` (ZZ>) M)) (e. N (opr M (...) N)))) stmt (eluzfz2b () () (<-> (e. N (` (ZZ>) M)) (e. N (opr M (...) N)))) stmt (elfz3t () () (-> (e. N (ZZ)) (e. N (opr N (...) N)))) stmt (elfz1eqt () () (-> (e. K (opr N (...) N)) (= K N))) stmt (fznt ((M k) (N k)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (br N (<) M) (= (opr M (...) N) ({/}))))) stmt (elfznnt () () (-> (e. K (opr (1) (...) N)) (e. K (NN)))) stmt (elfz2nn0t () () (-> (e. N A) (<-> (e. K (opr (0) (...) N)) (/\/\ (e. K (NN0)) (e. N (NN0)) (br K (<_) N))))) stmt (elfznn0t () () (-> (e. K (opr (0) (...) N)) (e. K (NN0)))) stmt (elfz3nn0t () () (-> (/\ (e. N A) (e. K (opr (0) (...) N))) (e. N (NN0)))) stmt (fznn0subt () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. (opr N (-) K) (NN0)))) stmt (fznn0sub2t () () (-> (/\ (e. N A) (e. K (opr (0) (...) N))) (e. (opr N (-) K) (opr (0) (...) N)))) stmt (fzaddelt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. J (opr M (...) N)) (e. (opr J (+) K) (opr (opr M (+) K) (...) (opr N (+) K)))))) stmt (fzsubelt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. J (opr M (...) N)) (e. (opr J (-) K) (opr (opr M (-) K) (...) (opr N (-) K)))))) stmt (fzoptht () () (-> (/\ (e. N (` (ZZ>) M)) (e. K A)) (<-> (= (opr M (...) N) (opr J (...) K)) (/\ (= M J) (= N K))))) stmt (fzss1t ((M k) (N k) (K k)) () (-> (/\ (e. K (` (ZZ>) M)) (e. N (ZZ))) (C_ (opr K (...) N) (opr M (...) N)))) stmt (fzss2t ((M k) (N k) (K k)) () (-> (/\ (e. N (` (ZZ>) K)) (e. M (ZZ))) (C_ (opr M (...) K) (opr M (...) N)))) stmt (fzssuzt ((M k) (N k)) () (C_ (opr M (...) N) (` (ZZ>) M))) stmt (fzssp1t () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (C_ (opr M (...) N) (opr M (...) (opr N (+) (1)))))) stmt (fzp1sst () () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (C_ (opr (opr M (+) (1)) (...) N) (opr M (...) N)))) stmt (fzelp1t () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. K (opr M (...) (opr N (+) (1)))))) stmt (fzelp1 () ((e. N (V))) (-> (e. K (opr M (...) N)) (e. K (opr M (...) (opr N (+) (1)))))) stmt (elfzp1 () () (-> (e. N (` (ZZ>) M)) (<-> (e. K (opr M (...) (opr N (+) (1)))) (\/ (e. K (opr M (...) N)) (= K (opr N (+) (1))))))) stmt (fzrevt () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. K (opr (opr J (-) N) (...) (opr J (-) M))) (e. (opr J (-) K) (opr M (...) N))))) stmt (fzrev2t () () (-> (/\ (/\ (e. M (ZZ)) (e. N (ZZ))) (/\ (e. J (ZZ)) (e. K (ZZ)))) (<-> (e. K (opr M (...) N)) (e. (opr J (-) K) (opr (opr J (-) N) (...) (opr J (-) M)))))) stmt (fzrev2it () () (-> (/\/\ (e. N A) (e. J (ZZ)) (e. K (opr M (...) N))) (e. (opr J (-) K) (opr (opr J (-) N) (...) (opr J (-) M))))) stmt (fzrev3t () () (-> (/\ (e. N A) (e. K (ZZ))) (<-> (e. K (opr M (...) N)) (e. (opr (opr M (+) N) (-) K) (opr M (...) N))))) stmt (fzrev3it () () (-> (/\ (e. N A) (e. K (opr M (...) N))) (e. (opr (opr M (+) N) (-) K) (opr M (...) N)))) stmt (fznn0t () () (-> (e. N (NN0)) (<-> (e. K (opr (0) (...) N)) (/\ (e. K (NN0)) (br K (<_) N))))) stmt (fz1sbct ((N k)) () (-> (e. N (ZZ)) (<-> (A.e. k (opr N (...) N) ph) ([/] N k ph)))) stmt (fzneuzt () () (-> (/\ (e. N (` (ZZ>) M)) (e. K (ZZ))) (-. (= (opr M (...) N) (` (ZZ>) K))))) stmt (fzrevralt ((j k) (j x) (K j) (k x) (K k) (K x) (M j) (M k) (M x) (N j) (N k) (N x) (k ph) (ph x)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (<-> (A.e. j (opr M (...) N) ph) (A.e. k (opr (opr K (-) N) (...) (opr K (-) M)) ([/] (opr K (-) (cv k)) j ph))))) stmt (fzrevral2t ((j k) (K j) (K k) (M j) (M k) (N j) (N k) (k ph)) () (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (e. K (ZZ))) (<-> (A.e. j (opr (opr K (-) N) (...) (opr K (-) M)) ph) (A.e. k (opr M (...) N) ([/] (opr K (-) (cv k)) j ph))))) stmt (fzrevral3t ((j k) (M j) (M k) (N j) (N k) (k ph)) () (-> (/\ (e. M (ZZ)) (e. N (ZZ))) (<-> (A.e. j (opr M (...) N) ph) (A.e. k (opr M (...) N) ([/] (opr (opr M (+) N) (-) (cv k)) j ph))))) stmt (fsequb ((j k) (j m) (j n) (j x) (j y) (F j) (k m) (k n) (k x) (k y) (F k) (m n) (m x) (m y) (F m) (n x) (n y) (F n) (x y) (F x) (F y) (M j) (M k) (M m) (M n) (M x) (M y) (N j) (N k) (N n) (N x) (N y)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR)))) (E.e. x (RR) (A.e. k (opr M (...) N) (br (` F (cv k)) (<) (cv x)))))) stmt (fsequb2 ((k x) (k y) (F k) (x y) (F x) (F y) (M k) (M x) (M y) (N k) (N x) (N y)) () (-> (/\ (e. N (` (ZZ>) M)) (:--> F (opr M (...) N) (RR))) (E.e. x (RR) (A.e. y (ran F) (br (cv y) (<_) (cv x)))))) stmt (limsupvalt ((k x) (k y) (k z) (F k) (x y) (x z) (F x) (y z) (F y) (F z)) () (-> (e. F A) (= (` (limsup) F) (sup ({|} x (E.e. k (ZZ) (= (cv x) (sup (i^i (" F (` (ZZ>) (cv k))) (RR*)) (RR*) (<))))) (RR*) (`' (<)))))) stmt (seq0fval ((f g) (f h) (S f) (g h) (S g) (S h) (F f) (F g) (F h)) ((e. S (V)) (e. F (V))) (= (opr S (seq0) F) (|` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) (NN0)))) stmt (seq0valt () ((e. S (V)) (e. F (V))) (-> (e. N (NN0)) (= (` (opr S (seq0) F) N) (` (opr (opr S (seq1) (opr F (shift) (1))) (shift) (-u (1))) N)))) stmt (seqzfval ((g h) (g k) (g x) (S g) (h k) (h x) (S h) (k x) (S k) (S x) (F g) (F h) (F k) (F x) (A k) (M g) (M h) (M k) (M x)) ((e. S (V)) (e. F (V))) (-> (e. M A) (= (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) ({e.|} k (ZZ) (br M (<_) (cv k))))))) stmt (seqzfval2 ((S k) (F k) (M k)) ((e. S (V)) (e. F (V))) (-> (e. M (ZZ)) (= (opr (<,> M S) (seq) F) (|` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) (` (ZZ>) M))))) stmt (seqzfn () ((e. S (V)) (e. F (V))) (-> (e. M (ZZ)) (Fn (opr (<,> M S) (seq) F) (` (ZZ>) M)))) stmt (seqzvalt ((S k) (F k) (A k) (M k) (N k)) ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M A) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) N) (` (opr (opr S (seq1) (opr F (shift) (opr (1) (-) M))) (shift) (opr M (-) (1))) N)))) stmt (seq1seqz ((S k) (F k)) ((e. S (V)) (e. F (V))) (= (opr S (seq1) F) (opr (<,> (1) S) (seq) F))) stmt (seq0seqz ((S k) (F k)) ((e. S (V)) (e. F (V))) (= (opr S (seq0) F) (opr (<,> (0) S) (seq) F))) stmt (seq1seq02t () ((e. S (V)) (e. F (V))) (-> (e. N (NN)) (= (` (opr S (seq1) (opr F (shift) (1))) N) (` (opr (opr S (seq0) F) (shift) (1)) N)))) stmt (seq1seq0t () ((e. S (V)) (e. F (V))) (-> (e. N (NN)) (= (` (opr S (seq1) F) N) (` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) N)))) stmt (seq1seq0 ((S k) (F k)) ((e. S (V)) (e. F (V))) (= (opr S (seq1) F) (|` (opr (opr S (seq0) (opr F (shift) (-u (1)))) (shift) (1)) (NN)))) stmt (seq0fn () ((e. S (V)) (e. F (V))) (Fn (opr S (seq0) F) (NN0))) stmt (seqz1 ((S k) (F k) (M k)) ((e. S (V)) (e. F (V))) (-> (e. M (ZZ)) (= (` (opr (<,> M S) (seq) F) M) (` F M)))) stmt (seqzp1 ((S k) (F k) (M k) (N k)) ((e. S (V)) (e. F (V))) (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M S) (seq) F) (opr N (+) (1))) (opr (` (opr (<,> M S) (seq) F) N) S (` F (opr N (+) (1))))))) stmt (seqzp1OLD ((S k) (F k) (M k) (N k)) ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) (opr N (+) (1))) (opr (` (opr (<,> M S) (seq) F) N) S (` F (opr N (+) (1))))))) stmt (seqzm1 () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<) N)) (= (` (opr (<,> M S) (seq) F) N) (opr (` (opr (<,> M S) (seq) F) (opr N (-) (1))) S (` F N))))) stmt (seq00 () ((e. S (V)) (e. F (V))) (= (` (opr S (seq0) F) (0)) (` F (0)))) stmt (seq0p1 () ((e. S (V)) (e. F (V))) (-> (e. N (NN0)) (= (` (opr S (seq0) F) (opr N (+) (1))) (opr (` (opr S (seq0) F) N) S (` F (opr N (+) (1))))))) stmt (seq01 () ((e. S (V)) (e. F (V))) (= (` (opr S (seq0) F) (1)) (opr (` F (0)) S (` F (1))))) stmt (seqzval2t () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. N (ZZ)) (br M (<_) N)) (= (` (opr (<,> M S) (seq) F) N) (` (opr (opr S (seq0) (opr F (shift) (-u M))) (shift) M) N)))) stmt (seqzfveq ((j k) (j m) (F j) (k m) (F k) (F m) (G j) (G k) (G m) (M j) (M k) (M m) (N j) (N k) (S j) (S m)) ((e. S (V)) (e. F (V)) (e. G (V))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (= (` F (cv k)) (` G (cv k))))) (= (` (opr (<,> M S) (seq) F) N) (` (opr (<,> M S) (seq) G) N)))) stmt (seqzeq ((k m) (F k) (F m) (G k) (G m) (M k) (M m) (S m)) ((e. S (V)) (e. F (V)) (e. G (V))) (-> (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (= (` F (cv k)) (` G (cv k))))) (= (opr (<,> M S) (seq) F) (opr (<,> M S) (seq) G)))) stmt (seqzrn ((j m) (j n) (j v) (j y) (B j) (m n) (m v) (m y) (B m) (n v) (n y) (B n) (v y) (B v) (B y) (C j) (C m) (C n) (C v) (C y) (F j) (F m) (F n) (F v) (F y) (M j) (M m) (M n) (M v) (M y) (S j) (S m) (S n) (S v) (S y)) ((e. S (V)) (e. F (V))) (-> (/\ (/\ (e. M (ZZ)) (e. (` F M) C)) (/\ (:--> (|` F (` (ZZ>) (opr M (+) (1)))) (` (ZZ>) (opr M (+) (1))) B) (:--> S (X. C B) C))) (C_ (ran (opr (<,> M S) (seq) F)) C))) stmt (seqzrn2 () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. M (ZZ)) (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C)) (C_ (ran (opr (<,> M S) (seq) F)) C))) stmt (seqzcl () ((e. S (V)) (e. F (V))) (-> (/\/\ (e. N (` (ZZ>) M)) (:--> F (` (ZZ>) M) C) (:--> S (X. C C) C)) (e. (` (opr (<,> M S) (seq) F) N) C))) stmt (seqzresval ((F m) (M m) (N m) (S m)) ((e. S (V)) (e. F (V))) (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M S) (seq) (|` F (opr M (...) N))) N) (` (opr (<,> M S) (seq) F) N)))) stmt (seqzres ((m n) (F m) (F n) (M m) (M n) (S m) (S n)) ((e. S (V)) (e. F (V))) (-> (e. M (ZZ)) (= (opr (<,> M S) (seq) (|` F (` (ZZ>) M))) (opr (<,> M S) (seq) F)))) stmt (seqzres2 ((n v) (n y) (v y) (k n) (k v) (k y) (F k) (F n) (F v) (F y) (M n) (M v) (M y) (S n) (S v) (S y)) ((e. S (V)) (e. F (V))) (-> (e. M (ZZ)) (= (opr (<,> M S) (seq) (|` ({<,>|} k y (= (cv y) (` F (cv k)))) (ZZ))) (opr (<,> M S) (seq) F)))) stmt (serzcl1OLD () ((:--> F (` (ZZ>) M) (CC))) (-> (e. N (` (ZZ>) M)) (e. (` (opr (<,> M (+)) (seq) F) N) (CC)))) stmt (serzreclOLD ((j k) (F j) (F k) (M j) (M k) (N j)) ((e. M (ZZ)) (:--> F (` (ZZ>) M) (RR))) (-> (/\ (e. N (ZZ)) (br M (<_) N)) (e. (` (opr (<,> M (+)) (seq) F) N) (RR)))) stmt (serzref ((F k) (M k)) ((e. M (ZZ)) (:--> F (` (ZZ>) M) (RR))) (:--> (opr (<,> M (+)) (seq) F) (` (ZZ>) M) (RR))) stmt (dfseq0 ((f g) (f h) (f x) (f y) (g h) (g x) (g y) (h x) (h y) (x y)) () (= (seq0) ({<<,>,>|} f g h (= (cv h) (opr (<,> (0) (cv f)) (seq) (cv g)))))) stmt (ser0cl ((N j) (j k) (F j) (F k)) ((:--> F (NN0) (CC))) (-> (e. N (NN0)) (e. (` (opr (+) (seq0) F) N) (CC)))) stmt (ser0f ((F j)) ((:--> F (NN0) (CC))) (:--> (opr (+) (seq0) F) (NN0) (CC))) stmt (ser00 ((A y) (k y) (B k) (B y)) ((= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) A)))) (e. B (V)) (-> (= (cv k) (0)) (= A B))) (= (` (opr (+) (seq0) F) (0)) B)) stmt (ser0p1 ((A y) (k y) (B k) (B y) (N k) (N y)) ((= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) A)))) (e. B (V)) (-> (= (cv k) (opr N (+) (1))) (= A B))) (-> (e. N (NN0)) (= (` (opr (+) (seq0) F) (opr N (+) (1))) (opr (` (opr (+) (seq0) F) N) (+) B)))) stmt (serzmulc ((j k) (j m) (C j) (k m) (C k) (C m) (F j) (F k) (F m) (G j) (G k) (G m) (M j) (M k) (M m) (N j)) ((e. 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B (NN))) (= (opr A (^) B) (` (opr (x.) (seq1) (X. (NN) ({} A))) B)))) stmt (exp1t () () (-> (e. A (CC)) (= (opr A (^) (1)) A))) stmt (expp1t () () (-> (/\ (e. A (CC)) (e. N (NN0))) (= (opr A (^) (opr N (+) (1))) (opr (opr A (^) N) (x.) A)))) stmt (expcllem ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B z) (F x) (F y) (F z) (F w)) ((C_ F (CC)) (-> (/\ (e. (cv x) F) (e. (cv y) F)) (e. (opr (cv x) (x.) (cv y)) F)) (e. (1) F)) (-> (/\ (e. A F) (e. B (NN0))) (e. (opr A (^) B) F))) stmt (nnexpclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (NN)) (e. N (NN0))) (e. (opr A (^) N) (NN)))) stmt (nn0expclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (NN0)) (e. N (NN0))) (e. (opr A (^) N) (NN0)))) stmt (zexpclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (ZZ)) (e. N (NN0))) (e. (opr A (^) N) (ZZ)))) stmt (qexpclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (QQ)) (e. N (NN0))) (e. (opr A (^) N) (QQ)))) stmt (reexpclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (RR)) (e. N (NN0))) (e. (opr A (^) N) (RR)))) stmt (expclt ((x y) (A x) (A y) (N x) (N y)) () (-> (/\ (e. A (CC)) (e. N (NN0))) (e. (opr A (^) N) (CC)))) stmt (expm1t () () (-> (/\ (e. A (CC)) (e. N (NN))) (= (opr A (^) N) (opr (opr A (^) (opr N (-) (1))) (x.) A)))) stmt (1expt ((j k) (N j) (N k)) () (-> (e. N (NN0)) (= (opr (1) (^) N) (1)))) stmt (expeq0t ((j k) (A j) (A k) (N j) (N k)) () (-> (/\ (e. A (CC)) (e. N (NN))) (<-> (= (opr A (^) N) (0)) (= A (0))))) stmt (expne0t () () (-> (/\ (e. A (CC)) (e. N (NN))) (<-> (=/= A (0)) (=/= (opr A (^) N) (0))))) stmt (expne0it () () (-> (/\/\ (e. A (CC)) (e. N (NN0)) (=/= A (0))) (=/= (opr A (^) N) (0)))) stmt (expgt0t ((j k) (A j) (A k) (N j) (N k)) () (-> (/\/\ (e. A (RR)) (e. N (NN0)) (br (0) (<) A)) (br (0) (<) (opr A (^) N)))) stmt (0expt () () (-> (e. N (NN)) (= (opr (0) (^) N) (0)))) stmt (expge0t () () (-> (/\/\ (e. A (RR)) (e. N (NN0)) (br (0) (<_) A)) (br (0) (<_) (opr A (^) N)))) stmt (expgt1t ((x y) (A x) (A y) (N x)) () (-> (/\/\ (e. A (RR)) (e. N (NN)) (br (1) (<) A)) (br (1) (<) (opr A (^) N)))) stmt (expge1t () () (-> (/\/\ (e. A (RR)) (e. N (NN)) (br (1) (<_) A)) (br (1) (<_) (opr A (^) N)))) stmt (mulexpt ((j k) (A j) (A k) (B j) (B k) (N j)) () (-> (/\/\ (e. A (CC)) (e. B (CC)) (e. N (NN0))) (= (opr (opr A (x.) B) (^) N) (opr (opr A (^) N) (x.) (opr B (^) N))))) stmt (recexpt ((j k) (A j) (A k) (N j)) () (-> (/\/\ (e. A (CC)) (e. N (NN0)) (=/= A (0))) (= (opr (opr (1) (/) A) (^) N) (opr (1) (/) (opr A (^) N))))) stmt (expaddt ((j k) (A j) (A k) (M j) (M k) (N j)) () (-> (/\/\ (e. A (CC)) (e. M (NN0)) (e. N (NN0))) (= (opr A (^) (opr M (+) N)) (opr (opr A (^) M) (x.) (opr A (^) N))))) stmt (expmult ((j k) (A j) (A k) (M j) (M k) (N j)) () (-> (/\/\ (e. A (CC)) (e. M (NN0)) (e. N (NN0))) (= (opr A (^) (opr M (x.) N)) (opr (opr A (^) M) (^) N)))) stmt (expsubt () () (-> (/\ (/\/\ (e. A (CC)) (e. M (NN0)) (e. N (NN0))) (/\ (=/= A (0)) (br N (<_) M))) (= (opr A (^) (opr M (-) N)) (opr (opr A (^) M) (/) (opr A (^) N))))) stmt (expordit () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (/\ (br (1) (<) A) (br M (<) N))) (br (opr A (^) M) (<) (opr A (^) N)))) stmt (expcant () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (br (1) (<) A)) (<-> (= (opr A (^) M) (opr A (^) N)) (= M N)))) stmt (expordt () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (br (1) (<) A)) (<-> (br M (<) N) (br (opr A (^) M) (<) (opr A (^) N))))) stmt (expwordit () () (-> (/\ (/\/\ (e. A (RR)) (e. M (NN0)) (e. N (NN0))) (/\ (br (1) (<_) A) (br M (<_) N))) (br (opr A (^) M) (<_) (opr A (^) N)))) stmt (expmwordit ((j k) (A j) (A k) (B j) (B k) (N j) (N k)) () (-> (/\ (/\/\ (e. A (RR)) (e. B (RR)) (e. N (NN0))) (/\ (br (0) (<_) A) (br A (<_) B))) (br (opr A (^) N) (<_) (opr B (^) N)))) stmt (expbndt () () (-> (/\/\ (e. A (RR)) (e. N (NN0)) (br (2) (<_) A)) (br (opr A (^) N) (<_) (opr (opr (2) (^) N) (x.) (opr (opr A (-) (1)) (^) N))))) stmt (sqvalt () () (-> (e. A (CC)) (= (opr A (^) (2)) (opr A (x.) A)))) stmt (sqclt () () (-> (e. A (CC)) (e. (opr A (^) (2)) (CC)))) stmt (sqval () ((e. A (CC))) (= (opr A (^) (2)) (opr A (x.) A))) stmt (sqcl () ((e. A (CC))) (e. (opr A (^) (2)) (CC))) stmt (sq00 () ((e. A (CC))) (<-> (= (opr A (^) (2)) (0)) (= A (0)))) stmt (sqmul () ((e. A (CC)) (e. B (CC))) (= (opr (opr A (x.) B) (^) (2)) (opr (opr A (^) (2)) (x.) (opr B (^) (2))))) stmt (sqdiv () ((e. A (CC)) (e. B (CC)) (=/= B (0))) (= (opr (opr A (/) B) (^) (2)) (opr (opr A (^) (2)) (/) (opr B (^) (2))))) stmt (sqreci () ((e. A (CC)) (=/= A (0))) (= (opr (opr (1) (/) A) (^) (2)) (opr (1) (/) (opr A (^) (2))))) stmt (sq0t () () (-> (e. A (CC)) (<-> (= (opr A (^) (2)) (0)) (= A (0))))) stmt (sqne0t () () (-> (e. A (CC)) (<-> (=/= (opr A (^) (2)) (0)) (=/= A (0))))) stmt (resqclt () () (-> (e. A (RR)) (e. (opr A (^) (2)) (RR)))) stmt (resqcl () ((e. A (RR))) (e. (opr A (^) (2)) (RR))) stmt (lt2sq () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2)))))) stmt (le2sq () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (opr A (^) (2)) (<_) (opr B (^) (2)))))) stmt (sq11 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (= A B)))) stmt (sqgt0 () ((e. A (RR))) (-> (=/= A (0)) (br (0) (<) (opr A (^) (2))))) stmt (sqge0 () ((e. A (RR))) (br (0) (<_) (opr A (^) (2)))) stmt (sq11t () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (= A B)))) stmt (lt2sqt () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2)))))) stmt (lt2sqtOLD () () (-> (/\ (e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (opr A (^) (2)) (<) (opr B (^) (2))))))) stmt (le2sqt () () (-> (/\ (/\ (e. A (RR)) (br (0) (<_) A)) (/\ (e. B (RR)) (br (0) (<_) B))) (<-> (br A (<_) B) (br (opr A (^) (2)) (<_) (opr B (^) (2)))))) stmt (sqge0t () () (-> (e. A (RR)) (br (0) (<_) (opr A (^) (2))))) stmt (sumsqne0 () ((e. A (RR)) (e. B (RR))) (<-> (\/ (=/= A (0)) (=/= B (0))) (=/= (opr (opr A (^) (2)) (+) (opr B (^) (2))) (0)))) stmt (sq0 () () (= (opr (0) (^) (2)) (0))) stmt (sq1 () () (= (opr (1) (^) (2)) (1))) stmt (sq2 () () (= (opr (2) (^) (2)) (4))) stmt (sq3 () () (= (opr (3) (^) (2)) (9))) stmt (cu2 () () (= (opr (2) (^) (3)) (8))) stmt (binom2 () ((e. A (CC)) (e. B (CC))) (= (opr (opr A (+) B) (^) (2)) (opr (opr (opr A (^) (2)) (+) (opr (2) (x.) (opr A (x.) B))) (+) (opr B (^) (2))))) stmt (binom2a () ((e. A (CC)) (e. B (CC))) (= (opr (opr A (+) B) (x.) (opr A (-) B)) (opr (opr A (^) (2)) (-) (opr B (^) (2))))) stmt (sqeqor () ((e. A (CC)) (e. B (CC))) (<-> (= (opr A (^) (2)) (opr B (^) (2))) (\/ (= A B) (= A (-u B))))) stmt (sq01t () () (-> (e. A (CC)) (<-> (= (opr A (^) (2)) A) (\/ (= A (0)) (= A (1)))))) stmt (bernneqOLD ((x y) (A x) (A y) (B x) (B y)) () (-> (/\/\ (e. A (RR)) (e. B (NN)) (br (-u (1)) (<_) A)) (br (opr (1) (+) (opr A (x.) B)) (<_) (opr (opr (1) (+) A) (^) B)))) stmt (bernneq ((j k) (A j) (A k) (N j) (N k)) () (-> (/\/\ (e. A (RR)) (e. N (NN0)) (br (-u (1)) (<_) A)) (br (opr (1) (+) (opr A (x.) N)) (<_) (opr (opr (1) (+) A) (^) N)))) stmt (discrlem1 () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (= D (-u (opr B (/) (opr (2) (x.) A)))) (br (0) (<) A)) (<-> (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C)) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0)))) stmt (discrlem2 () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (= D (-u (opr B (/) (opr (2) (x.) A)))) (-> (e. D (RR)) (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C)))) (-> (br (0) (<) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0)))) stmt (discrlem3 () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (= D (opr (opr C (+) (1)) (/) (-u B))) (-> (e. D (RR)) (br (0) (<_) (opr (opr (opr A (x.) (opr D (^) (2))) (+) (opr B (x.) D)) (+) C)))) (-> (= (0) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0)))) stmt (discrlem ((A x) (B x) (C x)) ((e. A (RR)) (e. B (RR)) (e. C (RR)) (A.e. x (RR) (br (0) (<_) (opr (opr (opr A (x.) (opr (cv x) (^) (2))) (+) (opr B (x.) (cv x))) (+) C)))) (-> (br (0) (<_) A) (br (opr (opr B (^) (2)) (-) (opr (4) (x.) (opr A (x.) C))) (<_) (0)))) stmt (nnsqcl () ((e. N (NN))) (e. (opr N (^) (2)) (NN))) stmt (nnlesq () ((e. N (NN))) (br N (<_) (opr N (^) (2)))) stmt (nnesq () ((e. N (NN))) (<-> (e. (opr N (/) (2)) (NN)) (e. (opr (opr N (^) (2)) (/) (2)) (NN)))) stmt (nn0le2msqt () ((e. A (NN0)) (e. B (NN0))) (<-> (br A (<_) B) (br (opr A (x.) A) (<_) (opr B (x.) B)))) stmt (nn0opthlem1 () ((e. A (NN0)) (e. C (NN0))) (<-> (br A (<) C) (br (opr (opr A (x.) A) (+) (opr (2) (x.) A)) (<) (opr C (x.) C)))) stmt (nn0opthlem2 () ((e. A (NN0)) (e. B (NN0)) (e. C (NN0)) (e. D (NN0))) (-> (/\ (br B (<_) A) (br D (<_) C)) (-> (br A (<) C) (-. (= (opr (opr A (x.) A) (+) B) (opr (opr C (x.) C) (+) D)))))) stmt (nn0opth () ((e. A (NN0)) (e. B (NN0)) (e. C (NN0)) (e. D (NN0))) (<-> (= (opr (opr (opr A (+) B) (x.) (opr A (+) B)) (+) B) (opr (opr (opr C (+) D) (x.) (opr C (+) D)) (+) D)) (/\ (= A C) (= B D)))) stmt (nn0opth2 () ((e. A (NN0)) (e. B (NN0)) (e. C (NN0)) (e. D (NN0))) (<-> (= (opr (opr (opr A (+) B) (^) (2)) (+) B) (opr (opr (opr C (+) D) (^) (2)) (+) D)) (/\ (= A C) (= B D)))) stmt (nn0opth2t () () (-> (/\ (/\ (e. A (NN0)) (e. B (NN0))) (/\ (e. C (NN0)) (e. D (NN0)))) (<-> (= (opr (opr (opr A (+) B) (^) (2)) (+) B) (opr (opr (opr C (+) D) (^) (2)) (+) D)) (/\ (= A C) (= B D))))) stmt (sqrval ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (= (` (sqr) A) (sup ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))) (RR) (<))))) stmt (sqr0 () () (= (` (sqr) (0)) (0))) stmt (sqrlem1 () ((e. A (RR)) (br (0) (<) A)) (br A (<) (opr (opr (1) (+) A) (x.) (opr (1) (+) A)))) stmt (sqrlem2 () ((e. A (RR)) (br (0) (<) A)) (br (opr (opr A (/) (opr (1) (+) A)) (x.) (opr A (/) (opr (1) (+) A))) (<) A)) stmt (sqrlem3 () ((e. A (RR)) (br (0) (<) A)) (br (0) (<) (opr A (/) (opr (1) (+) A)))) stmt (sqrlem4 ((A x) (S x) (D x) (A x)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (<-> (e. D S) (/\ (e. D (RR)) (/\ (br (0) (<_) D) (br (opr D (x.) D) (<_) A))))) stmt (sqrlem5 ((A x) (S x) (D x) (A x)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (-> (/\ (e. D (RR)) (/\ (br (0) (<) D) (br (opr D (x.) D) (<) A))) (e. D S))) stmt (sqrlem6 ((x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (/\/\ (C_ S (RR)) (=/= S ({/})) (E.e. x (RR) (A.e. y S (br (cv y) (<) (cv x)))))) stmt (sqrlem7 ((x y) (x z) (S x) (y z) (S y) (S z) (A z) (x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))) (= B (sup S (RR) (<)))) (e. B (RR))) stmt (sqrlem8 ((x y) (x z) (S x) (y z) (S y) (S z) (A z) (x y) (A x) (A y) (S x) (S y) (x y) (A x) (A y)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))) (= B (sup S (RR) (<)))) (br (0) (<) B)) stmt (sqrlem9 () ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (e. C (RR)) (br (0) (<) B) (br A (<) (opr B (x.) B)) (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (br (0) (<) C)) stmt (sqrlem10 () ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (e. C (RR)) (br (0) (<) B) (br A (<) (opr B (x.) B)) (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (br C (<) B)) stmt (sqrlem11 () ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (e. C (RR)) (br (0) (<) B) (br A (<) (opr B (x.) B)) (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1))))) (br A (<) (opr C (x.) C))) stmt (sqrlem12 ((A x) (S x) (D x) (A x)) ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (e. C (RR)) (br (0) (<) B) (br A (<) (opr B (x.) B)) (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (-> (e. D S) (br D (<) C))) stmt (sqrlem13 ((x y) (A x) (A y) (x z) (S x) (y z) (S y) (S z) (C y) (C z) (x y) (A x) (A y)) ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (e. C (RR)) (br (0) (<) B) (br A (<) (opr B (x.) B)) (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (-> (= B (sup S (RR) (<))) (-. (br C (<) B)))) stmt (sqrlem14 ((A x) (S x) (A x)) ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (e. C (RR)) (br (0) (<) B) (br A (<) (opr B (x.) B)) (= C (opr (opr B (+) (opr A (/) B)) (/) (opr (1) (+) (1)))) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (-. (= B (sup S (RR) (<))))) stmt (sqrlem15 () ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (br (0) (<) B) (e. C (RR)) (br (0) (<) C)) (br B (<) (opr B (+) C))) stmt (sqrlem16 () ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (br (0) (<) B) (e. C (RR)) (br (0) (<) C) (br C (<) B)) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (br (opr (opr B (+) C) (x.) (opr B (+) C)) (<) A))) stmt (sqrlem17 ((A x) (B x) (C x) (S x) (A x)) ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (br (0) (<) B) (e. C (RR)) (br (0) (<) C) (br C (<) B) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (e. (opr B (+) C) S))) stmt (sqrlem18 ((x y) (A x) (A y) (x z) (B x) (y z) (B y) (B z) (C x) (S x) (S y) (S z) (x y) (A x) (A y)) ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (br (0) (<) B) (e. C (RR)) (br (0) (<) C) (br C (<) B) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (-> (br C (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B))) (-. (= B (sup S (RR) (<)))))) stmt (sqrlem19 () ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (br (0) (<) B) (br (opr B (x.) B) (<) A)) (br (0) (<) (opr (opr A (-) (opr B (x.) B)) (/) (opr (opr (opr (1) (+) (1)) (+) (1)) (x.) B)))) stmt (sqrlem20 ((w x) (A x) (A w) (B x) (B w) (S x) (S w) (A x)) ((e. A (RR)) (br (0) (<) A) (e. B (RR)) (br (0) (<) B) (br (opr B (x.) B) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A))))) (-. (= B (sup S (RR) (<))))) stmt (sqrlem21 ((A x) (B x) (S x) (A x)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))) (= B (sup S (RR) (<)))) (-. (br A (<) (opr B (x.) B)))) stmt (sqrlem22 ((A x) (B x) (S x) (A x)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))) (= B (sup S (RR) (<)))) (-. (br (opr B (x.) B) (<) A))) stmt (sqrlem23 ((A x) (B x) (S x) (A x)) ((e. A (RR)) (br (0) (<) A) (= S ({e.|} x (RR) (/\ (br (0) (<_) (cv x)) (br (opr (cv x) (x.) (cv x)) (<_) A)))) (= B (sup S (RR) (<)))) (= (opr B (x.) B) A)) stmt (sqrlem24 ((x y) (A x) (A y)) ((e. A (RR)) (br (0) (<) A)) (e. (` (sqr) A) (RR))) stmt (sqrgt0i ((x y) (A x) (A y)) ((e. A (RR)) (br (0) (<) A)) (br (0) (<) (` (sqr) A))) stmt (sqrlem26 ((x y) (A x) (A y)) ((e. A (RR)) (br (0) (<) A)) (= (opr (` (sqr) A) (x.) (` (sqr) A)) A)) stmt (sqrth () ((e. A (RR))) (-> (br (0) (<_) A) (= (opr (` (sqr) A) (x.) (` (sqr) A)) A))) stmt (sqrcl () ((e. A (RR))) (-> (br (0) (<_) A) (e. (` (sqr) A) (RR)))) stmt (sqrgt0 () ((e. A (RR))) (-> (br (0) (<) A) (br (0) (<) (` (sqr) A)))) stmt (sqrge0 () ((e. A (RR))) (-> (br (0) (<_) A) (br (0) (<_) (` (sqr) A)))) stmt (sqr11 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (` (sqr) A) (` (sqr) B)) (= A B)))) stmt (sqrmuli () ((e. A (RR)) (e. B (RR)) (br (0) (<_) A) (br (0) (<_) B)) (= (` (sqr) (opr A (x.) B)) (opr (` (sqr) A) (x.) (` (sqr) B)))) stmt (sqrmul () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (= (` (sqr) (opr A (x.) B)) (opr (` (sqr) A) (x.) (` (sqr) B))))) stmt (sqrmsq2 () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (= (` (sqr) A) B) (= A (opr B (x.) B))))) stmt (sqrle () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<_) B) (br (` (sqr) A) (<_) (` (sqr) B))))) stmt (sqrlt () ((e. A (RR)) (e. B (RR))) (-> (/\ (br (0) (<_) A) (br (0) (<_) B)) (<-> (br A (<) B) (br (` (sqr) A) (<) (` (sqr) B))))) stmt (sqrmsq () ((e. A (RR))) (-> (br (0) (<_) A) (= (` (sqr) (opr A (x.) A)) A))) stmt (sqrclt () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (e. (` (sqr) A) (RR)))) stmt (sqrgt0t () () (-> (/\ (e. A (RR)) (br (0) (<) A)) (br (0) (<) (` (sqr) A)))) stmt (sqrge0t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (br (0) (<_) (` (sqr) A)))) stmt (sqrlet () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (br (0) (<_) A) (br (0) (<_) B))) (<-> (br A (<_) B) (br (` (sqr) A) (<_) (` (sqr) B))))) stmt (sqr00t () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (<-> (= (` (sqr) A) (0)) (= A (0))))) stmt (sqr1 () () (= (` (sqr) (1)) (1))) stmt (sqr4 () () (= (` (sqr) (4)) (2))) stmt (sqr9 () () (= (` (sqr) (9)) (3))) stmt (sqr2gt1lt2 () () (/\ (br (1) (<) (` (sqr) (2))) (br (` (sqr) (2)) (<) (2)))) stmt (sqrsq () ((e. A (RR))) (-> (br (0) (<_) A) (= (` (sqr) (opr A (^) (2))) A))) stmt (sqsqr () ((e. A (RR))) (-> (br (0) (<_) A) (= (opr (` (sqr) A) (^) (2)) A))) stmt (sqsqrt () () (-> (/\ (e. A (RR)) (br (0) (<_) A)) (= (opr (` (sqr) A) (^) (2)) A))) stmt (sqr2irrlem1 () ((e. A (NN)) (e. B (NN))) (-> (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))) (/\ (/\ (br B (<) A) (e. (opr A (/) (2)) (NN))) (= (opr B (^) (2)) (opr (2) (x.) (opr (opr A (/) (2)) (^) (2))))))) stmt (sqr2irrlem2 () () (-> (/\ (e. A (NN)) (e. B (NN))) (-> (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))) (/\ (/\ (br B (<) A) (e. (opr A (/) (2)) (NN))) (= (opr B (^) (2)) (opr (2) (x.) (opr (opr A (/) (2)) (^) (2)))))))) stmt (sqr2irrlem3 ((x y) (x z) (w x) (y z) (w y) (w z)) () (-. (E.e. x (NN) (E.e. y (NN) (= (opr (cv x) (^) (2)) (opr (2) (x.) (opr (cv y) (^) (2)))))))) stmt (sqr2irrlem4 () ((e. A (NN)) (e. B (NN))) (<-> (= (` (sqr) (2)) (opr A (/) B)) (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2)))))) stmt (sqr2irrlem5 () () (-> (/\ (e. A (NN)) (e. B (NN))) (<-> (= (` (sqr) (2)) (opr A (/) B)) (= (opr A (^) (2)) (opr (2) (x.) (opr B (^) (2))))))) stmt (sqr2irr ((x y)) () (e/ (` (sqr) (2)) (QQ))) stmt (sqr2re () () (e. (` (sqr) (2)) (RR))) stmt (ine0 () () (=/= (i) (0))) stmt (isqm1 () () (= (opr (i) (^) (2)) (-u (1)))) stmt (itimesi () () (= (opr (i) (x.) (i)) (-u (1)))) stmt (irec () () (= (opr (1) (/) (i)) (-u (i)))) stmt (inelr () () (-. (e. (i) (RR)))) stmt (crulem () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (e. D (RR))) (-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (= B D))) stmt (cru () ((e. A (RR)) (e. B (RR)) (e. C (RR)) (e. D (RR))) (<-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (/\ (= A C) (= B D)))) stmt (crut () () (-> (/\ (/\ (e. A (RR)) (e. B (RR))) (/\ (e. C (RR)) (e. D (RR)))) (<-> (= (opr A (+) (opr B (x.) (i))) (opr C (+) (opr D (x.) (i)))) (/\ (= A C) (= B D))))) stmt (crne0 () ((e. A (RR)) (e. B (RR))) (<-> (\/ (=/= A (0)) (=/= B (0))) (=/= (opr A (+) (opr B (x.) (i))) (0)))) stmt (crmul () ((e. A (CC)) (e. B (CC)) (e. C (CC)) (e. D (CC))) (= (opr (opr A (+) (opr B (x.) (i))) (x.) (opr C (+) (opr D (x.) (i)))) (opr (opr (opr A (x.) C) (-) (opr B (x.) D)) (+) (opr (opr (opr A (x.) D) (+) (opr B (x.) C)) (x.) (i))))) stmt (crrecz () ((e. A (RR)) (e. B (RR))) (-> (\/ (=/= A (0)) (=/= B (0))) (= (opr (1) (/) (opr A (+) (opr B (x.) (i)))) (opr (opr A (-) (opr B (x.) (i))) (/) (opr (opr A (^) (2)) (+) (opr B (^) (2))))))) stmt (creur ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (E!e. x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))))))) stmt (creui ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (E!e. y (RR) (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))))))) stmt (rimul () () (-> (/\ (e. A (RR)) (e. (opr A (x.) (i)) (RR))) (= A (0)))) stmt (nthruc () () (/\ (/\ (C: (NN) (ZZ)) (C: (ZZ) (QQ))) (/\ (C: (QQ) (RR)) (C: (RR) (CC))))) stmt (nthruz () () (/\ (C: (NN) (NN0)) (C: (NN0) (ZZ)))) stmt (revalt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (= (` (Re) A) (U. ({e.|} x (RR) (E.e. y (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))))))))) stmt (imvalt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (CC)) (= (` (Im) A) (U. ({e.|} y (RR) (E.e. x (RR) (= A (opr (cv x) (+) (opr (cv y) (x.) (i)))))))))) stmt (reclt ((x y) (A x) (A y)) () (-> (e. A (CC)) (e. (` (Re) A) (RR)))) stmt (imclt ((x y) (A x) (A y)) () (-> (e. A (CC)) (e. (` (Im) A) (RR)))) stmt (replimt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= A (opr (` (Re) A) (+) (opr (` (Im) A) (x.) (i)))))) stmt (cjvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (*) A) (opr (` (Re) A) (-) (opr (` (Im) A) (x.) (i)))))) stmt (absvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (abs) A) (` (sqr) (opr A (x.) (` (*) A)))))) stmt (cjclt () () (-> (e. A (CC)) (e. (` (*) A) (CC)))) stmt (recl () ((e. A (CC))) (e. (` (Re) A) (RR))) stmt (imcl () ((e. A (CC))) (e. (` (Im) A) (RR))) stmt (cjcl () ((e. A (CC))) (e. (` (*) A) (CC))) stmt (replim () ((e. A (CC))) (= A (opr (` (Re) A) (+) (opr (` (Im) A) (x.) (i))))) stmt (crret () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (Re) (opr A (+) (opr B (x.) (i)))) A))) stmt (crimt () () (-> (/\ (e. A (RR)) (e. B (RR))) (= (` (Im) (opr A (+) (opr B (x.) (i)))) B))) stmt (crre () ((e. A (RR)) (e. B (RR))) (= (` (Re) (opr A (+) (opr B (x.) (i)))) A)) stmt (crim () ((e. A (RR)) (e. B (RR))) (= (` (Im) (opr A (+) (opr B (x.) (i)))) B)) stmt (imret () () (-> (e. A (CC)) (= (` (Im) A) (` (Re) (opr (-u (i)) (x.) A))))) stmt (reim0t () () (-> (e. A (RR)) (= (` (Im) A) (0)))) stmt (reim0bt () () (-> (e. A (CC)) (<-> (e. A (RR)) (= (` (Im) A) (0))))) stmt (cjcj () ((e. A (CC))) (= (` (*) (` (*) A)) A)) stmt (reim0b () ((e. A (CC))) (<-> (e. A (RR)) (= (` (Im) A) (0)))) stmt (rereb () ((e. A (CC))) (<-> (e. A (RR)) (= (` (Re) A) A))) stmt (cjreb () ((e. A (CC))) (<-> (e. A (RR)) (= (` (*) A) A))) stmt (recj () ((e. A (CC))) (= (` (Re) (` (*) A)) (` (Re) A))) stmt (imcj () ((e. A (CC))) (= (` (Im) (` (*) A)) (-u (` (Im) A)))) stmt (readd () ((e. A (CC)) (e. B (CC))) (= (` (Re) (opr A (+) B)) (opr (` (Re) A) (+) (` (Re) B)))) stmt (imadd () ((e. A (CC)) (e. B (CC))) (= (` (Im) (opr A (+) B)) (opr (` (Im) A) (+) (` (Im) B)))) stmt (remul () ((e. A (CC)) (e. B (CC))) (= (` (Re) (opr A (x.) B)) (opr (opr (` (Re) A) (x.) (` (Re) B)) (-) (opr (` (Im) A) (x.) (` (Im) B))))) stmt (immul () ((e. A (CC)) (e. B (CC))) (= (` (Im) (opr A (x.) 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(sum_ k A B) (V))) stmt (sumeq1 ((k m) (k n) (k x) (k y) (m n) (m x) (m y) (n x) (n y) (x y) (A m) (A n) (A x) (A y) (B m) (B n) (B x) (B y) (C m) (C n) (C x) (C y)) () (-> (= A B) (= (sum_ k A C) (sum_ k B C)))) stmt (hbsum1 ((m n) (m x) (m y) (m z) (A m) (n x) (n y) (n z) (A n) (x y) (x z) (A x) (y z) (A y) (A z) (B m) (B n) (B x) (B y) (B z) (k m) (k n) (k x) (k y) (k z)) ((-> (e. (cv x) A) (A. k (e. (cv x) A)))) (-> (e. (cv x) (sum_ k A B)) (A. k (e. (cv x) (sum_ k A B))))) stmt (hbsum ((m n) (m w) (m y) (m z) (A m) (n w) (n y) (n z) (A n) (w y) (w z) (A w) (y z) (A y) (A z) (B m) (B n) (B w) (B y) (B z) (k m) (k n) (k w) (k x) (k y) (k z) (m x) (n x) (w x) (x y) (x z)) ((-> (e. (cv y) A) (A. x (e. (cv y) A))) (-> (e. (cv y) B) (A. x (e. (cv y) B)))) (-> (e. (cv y) (sum_ k A B)) (A. x (e. (cv y) (sum_ k A B))))) stmt (sumeq2 ((k m) (k n) (k x) (k y) (A k) (m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (B m) (B n) (B x) (B y) (C m) (C n) (C x) (C y)) () (-> (A.e. k A (= B C)) (= (sum_ k A B) (sum_ k A C)))) stmt (cbvsum ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (B m) (B n) (B x) (B y) (C m) (C n) (C x) (C y) (j k) (j m) (j n) (j x) (j y) (k m) (k n) (k x) (k y)) ((-> (e. (cv x) B) (A. k (e. (cv x) B))) (-> (e. (cv x) C) (A. j (e. (cv x) C))) (-> (= (cv j) (cv k)) (= B C))) (= (sum_ j A B) (sum_ k A C))) stmt (sumeq1i () ((= A B)) (= (sum_ k A C) (sum_ k B C))) stmt (sumeq2i ((A k)) ((-> (e. (cv k) A) (= B C))) (= (sum_ k A B) (sum_ k A C))) stmt (sumeq1d () ((-> ph (= A B))) (-> ph (= (sum_ k A C) (sum_ k B C)))) stmt (sumeq2d ((A k)) ((-> ph (A.e. k A (= B C)))) (-> ph (= (sum_ k A B) (sum_ k A C)))) stmt (sumeq2dv ((A k) (k ph)) ((-> (/\ ph (e. (cv k) A)) (= B C))) (-> ph (= (sum_ k A B) (sum_ k A C)))) stmt (sumeq2sdv ((A k) (k ph)) ((-> ph (= B C))) (-> ph (= (sum_ k A B) (sum_ k A C)))) stmt (2sumeq2d ((A j) (B k)) ((-> ph (A.e. j A (A.e. k B (= C D))))) (-> ph (= (sum_ j A (sum_ k B C)) (sum_ j A (sum_ k B D))))) stmt (2sumeq2dv ((j k) (A j) (A k) (B k) (j ph) (k ph)) ((-> (/\/\ ph (e. (cv j) A) (e. (cv k) B)) (= C D))) (-> ph (= (sum_ j A (sum_ k B C)) (sum_ j A (sum_ k B D))))) stmt (sumeq12d ((A k)) ((-> ph (= A B)) (-> ph (A.e. k A (= C D)))) (-> ph (= (sum_ k A C) (sum_ k B D)))) stmt (sumeq12rd ((B k)) ((-> ph (= A B)) (-> ph (A.e. k B (= C D)))) (-> ph (= (sum_ k A C) (sum_ k B D)))) stmt (sumeqfv ((A y) (k y) (B k) (B y) (C k)) ((e. A (V)) (= F ({<,>|} k y (/\ (e. (cv k) B) (= (cv y) A))))) (-> (C_ C B) (= (sum_ k C (` F (cv k))) (sum_ k C A)))) stmt (dffsum ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (M m) (M n) (M x) (M y) (N m) (N n) (N x) (N y) (k m) (k n) (k x) (k y)) () (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) A) (` (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) N)))) stmt (fsumserz ((k y) (F k) (F y) (M y) (N y)) ((e. F (V))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) (` F (cv k))) (` (opr (<,> M (+)) (seq) F) N)))) stmt (fsumserzf ((j x) (F j) (F x) (M j) (M x) (N x) (j k) (k x)) ((e. F (V)) (-> (e. (cv x) F) (A. k (e. (cv x) F)))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) (` F (cv k))) (` (opr (<,> M (+)) (seq) F) N)))) stmt (fsumser0f ((F x) (N x) (k x)) ((e. F (V)) (-> (e. (cv x) F) (A. k (e. (cv x) F)))) (-> (e. N (NN0)) (= (sum_ k (opr (0) (...) N) (` F (cv k))) (` (opr (+) (seq0) F) N)))) stmt (fsumser1f ((F x) (N x) (k x)) ((e. F (V)) (-> (e. (cv x) F) (A. k (e. (cv x) F)))) (-> (e. N (NN)) (= (sum_ k (opr (1) (...) N) (` F (cv k))) (` (opr (+) (seq1) F) N)))) stmt (fsumserz2 ((A y) (F x) (k x) (k y) (M k) (x y) (M x) (M y) (N k) (N x)) ((e. A (V)) (= F ({<,>|} k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A))))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) A) (` (opr (<,> M (+)) (seq) F) N)))) stmt (serzfsum ((A y) (n x) (n z) (F n) (x z) (F x) (F z) (k n) (k x) (k y) (k z) (M k) (n y) (M n) (x y) (M x) (y z) (M y) (M z)) ((e. A (V)) (= F ({<,>|} k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A)))) (= G ({<,>|} n z (/\ (e. (cv n) (` (ZZ>) M)) (= (cv z) (sum_ k (opr M (...) (cv n)) A)))))) (-> (e. M (ZZ)) (= (opr (<,> M (+)) (seq) F) G))) stmt (fsum1 ((x y) (A x) (A y) (k y) (B k) (B y) (k x) (M k) (M x) (M y)) ((-> (= (cv k) M) (= A B))) (-> (/\ (e. B C) (e. M (ZZ))) (= (sum_ k (opr M (...) M) A) B))) stmt (fsump1 ((x y) (A x) (A y) (k y) (B k) (B y) (k x) (M k) (M x) (N k) (N x) (N y)) ((e. A (V)) (e. B (V)) (-> (= (cv k) (opr N (+) (1))) (= A B))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) (opr N (+) (1))) A) (opr (sum_ k (opr M (...) N) A) (+) B)))) stmt (fsum1f ((j y) (j z) (A j) (y z) (A y) (A z) (j x) (B j) (B x) (j k) (M j) (k z) (M k) (M z) (k x) (k y) (x y)) ((-> (e. (cv x) B) (A. k (e. (cv x) B))) (-> (= (cv k) M) (= A B))) (-> (/\ (e. B C) (e. M (ZZ))) (= (sum_ k (opr M (...) M) A) B))) stmt (fsum1slem ((x y) (A x) (A y) (M x) (k x) (k y)) ((e. A (V))) (-> (e. M (ZZ)) (= (sum_ k (opr M (...) M) A) ([_/]_ M k A)))) stmt (fsum1s ((k x) (M k) (M x) (A x) (B x)) () (-> (/\ (e. M (ZZ)) (A.e. k (opr M (...) M) (e. A B))) (= (sum_ k (opr M (...) M) A) ([_/]_ M k A)))) stmt (fsum1s2 ((M k)) () (-> (/\ (e. M (ZZ)) (e. ([_/]_ M k A) B)) (= (sum_ k (opr M (...) M) A) ([_/]_ M k A)))) stmt (fsump1f ((j y) (j z) (A j) (y z) (A y) (A z) (j x) (B j) (B x) (M j) (M z) (j k) (N j) (k z) (N k) (N z) (k x) (k y) (x y)) ((e. A (V)) (e. B (V)) (-> (e. (cv x) B) (A. k (e. (cv x) B))) (-> (= (cv k) (opr N (+) (1))) (= A B))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) (opr N (+) (1))) A) (opr (sum_ k (opr M (...) N) A) (+) B)))) stmt (fsump1slem ((j x) (A j) (A x) (M j) (N j) (j k) (k x)) ((e. A (V))) (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) (opr N (+) (1))) A) (opr (sum_ k (opr M (...) N) A) (+) ([_/]_ (opr N (+) (1)) k A))))) stmt (fsump1s ((k x) (M k) (M x) (N k) (N x) (A x) (B x)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) (opr N (+) (1))) (e. A B))) (= (sum_ k (opr M (...) (opr N (+) (1))) A) (opr (sum_ k (opr M (...) N) A) (+) ([_/]_ (opr N (+) (1)) k A))))) stmt (fsumcllem ((j m) (j x) (j y) (A j) (m x) (m y) (A m) (x y) (A x) (A y) (j k) (C j) (k m) (k x) (k y) (C k) (C m) (C x) (C y) (M j) (M k) (M m) (M x) (M y) (N j) (N k)) ((C_ C (CC)) (-> (/\ (e. (cv x) C) (e. (cv y) C)) (e. (opr (cv x) (+) (cv y)) C))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. A C))) (e. (sum_ k (opr M (...) N) A) C))) stmt (fsumclt ((x y) (A x) (A y) (k x) (M x) (k y) (M y) (M k) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. A (CC)))) (e. (sum_ k (opr M (...) N) A) (CC)))) stmt (fsum0clt ((N k)) () (-> (/\ (e. N (NN0)) (A.e. k (opr (0) (...) N) (e. A (CC)))) (e. (sum_ k (opr (0) (...) N) A) (CC)))) stmt (fsumreclt ((x y) (A x) (A y) (k x) (M x) (k y) (M y) (M k) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. A (RR)))) (e. (sum_ k (opr M (...) N) A) (RR)))) stmt (fsum1ps ((j m) (A j) (A m) (j k) (M j) (k m) (M k) (M m) (N j) (N k)) () (-> (/\/\ (e. N (` (ZZ>) M)) (br M (<) N) (A.e. k (opr M (...) N) (e. A (CC)))) (= (sum_ k (opr M (...) N) A) (opr ([_/]_ M k A) (+) (sum_ k (opr (opr M (+) (1)) (...) N) A))))) stmt (fsum1p ((k x) (B x) (B k) (M x) (M k) (N k)) ((-> (= (cv k) M) (= A B))) (-> (/\/\ (e. N (` (ZZ>) M)) (br M (<) N) (A.e. k (opr M (...) N) (e. A (CC)))) (= (sum_ k (opr M (...) N) A) (opr B (+) (sum_ k (opr (opr M (+) (1)) (...) N) A))))) stmt (fsumsplit ((j m) (A j) (A m) (K j) (j k) (M j) (k m) (M k) (M m) (N j) (N k) (N m)) () (-> (/\/\ (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1)))) (A.e. k (opr M (...) N) (e. A (CC)))) (= (sum_ k (opr M (...) N) A) (opr (sum_ k (opr M (...) K) A) (+) (sum_ k (opr (opr K (+) (1)) (...) N) A))))) stmt (fsum0split ((N k)) () (-> (/\/\ (e. N (ZZ)) (e. K (opr (1) (...) N)) (A.e. k (opr (0) (...) N) (e. A (CC)))) (= (sum_ k (opr (0) (...) N) A) (opr (sum_ k (opr (0) (...) (opr N (-) K)) A) (+) (sum_ k (opr (opr (opr N (-) K) (+) (1)) (...) N) A))))) stmt (fsumadd ((j m) (A j) (A m) (B j) (B m) (j k) (M j) (k m) (M k) (M m) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (/\ (e. A (CC)) (e. B (CC))))) (= (sum_ k (opr M (...) N) (opr A (+) B)) (opr (sum_ k (opr M (...) N) A) (+) (sum_ k (opr M (...) N) B))))) stmt (csbfsumlem ((k x) (m x) (n x) (A x) (k m) (k n) (A k) (m n) (A m) (A n) (B m) (B n) (M x) (M k) (M m) (M n) (N x) (N n)) ((e. A (V)) (e. B (V))) (-> (e. N (` (ZZ>) M)) (= ([_/]_ A x (sum_ k (opr M (...) N) B)) (sum_ k (opr M (...) N) ([_/]_ A x B))))) stmt (csbfsum ((k y) (A k) (A y) (B y) (k x) (x y) (M x) (M k) (M y) (N x) (N k) (N y)) () (-> (/\/\ (e. A C) (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. ([_/]_ A x B) (CC)))) (= ([_/]_ A x (sum_ k (opr M (...) N) B)) (sum_ k (opr M (...) N) ([_/]_ A x B))))) stmt (fsumcom ((k n) (A k) (A n) (j k) (j m) (j n) (J j) (k m) (J k) (m n) (J m) (J n) (K j) (K k) (K m) (M j) (M k) (M m) (M n) (N j) (N k) (N m) (N n)) () (-> (/\/\ (e. K (` (ZZ>) J)) (e. N (` (ZZ>) M)) (A.e. j (opr J (...) K) (A.e. m (opr M (...) N) (e. A (CC))))) (= (sum_ j (opr J (...) K) (sum_ m (opr M (...) N) A)) (sum_ m (opr M (...) N) (sum_ j (opr J (...) K) A))))) stmt (fsumrev ((k m) (k n) (A k) (m n) (A m) (A n) (j k) (j m) (j n) (K j) (K k) (K m) (K n) (M j) (M k) (M m) (M n) (N j) (N k) (N m)) () (-> (/\/\ (e. N (` (ZZ>) M)) (e. K (ZZ)) (A.e. j (opr M (...) N) (e. A (CC)))) (= (sum_ j (opr M (...) N) A) (sum_ k (opr (opr K (-) N) (...) (opr K (-) M)) ([_/]_ (opr K (-) (cv k)) j A))))) stmt (fsumshft ((k m) (A k) (A m) (j k) (j m) (K j) (K k) (K m) (M j) (M k) (M m) (N j) (N k) (N m)) () (-> (/\/\ (e. N (` (ZZ>) M)) (e. K (ZZ)) (A.e. j (opr M (...) N) (e. A (CC)))) (= (sum_ j (opr M (...) N) A) (sum_ k (opr (opr M (+) K) (...) (opr N (+) K)) ([_/]_ (opr (cv k) (-) K) j A))))) stmt (fsummulc1 ((j k) (j m) (C j) (k m) (C k) (C m) (A j) (A m) (M j) (M k) (M m) (N j) (N k)) () (-> (/\/\ (e. N (` (ZZ>) M)) (e. C (CC)) (A.e. k (opr M (...) N) (e. A (CC)))) (= (opr C (x.) (sum_ k (opr M (...) N) A)) (sum_ k (opr M (...) N) (opr C (x.) A))))) stmt (fsummulc2 ((C k) (M k) (N k)) () (-> (/\/\ (e. N (` (ZZ>) M)) (e. C (CC)) (A.e. k (opr M (...) N) (e. A (CC)))) (= (opr (sum_ k (opr M (...) N) A) (x.) C) (sum_ k (opr M (...) N) (opr A (x.) C))))) stmt (fsumdivc ((C k) (M k) (N k)) () (-> (/\ (/\ (e. N (` (ZZ>) M)) (e. C (CC))) (/\ (=/= C (0)) (A.e. k (opr M (...) N) (e. A (CC))))) (= (opr (sum_ k (opr M (...) N) A) (/) C) (sum_ k (opr M (...) N) (opr A (/) C))))) stmt (fsum2mul ((A m) (B j) (J j) (K j) (j m) (M j) (M m) (N j) (N m)) () (-> (/\ (/\ (e. K (` (ZZ>) J)) (A.e. j (opr J (...) K) (e. A (CC)))) (/\ (e. N (` (ZZ>) M)) (A.e. m (opr M (...) N) (e. B (CC))))) (= (sum_ j (opr J (...) K) (sum_ m (opr M (...) N) (opr A (x.) B))) (opr (sum_ j (opr J (...) K) A) (x.) (sum_ m (opr M (...) N) B))))) stmt (fsumconst ((j k) (j m) (A j) (k m) (A k) (A m) (M j) (M k) (M m) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (e. A (CC))) (= (sum_ k (opr M (...) N) A) (opr (opr (opr N (-) M) (+) (1)) (x.) A)))) stmt (fsum0 ((M k) (N k)) () (-> (e. N (` (ZZ>) M)) (= (sum_ k (opr M (...) N) (0)) (0)))) stmt (fsumcmp ((j m) (A j) (A m) (B j) (B m) (j k) (M j) (k m) (M k) (M m) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (/\/\ (e. A (RR)) (e. B (RR)) (br A (<_) B)))) (br (sum_ k (opr M (...) N) A) (<_) (sum_ k (opr M (...) N) B)))) stmt (fsumabs ((j m) (A j) (A m) (j k) (M j) (k m) (M k) (M m) (N j) (N k)) () (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. A (CC)))) (br (` (abs) (sum_ k (opr M (...) N) A)) (<_) (sum_ k (opr M (...) N) (` (abs) A))))) stmt (serzclt ((F k) (M k) (N k)) ((e. F (V))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (e. (` (opr (<,> M (+)) (seq) F) N) (CC)))) stmt (ser0clt ((F k) (N k)) ((e. F (V))) (-> (/\ (e. N (NN0)) (A.e. k (opr (0) (...) N) (e. (` F (cv k)) (CC)))) (e. (` (opr (+) (seq0) F) N) (CC)))) stmt (ser1clt ((F k) (N k)) ((e. F (V))) (-> (/\ (e. N (NN)) (A.e. k (opr (1) (...) N) (e. (` F (cv k)) (CC)))) (e. (` (opr (+) (seq1) F) N) (CC)))) stmt (serzcl2t ((j k) (F j) (F k) (M j) (M k) (N j)) ((e. F (V))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (CC)))) (e. (` (opr (<,> M (+)) (seq) F) N) (CC)))) stmt (serzreclt ((F k) (M k) (N k)) ((e. F (V))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (e. (` F (cv k)) (RR)))) (e. (` (opr (<,> M (+)) (seq) F) N) (RR)))) stmt (serz1p ((F k) (M k) (N k)) ((e. F (V))) (-> (/\/\ (e. N (` (ZZ>) M)) (br M (<) N) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (` (opr (<,> M (+)) (seq) F) N) (opr (` F M) (+) (` (opr (<,> (opr M (+) (1)) (+)) (seq) F) N))))) stmt (serz0 ((M k) (N k)) () (-> (e. N (` (ZZ>) M)) (= (` (opr (<,> M (+)) (seq) (X. (` (ZZ>) M) ({} (0)))) N) (0)))) stmt (serzcmp ((F k) (G k) (M k) (N k)) ((e. F (V)) (e. G (V))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (/\/\ (e. (` F (cv k)) (RR)) (e. (` G (cv k)) (RR)) (br (` F (cv k)) (<_) (` G (cv k)))))) (br (` (opr (<,> M (+)) (seq) F) N) (<_) (` (opr (<,> M (+)) (seq) G) N)))) stmt (serzcmp0 ((F k) (M k) (N k)) ((e. F (V))) (-> (/\ (e. N (` (ZZ>) M)) (A.e. k (opr M (...) N) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k)))))) (br (0) (<_) (` (opr (<,> M (+)) (seq) F) N)))) stmt (serzsplit ((F k) (M k) (N k)) ((e. F (V))) (-> (/\/\ (e. N (ZZ)) (e. K (opr M (...) (opr N (-) (1)))) (A.e. k (opr M (...) N) (e. (` F (cv k)) (CC)))) (= (` (opr (<,> M (+)) (seq) F) N) (opr (` (opr (<,> M (+)) (seq) F) K) (+) (` (opr (<,> (opr K (+) (1)) (+)) (seq) F) N))))) stmt (binomlem1 ((k x) (A k) (A x) (B k) (B x) (N k) (N x)) ((e. A (CC)) (e. B (CC))) (-> (e. N (NN)) (= (opr (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))) (x.) A) (opr (opr A (^) (opr N (+) (1))) (+) (sum_ k (opr (1) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))))))) stmt (binomlem2 ((j k) (j x) (A j) (k x) (A k) (A x) (B j) (B k) (B x) (N j) (N k) (N x)) ((e. A (CC)) (e. B (CC))) (-> (e. N (NN)) (= (opr (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))) (x.) B) (opr (sum_ k (opr (1) (...) N) (opr (opr N (C.) (opr (cv k) (-) (1))) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (+) (opr B (^) (opr N (+) (1))))))) stmt (binomlem3 ((A k) (B k) (N k)) ((e. A (CC)) (e. B (CC))) (-> (e. N (NN)) (= (sum_ k (opr (1) (...) (opr N (+) (1))) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (opr (sum_ k (opr (1) (...) N) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (+) (opr B (^) (opr N (+) (1))))))) stmt (binomlem4 ((k x) (A k) (A x) (B k) (B x) (N k) (N x)) ((e. A (CC)) (e. B (CC))) (-> (e. N (NN0)) (= (sum_ k (opr (0) (...) (opr N (+) (1))) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))) (opr (opr A (^) (opr N (+) (1))) (+) (sum_ k (opr (1) (...) (opr N (+) (1))) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k))))))))) stmt (binomlem5 ((A k) (B k) (N k)) ((e. A (CC)) (e. B (CC))) (-> (e. N (NN)) (= (opr (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k))))) (x.) (opr A (+) B)) (sum_ k (opr (0) (...) (opr N (+) (1))) (opr (opr (opr N (+) (1)) (C.) (cv k)) (x.) (opr (opr A (^) (opr (opr N (+) (1)) (-) (cv k))) (x.) (opr B (^) (cv k)))))))) stmt (binomlem6 ((j k) (j x) (A j) (k x) (A k) (A x) (B j) (B k) (B x) (N j) (N k) (N x)) ((e. A (CC)) (e. B (CC))) (-> (e. N (NN)) (= (opr (opr A (+) B) (^) N) (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k)))))))) stmt (binom ((A k) (B k) (N k)) ((e. A (CC)) (e. B (CC))) (-> (e. N (NN0)) (= (opr (opr A (+) B) (^) N) (sum_ k (opr (0) (...) N) (opr (opr N (C.) (cv k)) (x.) (opr (opr A (^) (opr N (-) (cv k))) (x.) (opr B (^) (cv k)))))))) stmt (climcvgc1 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x) (B j) (B k) (B x)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))) stmt (climcvg1 ((j k) (F j) (F k) (A j) (A k) (B j) (B k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (clim1 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((e. F (V))) (-> (e. A (CC)) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))) stmt (clim1a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x)) ((e. F (V))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (clim2a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x) (M j) (M k) (M m) (M n) (M x)) ((e. F (V)) (e. M (ZZ))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))) stmt (clim2 ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((e. F (V)) (e. M (ZZ))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br M (<_) (cv k)) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))) stmt (climcvg2 ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) ((e. M (ZZ))) (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))) stmt (climcvg2z ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (e. M (ZZ)) (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) (cv j)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))) stmt (climcvgc2z ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))) stmt (climcvg2zb ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (climcvg2zbOLD ((j k) (j m) (F j) (k m) (F k) (F m) (A j) (A k) (A m) (B j) (B k) (B m) (M j) (M k) (M m)) () (-> (/\ (e. M (ZZ)) (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (clim2az ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((e. F (V))) (-> (/\ (e. M (ZZ)) (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC))))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (` (ZZ>) M) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (clim3az ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x) (M j) (M k) (M m) (M n) (M x)) ((e. F (V))) (-> (/\ (e. M (ZZ)) (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC))))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) M) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (clim3z ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. A (CC)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (CC)))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) M) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (clim3zOLD ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x)) ((e. F (V))) (-> (/\ (e. M (ZZ)) (/\ (e. A (CC)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (` (ZZ>) M) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (clim3a ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A j) (A k) (A m) (A n) (A x) (N j) (N k) (N m) (N n) (N x)) ((e. F (V)) (e. N (ZZ))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (clim3 ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (N j) (N k) (N m) (N x)) ((e. F (V)) (e. N (ZZ))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br N (<_) (cv k)) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (clim3b ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (N j) (N k) (N m) (N x)) ((e. F (V)) (e. N (ZZ))) (-> (e. A (CC)) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))) stmt (climcvg3 ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (N j) (N k)) ((e. N (ZZ))) (-> (/\ (/\ (e. A (CC)) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (climcvg3z ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (N j) (N k)) () (-> (/\ (/\ (e. A (CC)) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (clim4a ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x) (N j) (N k) (N m) (N x)) ((e. F (V)) (e. M (ZZ)) (e. N (ZZ))) (-> (/\ (e. A (CC)) (E.e. m (ZZ) (A.e. k (ZZ) (-> (br (cv m) (<_) (cv k)) (e. (` F (cv k)) (CC)))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))) stmt (clim4 ((j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (A j) (A k) (A m) (A x) (M j) (M k) (M m) (M x) (N j) (N k) (N m) (N x)) ((e. F (V)) (e. M (ZZ)) (e. N (ZZ))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (/\ (br M (<_) (cv k)) (br N (<_) (cv k))) (e. (` F (cv k)) (CC))))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x))))))))))) stmt (climcvg4 ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) ((e. M (ZZ))) (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (ZZ) (/\ (br M (<_) (cv j)) (A.e. k (ZZ) (-> (/\ (br N (<_) (cv k)) (br (cv j) (<_) (cv k))) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))) stmt (climcvgc4z ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))) stmt (climcvg4z ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (clim0cvg4z ((j k) (F j) (F k) (A j) (A k) (M j) (M k)) () (-> (/\ (/\ (e. M (ZZ)) (br F (~~>) (0))) (/\ (e. A (RR)) (br (0) (<) A))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (` F (cv k))) (<) A)))))) stmt (climcvg4uzOLD ((j k) (F j) (F k) (A j) (A k) (B j) (B k) (M j) (M k)) () (-> (/\ (e. M (ZZ)) (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B)))) (E.e. j (` (ZZ>) M) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (climcvgc5z ((j k) (A j) (A k) (B j) (B k) (F j) (F k) (K j) (K k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. K (ZZ)) (e. A C)) (/\/\ (br F (~~>) A) (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (/\ (br K (<_) (cv j)) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) (CC)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))))) stmt (climcvg5z ((j k) (A j) (A k) (B j) (B k) (F j) (F k) (K j) (K k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. K (ZZ)) (e. A C)) (/\/\ (br F (~~>) A) (e. B (RR)) (br (0) (<) B))) (E.e. j (` (ZZ>) M) (/\ (br K (<_) (cv j)) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B))))))) stmt (clim0cvg5z ((j k) (A j) (A k) (F j) (F k) (K j) (K k) (M j) (M k)) () (-> (/\ (/\/\ (e. M (ZZ)) (e. K (ZZ)) (br F (~~>) (0))) (/\ (e. A (RR)) (br (0) (<) A))) (E.e. j (` (ZZ>) M) (/\ (br K (<_) (cv j)) (A.e. k (` (ZZ>) N) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (` F (cv k))) (<) A))))))) stmt (climnn0 ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((e. F (V))) (-> (/\ (e. A (CC)) (A.e. k (NN0) (e. (` F (cv k)) (CC)))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (climnn ((j k) (j x) (F j) (k x) (F k) (F x) (A j) (A k) (A x)) ((e. F (V))) (-> (/\ (e. A (CC)) (A.e. k (NN) (e. (` F (cv k)) (CC)))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (clim0nn ((j k) (j x) (F j) (k x) (F k) (F x)) ((e. F (V))) (-> (A.e. k (NN) (e. (` F (cv k)) (CC))) (<-> (br F (~~>) (0)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (` F (cv k))) (<) (cv x)))))))))) stmt (climnn0le ((j k) (j x) (j y) (F j) (k x) (k y) (F k) (x y) (F x) (F y) (A j) (A k) (A x) (A y)) ((e. F (V))) (-> (/\ (e. A (CC)) (A.e. k (NN0) (e. (` F (cv k)) (CC)))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<_) (cv x)))))))))) stmt (climfnn ((f j) (f k) (f x) (A f) (j k) (j x) (A j) (k x) (A k) (A x) (F f) (F j) (F k) (F x)) () (-> (/\ (:--> F (NN) (CC)) (e. A (CC))) (<-> (br F (~~>) A) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) (cv x)))))))))) stmt (climcvgnn ((j k) (A j) (A k) (B j) (B k) (F j) (F k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (climcvgnn0 ((j k) (A j) (A k) (B j) (B k) (F j) (F k)) () (-> (/\ (/\ (e. A C) (br F (~~>) A)) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. j (NN0) (A.e. k (NN0) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` F (cv k)) (-) A)) (<) B)))))) stmt (climconst ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F k) (F x) (M j) (M k) (M x)) ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (e. A (CC)) (A.e. k (` (ZZ>) M) (= (` F (cv k)) A))) (br F (~~>) A))) stmt (climconstyOLD ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F k) (F x) (N j) (N k) (N x)) ((e. F (V)) (e. N (ZZ))) (-> (/\ (e. A (CC)) (A.e. k (ZZ) (-> (br N (<_) (cv k)) (= (` F (cv k)) A)))) (br F (~~>) A))) stmt (climconst2 ((A k)) () (-> (e. A (CC)) (br (X. (ZZ) ({} A)) (~~>) A))) stmt (clim0 () () (br (X. (ZZ) ({} (0))) (~~>) (0))) stmt (climunii ((j k) (j m) (A j) (k m) (A k) (A m) (B j) (B k) (B m) (F j) (F k) (F m)) ((e. A (V)) (e. B (V)) (/\ (br F (~~>) A) (br F (~~>) B))) (= A B)) stmt (climuni () ((e. A (V)) (e. B (V))) (-> (/\ (br F (~~>) A) (br F (~~>) B)) (= A B))) stmt (climeu ((A y) (x y) (F x) (F y)) ((e. A (V))) (-> (br F (~~>) A) (E! x (br F (~~>) (cv x))))) stmt (climreu ((F x)) ((e. A (V))) (-> (br F (~~>) A) (E!e. x (CC) (br F (~~>) (cv x))))) stmt (2climnn ((x y) (k x) (j x) (A x) (k y) (j y) (A y) (j k) (A k) (A j) (F j) (F k) (F x) (F y) (G x) (G y) (G k) (G j)) ((e. G (V))) (-> (/\ (/\ (e. A (CC)) (A.e. k (NN) (/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC))))) (/\ (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (NN) (A.e. k (NN) (-> (br (cv j) (<_) (cv k)) (br (` (abs) (opr (` G (cv k)) (-) (` F (cv k)))) (<) (cv x))))))) (br F (~~>) A))) (br G (~~>) A))) stmt (climshft ((j t) (j m) (j n) (j x) (A j) (m t) (n t) (t x) (A t) (m n) (m x) (A m) (n x) (A n) (A x) (M j) (M t) (M m) (M n) (M x) (F j) (F t) (F m) (F n) (F x)) ((e. F (V)) (e. M (ZZ))) (-> (e. A (CC)) (<-> (br (opr F (shift) M) (~~>) A) (br F (~~>) A)))) stmt (climres ((j t) (j x) (A j) (t x) (A t) (A x) (j k) (M j) (k t) (k x) (M k) (M t) (M x) (F j) (F t) (F x)) ((e. F (V)) (e. M (ZZ))) (-> (e. A (CC)) (<-> (br (|` F (` (ZZ>) M)) (~~>) A) (br F (~~>) A)))) stmt (climuz0 () ((e. M (ZZ))) (br (X. (` (ZZ>) M) ({} (0))) (~~>) (0))) stmt (climrecl ((j m) (A j) (A m) (j k) (F j) (k m) (F k) (F m) (M j) (M k) (M m)) ((e. A (V))) (-> (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR)))) (e. A (RR)))) stmt (climfnrcl ((F k)) ((e. A (V)) (:--> F (NN) (RR)) (br F (~~>) A)) (e. A (RR))) stmt (climge0 ((j m) (A j) (A m) (j k) (F j) (k m) (F k) (F m) (M j) (M k) (M m)) ((e. A (V))) (-> (/\/\ (e. M (ZZ)) (br F (~~>) A) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (RR)) (br (0) (<_) (` F (cv k)))))) (br (0) (<_) A))) stmt (climaddlem1 ((ph t) (A t) (B t) (k t) (F k) (F t) (G k) (G t) (H k) (H t) (M k) (M t)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (V)) (e. B (V)) (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))))) (-> (/\ (e. (cv t) (` (ZZ>) M)) ph) (/\/\ (e. (` F (cv t)) (CC)) (e. (` G (cv t)) (CC)) (= (` H (cv t)) (opr (` F (cv t)) (+) (` G (cv t))))))) stmt (climaddlem2 ((F k) (G k) (H k) (M k)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (V)) (e. B (V)) (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))))) (-> ph (/\ (e. A (CC)) (e. B (CC))))) stmt (climaddlem3 ((f t) (f u) (f v) (f ph) (t u) (t v) (ph t) (u v) (ph u) (ph v) (f g) (f h) (f t) (f u) (f v) (A f) (g h) (g t) (g u) (g v) (A g) (h t) (h u) (h v) (A h) (t u) (t v) (A t) (u v) (A u) (A v) (B f) (B g) (B h) (B t) (B u) (B v) (f k) (F f) (g k) (F g) (k t) (k u) (k v) (F k) (F t) (F u) (F v) (G f) (G g) (G k) (G t) (G u) (G v) (H f) (h k) (H h) (H k) (H t) (H u) (H v) (M f) (M g) (M h) (M k) (M t) (M u) (M v)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (V)) (e. B (V)) (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k))))))))) (-> (/\ (e. M (ZZ)) ph) (br H (~~>) (opr A (+) B)))) stmt (climadd ((F k) (G k) (H k) (M k)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (V)) (e. B (V))) (-> (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (+) (` G (cv k)))))))) (br H (~~>) (opr A (+) B)))) stmt (climaddc1 ((A k) (C k) (F k) (G k) (M k)) ((e. F (V)) (e. G (V)) (e. A (V)) (e. C (V))) (-> (/\ (/\ (br F (~~>) A) (e. C (CC))) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr (` F (cv k)) (+) C)))))) (br G (~~>) (opr A (+) C)))) stmt (climaddc2 ((A k) (C k) (F k) (G k) (M k)) ((e. F (V)) (e. G (V)) (e. A (V)) (e. C (V))) (-> (/\ (/\ (br F (~~>) A) (e. C (CC))) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr C (+) (` F (cv k)))))))) (br G (~~>) (opr C (+) A)))) stmt (climmullem1 () () (-> (/\ (e. A (CC)) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (/\ (e. (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))) (RR)) (br (0) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))))) stmt (climmullem2 () () (-> (/\ (e. B (CC)) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (/\ (e. (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))))) (RR)) (br (0) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B)))))))))) stmt (climmullem3 () () (-> (/\ (/\/\ (/\ (e. F (CC)) (e. G (CC))) (/\ (e. A (CC)) (e. B (CC))) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (/\ (br (` (abs) (opr F (-) A)) (<) (1)) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A)))))) (br (` (abs) (opr F (x.) (opr G (-) B))) (<) (opr (cv v) (/) (2))))) stmt (climmullem4 () () (-> (/\ (/\/\ (/\ (e. F (CC)) (e. G (CC))) (/\ (e. A (CC)) (e. B (CC))) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (br (` (abs) (opr F (-) A)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))) (br (` (abs) (opr (opr F (-) A) (x.) B)) (<) (opr (cv v) (/) (2))))) stmt (climmullem5 () () (-> (/\/\ (/\ (e. F (CC)) (e. G (CC))) (/\ (e. A (CC)) (e. B (CC))) (/\ (e. (cv v) (RR)) (br (0) (<) (cv v)))) (-> (/\ (br (` (abs) (opr F (-) A)) (<) (opr (1) (/) (opr (1) (+) (opr (1) (/) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) B))))))) (br (` (abs) (opr G (-) B)) (<) (opr (opr (cv v) (/) (2)) (/) (opr (1) (+) (` (abs) A))))) (br (` (abs) (opr (opr F (x.) G) (-) (opr A (x.) B))) (<) (cv v))))) stmt (climmullem6 ((ph t) (A t) (B t) (k t) (F k) (F t) (G k) (G t) (H k) (H t) (M k) (M t)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (V)) (e. B (V)) (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k))))))))) (-> (/\ (e. (cv t) (` (ZZ>) M)) ph) (/\/\ (e. (` F (cv t)) (CC)) (e. (` G (cv t)) (CC)) (= (` H (cv t)) (opr (` F (cv t)) (x.) (` G (cv t))))))) stmt (climmullem7 ((F k) (G k) (H k) (M k)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (V)) (e. B (V)) (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k))))))))) (-> ph (/\ (e. A (CC)) (e. B (CC))))) stmt (climmullem8 ((f t) (f u) (f v) (f ph) (t u) (t v) (ph t) (u v) (ph u) (ph v) (f g) (f h) (f t) (f u) (f v) (A f) (g h) (g t) (g u) (g v) (A g) (h t) (h u) (h v) (A h) (t u) (t v) (A t) (u v) (A u) (A v) (B f) (B g) (B h) (B t) (B u) (B v) (f k) (F f) (g k) (F g) (k t) (k u) (k v) (F k) (F t) (F u) (F v) (G f) (G g) (G k) (G t) (G u) (G v) (H f) (h k) (H h) (H k) (H t) (H u) (H v) (M f) (M g) (M h) (M k) (M t) (M u) (M v)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (V)) (e. B (V)) (<-> ph (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k))))))))) (-> (/\ (e. M (ZZ)) ph) (br H (~~>) (opr A (x.) B)))) stmt (climmul ((F k) (G k) (H k) (M k)) ((e. F (V)) (e. G (V)) (e. H (V)) (e. A (V)) (e. B (V))) (-> (/\ (/\ (br F (~~>) A) (br G (~~>) B)) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\/\ (e. (` F (cv k)) (CC)) (e. (` G (cv k)) (CC)) (= (` H (cv k)) (opr (` F (cv k)) (x.) (` G (cv k)))))))) (br H (~~>) (opr A (x.) B)))) stmt (climmulc2 ((A x) (k x) (C k) (C x) (F k) (F x) (G k) (G x) (M k) (M x)) ((e. F (V)) (e. G (V)) (e. A (V))) (-> (/\ (/\ (e. C (CC)) (br F (~~>) A)) (/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr C (x.) (` F (cv k)))))))) (br G (~~>) (opr C (x.) A)))) stmt (climaddc ((B k) (A k) (F k) (G k)) ((e. A (CC)) (e. B (V)) (br F (~~>) B) (Fn G (NN)) (-> (e. (cv k) (NN)) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr A (+) (` F (cv k))))))) (br G (~~>) (opr A (+) B))) stmt (climmulc ((A k) (F k) (G k)) ((e. A (CC)) (e. B (V)) (br F (~~>) B) (Fn G (NN)) (-> (e. (cv k) (NN)) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (opr A (x.) (` F (cv k))))))) (br G (~~>) (opr A (x.) B))) stmt (clim2serz ((A k) (j k) (F j) (F k) (M j) (M k) (N j) (N k)) ((e. F (V)) (e. A (V)) (br (opr (<,> M (+)) (seq) F) (~~>) A) (:--> F (` (ZZ>) M) (CC))) (-> (e. N (` (ZZ>) M)) (br (opr (<,> (opr N (+) (1)) (+)) (seq) F) (~~>) (opr A (-) (` (opr (<,> M (+)) (seq) F) N))))) stmt (climcj ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F k) (F x) (G j) (G k) (G x) (M j) (M k) (M x)) ((e. A (V)) (e. G (V)) (e. M (ZZ)) (br F (~~>) A) (-> (e. (cv k) (` (ZZ>) M)) (/\ (e. (` F (cv k)) (CC)) (= (` G (cv k)) (` (*) (` F (cv k))))))) (br G (~~>) (` (*) A))) stmt (climubi ((j m) (j w) (A j) (m w) (A m) (A w) (j k) (F j) (k m) (k w) (F k) (F m) (F w) (N j) (N m) (N w)) ((e. A (V)) (:--> F (NN) (RR)) (-> (e. (cv k) (NN)) (br (` F (cv k)) (<_) (` F (opr (cv k) (+) (1))))) (br F (~~>) A) (e. N (NN))) (br (` F N) (<_) A)) stmt (climub ((F k)) ((e. A (V)) (:--> F (NN) (RR)) (-> (e. (cv k) (NN)) (br (` F (cv k)) (<_) (` F (opr (cv k) (+) (1))))) (br F (~~>) A)) (-> (e. N (NN)) (br (` F N) (<_) A))) stmt (climsup ((j k) (j x) (j y) (j z) (F j) (k x) (k y) (k z) (F k) (x y) (x z) (F x) (y z) (F y) (F z)) ((:--> F (NN) (RR)) (-> (e. (cv k) (NN)) (br (` F (cv k)) (<_) (` F (opr (cv k) (+) (1))))) (E.e. x (RR) (A.e. k (NN) (br (` F (cv k)) (<_) (cv x))))) (br F (~~>) (sup (ran F) (RR) (<)))) stmt (climcau ((x y) (x z) (w x) (v x) (F x) (y z) (w y) (v y) (F y) (w z) (v z) (F z) (v w) (F w) (F v) (A y) (A z) (A w) (A v)) ((e. A (V)) (br F (~~>) A) (:--> F (NN) (CC))) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. y (NN) (A.e. z (NN) (-> (br (cv y) (<) (cv z)) (br (` (abs) (opr (` F (cv z)) (-) (` F (cv y)))) (<) (cv x)))))))) stmt (caucvglem1 ((v y) (u y) (A y) (u v) (A v) (A u) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v) (S z) (S w)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (<-> (e. A S) (/\ (e. A (RR)) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br A (<) (` F (cv y))))))))) stmt (caucvglem2 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (/\/\ (C_ S (RR)) (-. (= S ({/}))) (E.e. x (RR) (A.e. f S (br (cv f) (<_) (cv x)))))) stmt (caucvglem3 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (e. (sup S (RR) (<)) (RR))) stmt (caucvglem4 ((v y) (u y) (t y) (A y) (u v) (t v) (A v) (t u) (A u) (A t) (x y) (x z) (w x) (u x) (v x) (t x) (f x) (F x) (y z) (w y) (u y) (v y) (t y) (f y) (F y) (w z) (u z) (v z) (t z) (f z) (F z) (u w) (v w) (t w) (f w) (F w) (u v) (t u) (f u) (F u) (t v) (f v) (F v) (f t) (F t) (F f) (S x) (S z) (S w) (S t) (S f)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (-> (e. A (RR)) (-> (br A (<) (sup S (RR) (<))) (e. A S)))) stmt (caucvglem5 ((x y) (x z) (w x) (u x) (v x) (f x) (F x) (y z) (w y) (u y) (v y) (f y) (F y) (w z) (u z) (v z) (f z) (F z) (u w) (v w) (f w) (F w) (u v) (f u) (F u) (f v) (F v) (F f) (S x) (S z) (S w) (S f)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (-> (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) (-> (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2))))) (br (-u (opr (cv x) (/) (2))) (<_) (opr (sup S (RR) (<)) (-) (` F (cv w))))))) stmt (caucvglem6 ((x y) (x z) (w x) (u x) (v x) (F x) (y z) (w y) (u y) (v y) (F y) (w z) (u z) (v z) (F z) (u w) (v w) (F w) (u v) (F u) (F v) (S x) (S z) (S w)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (-> (/\ (e. (cv x) (RR)) (e. (cv w) (NN))) (-> (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (opr (cv x) (/) (2))))) (br (opr (sup S (RR) (<)) (-) (` F (cv w))) (<_) (opr (cv x) (/) (2)))))) stmt (caucvg ((x y) (x z) (w x) (u x) (v x) (t x) (F x) (y z) (w y) (u y) (v y) (t y) (F y) (w z) (u z) (v z) (t z) (F z) (u w) (v w) (t w) (F w) (u v) (t u) (F u) (t v) (F v) (F t) (S x) (S z) (S w) (S t)) ((:--> F (NN) (RR)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` F (cv y))))))))) (br F (~~>) (sup S (RR) (<)))) stmt (caucvg3a ((v x) (u x) (F x) (u v) (F v) (F u) (x y) (x z) (w x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (G v) (G u) (H x) (H y) (H z) (H w) (H v) (H u) (D x) (D u) (D v) (R x) (R z) (R w) (S z) (S w)) ((:--> F (NN) (CC)) (A.e. z (RR) (-> (br (0) (<) (cv z)) (E.e. w (NN) (A.e. y (NN) (-> (br (cv w) (<) (cv y)) (br (` (abs) (opr (` F (cv y)) (-) (` F (cv w)))) (<) (cv z))))))) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` F (cv x))))) (= R ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` G (cv y)))))))) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (Im) (` F (cv x))))) (= S ({e.|} u (RR) (E.e. v (NN) (A.e. y (NN) (-> (br (cv v) (<_) (cv y)) (br (cv u) (<) (` H (cv y)))))))) (Fn D (NN)) (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` H (cv x)))))) (br F (~~>) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<)))))) stmt (ser1f0 ((j k) (j m) (j n) (j x) (F j) (k m) (k n) (k x) (F k) (m n) (m x) (F m) (n x) (F n) (F x) (A k) (A m) (A n)) ((:--> F (NN) (CC)) (e. A (CC)) (br (opr (+) (seq1) F) (~~>) A)) (br F (~~>) (0))) stmt (ser1const ((j k) (A j) (A k) (N j)) ((e. A (CC))) (-> (e. N (NN)) (= (` (opr (+) (seq1) (X. (NN) ({} A))) N) (opr N (x.) A)))) stmt (ser10 () () (-> (e. N (NN)) (= (` (opr (+) (seq1) (X. (NN) ({} (0)))) N) (0)))) stmt (ser1clim0 ((j k) (j x) (k x)) () (br (opr (+) (seq1) (X. (NN) ({} (0)))) (~~>) (0))) stmt (ser1cmp ((A y) (x y) (x z) (F x) (y z) (F y) (F z) (G x) (G y) (G z)) ((:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (` G (cv x)) (<_) (` F (cv x))))) (-> (e. A (NN)) (br (` (opr (+) (seq1) G) A) (<_) (` (opr (+) (seq1) F) A)))) stmt (ser1cmp0 ((F x)) ((:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x))))) (-> (e. A (NN)) (br (0) (<_) (` (opr (+) (seq1) F) A)))) stmt (ser1cmp2lem ((A z) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F z) (G x) (G z) (S z)) ((:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))) (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)))) (-> (/\ (e. A (NN)) (e. B (NN))) (-> (br A (<_) B) (br (` (opr (+) (seq1) G) A) (<_) (opr S (+) (` (opr (+) (seq1) F) A)))))) stmt (ser1cmp2 ((A z) (x y) (w x) (x z) (B x) (w y) (y z) (B y) (w z) (B w) (B z) (F x) (F w) (F z) (G x) (G w) (G z) (S w) (S z)) ((:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))) (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<)))) (-> (/\ (e. A (NN)) (e. B (NN))) (br (` (opr (+) (seq1) G) A) (<_) (opr S (+) (` (opr (+) (seq1) F) A))))) stmt (cvgcmp2lem ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w) (S x) (S z) (S w) (H x) (H z)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x)))) (br (opr (+) (seq1) F) (~~>) A) (e. B (NN)) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x)))) (= S (sup (ran (|` (opr (+) (seq1) G) ({e.|} y (NN) (br (cv y) (<_) B)))) (RR) (<))) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (opr S (+) (` (opr (+) (seq1) F) (cv x)))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) stmt (cvgcmp2 ((x z) (A x) (A z) (x y) (w x) (B x) (y z) (w y) (B y) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x)))) (br (opr (+) (seq1) F) (~~>) A) (e. B (NN)) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) stmt (cvgcmp2clem ((A x) (B x) (F x) (G x) (S x) (C x)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x)))) (br (opr (+) (seq1) F) (~~>) A) (e. B (NN)) (e. C (RR)) (br (0) (<_) C) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (opr C (x.) (` F (cv x))))) (Fn S (NN)) (-> (e. (cv x) (NN)) (= (` S (cv x)) (opr C (x.) (` F (cv x)))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) stmt (cvgcmp2c ((x z) (A x) (A z) (w x) (B x) (w z) (B z) (B w) (F x) (F z) (F w) (G x) (G z) (G w) (C x) (C z) (C w)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (-> (e. (cv x) (NN)) (br (0) (<_) (` G (cv x)))) (br (opr (+) (seq1) F) (~~>) A) (e. B (NN)) (e. C (RR)) (br (0) (<_) C) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` G (cv x)) (<_) (opr C (x.) (` F (cv x)))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) stmt (cvgcmp ((A x) (F x) (G x)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>) A)) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) stmt (cvgcmpub ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (G x) (G y) (G z) (G w)) ((e. A (V)) (:--> F (NN) (RR)) (:--> G (NN) (RR)) (-> (e. (cv x) (NN)) (/\ (br (0) (<_) (` G (cv x))) (br (` G (cv x)) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>) A) (e. B (V)) (br (opr (+) (seq1) G) (~~>) B)) (br B (<_) A)) stmt (cvgcmp3c ((Q x) (A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (G x) (y z) (w y) (v y) (u y) (G y) (w z) (v z) (u z) (G z) (v w) (u w) (G w) (u v) (G v) (G u) (H x) (H z) (H w) (H v) (H u) (T x) (T y) (T u) (R x) (R z) (R w) (S z) (S w) (D x) (D y) (D u) (C x) (C y) (C z) (C w) (C v) (C u)) ((:--> F (NN) (RR)) (:--> T (NN) (CC)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (e. A (V)) (br (opr (+) (seq1) F) (~~>) A) (Fn H (NN)) (-> (e. (cv x) (NN)) (= (` H (cv x)) (` (abs) (` T (cv x))))) (Fn G (NN)) (-> (e. (cv x) (NN)) (= (` G (cv x)) (` (Re) (` (opr (+) (seq1) T) (cv x))))) (= R ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` G (cv v)))))))) (Fn C (NN)) (-> (e. (cv x) (NN)) (= (` C (cv x)) (` (Im) (` (opr (+) (seq1) T) (cv x))))) (= S ({e.|} u (RR) (E.e. y (NN) (A.e. v (NN) (-> (br (cv y) (<_) (cv v)) (br (cv u) (<) (` C (cv v)))))))) (Fn D (NN)) (-> (e. (cv x) (NN)) (= (` D (cv x)) (opr (i) (x.) (` C (cv x))))) (e. B (NN)) (e. Q (RR)) (br (0) (<_) Q) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` T (cv x))) (<_) (opr Q (x.) (` F (cv x)))))) (br (opr (+) (seq1) T) (~~>) (opr (sup R (RR) (<)) (+) (opr (i) (x.) (sup S (RR) (<)))))) stmt (cvgcmp3ce ((A x) (B x) (F x) (x y) (x z) (w x) (v x) (u x) (t x) (h x) (g x) (G x) (y z) (w y) (v y) (u y) (t y) (h y) (g y) (G y) (w z) (v z) (u z) (t z) (h z) (g z) (G z) (v w) (u w) (t w) (h w) (g w) (G w) (u v) (t v) (h v) (g v) (G v) (t u) (h u) (g u) (G u) (h t) (g t) (G t) (g h) (G h) (G g) (C x)) ((e. A (V)) (e. B (NN)) (:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (br (opr (+) (seq1) F) (~~>) A) (e. C (RR)) (br (0) (<_) C) (:--> G (NN) (CC)) (-> (/\ (e. (cv x) (NN)) (br B (<) (cv x))) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x)))))) (E. y (br (opr (+) (seq1) G) (~~>) (cv y)))) stmt (cvgcmp3cetlem1 ((A z) (x z) (B x) (B z) (F x) (F z) (x y) (G x) (y z) (G y) (G z) (C x) (C z) (ph y) (ph z)) ((e. A (V)) (e. B (NN)) (:--> F (NN) (RR)) (-> (e. (cv x) (NN)) (br (0) (<_) (` F (cv x)))) (br (opr (+) (seq1) F) (~~>) A) (<-> ph (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x)))))))))) (-> ph (E. y (br (opr (+) (seq1) G) (~~>) (cv y))))) stmt (cvgcmp3cetlem2 ((x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (G x) (G y) (G z) (C x) (C y) (C z) (ph y) (ph z)) ((e. A (V)) (e. B (NN)) (<-> ph (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>) A)))) (-> (/\ ph (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) G) (~~>) (cv y))))) stmt (cvgcmp3cet ((A y) (x y) (B x) (B y) (F x) (F y) (G x) (G y) (C x) (C y)) ((e. A (V))) (-> (/\ (e. B (NN)) (/\ (/\ (/\ (:--> F (NN) (RR)) (A.e. x (NN) (br (0) (<_) (` F (cv x))))) (br (opr (+) (seq1) F) (~~>) A)) (/\ (/\ (e. C (RR)) (br (0) (<_) C)) (/\ (:--> G (NN) (CC)) (A.e. x (NN) (-> (br B (<) (cv x)) (br (` (abs) (` G (cv x))) (<_) (opr C (x.) (` F (cv x)))))))))) (E. y (br (opr (+) (seq1) G) (~~>) (cv y))))) stmt (dfisum ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (M m) (M n) (M x) (M y) (k m) (k n) (k x) (k y)) () (-> (e. M (ZZ)) (= (sum_ k (` (ZZ>) M) A) (U. ({|} x (br (opr (<,> M (+)) (seq) (|` ({<,>|} k y (= (cv y) A)) (ZZ))) (~~>) (cv x))))))) stmt (isumvalt ((k x) (k y) (F k) (x y) (F x) (F y) (M x) (M y)) ((e. F (V))) (-> (e. M (ZZ)) (= (sum_ k (` (ZZ>) M) (` F (cv k))) (U. ({|} x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x))))))) stmt (isumvaltf ((j x) (j y) (j z) (F j) (x y) (x z) (F x) (y z) (F y) (F z) (M x) (M z) (j k) (k y) (k z)) ((e. F (V)) (-> (e. (cv y) F) (A. k (e. (cv y) F)))) (-> (e. M (ZZ)) (= (sum_ k (` (ZZ>) M) (` F (cv k))) (U. ({|} x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x))))))) stmt (isumclimtf ((A x) (x y) (F x) (F y) (M x) (k y)) ((-> (e. (cv y) F) (A. k (e. (cv y) F))) (e. F (V)) (e. A (V))) (-> (/\ (e. M (ZZ)) (br (opr (<,> M (+)) (seq) F) (~~>) A)) (= (sum_ k (` (ZZ>) M) (` F (cv k))) A))) stmt (isumclimt ((k x) (F x) (F k)) ((e. F (V)) (e. A (V))) (-> (/\ (e. M (ZZ)) (br (opr (<,> M (+)) (seq) F) (~~>) A)) (= (sum_ k (` (ZZ>) M) (` F (cv k))) A))) stmt (isumclim2tf ((x y) (F x) (F y) (M x) (k x) (k y)) ((-> (e. (cv y) F) (A. k (e. (cv y) F))) (e. F (V))) (-> (/\ (e. M (ZZ)) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (br (opr (<,> M (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) M) (` F (cv k)))))) stmt (isumclim2t ((k x) (k y) (F k) (x y) (F x) (F y) (M x)) ((e. F (V))) (-> (/\ (e. M (ZZ)) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (br (opr (<,> M (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) M) (` F (cv k)))))) stmt (isumclim3t ((x y) (x z) (A x) (y z) (A y) (A z) (k x) (k y) (k z) (M k) (M x) (M y) (M z) (F z)) ((e. A (V)) (= F ({<,>|} k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A))))) (-> (/\ (e. M (ZZ)) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (br (opr (<,> M (+)) (seq) F) (~~>) (sum_ k (` (ZZ>) M) A)))) stmt (isumclim4t ((y z) (A y) (A z) (k y) (k z) (M k) (M y) (M z) (F z)) ((e. A (V)) (= F ({<,>|} k y (/\ (e. (cv k) (` (ZZ>) M)) (= (cv y) A)))) (e. B (V))) (-> (/\ (e. M (ZZ)) (br (opr (<,> M (+)) (seq) F) (~~>) B)) (= (sum_ k (` (ZZ>) M) A) B))) stmt (isumclim5t ((j x) (j y) (j z) (A j) (x y) (x z) (A x) (y z) (A y) (A z) (j k) (M j) (k x) (k y) (k z) (M k) (M x) (M y) (M z)) ((e. A (V)) (= F ({<,>|} j y (/\ (e. (cv j) (` (ZZ>) M)) (= (cv y) (sum_ k (opr M (...) (cv j)) A)))))) (-> (/\ (e. M (ZZ)) (E. x (br F (~~>) (cv x)))) (br F (~~>) (sum_ k (` (ZZ>) M) A)))) stmt (isum1p ((j k) (j n) (j x) (F j) (k n) (k x) (F k) (n x) (F n) (F x) (M j) (M k) (M n) (M x)) ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (CC))) (E. x (br (opr (<,> (opr M (+) (1)) (+)) (seq) F) (~~>) (cv x)))) (= (sum_ k (` (ZZ>) M) (` F (cv k))) (opr (` F M) (+) (sum_ k (` (ZZ>) (opr M (+) (1))) (` F (cv k))))))) stmt (isumnn0nn ((k x) (F x) (F k)) ((e. F (V))) (-> (/\ (A.e. k (NN0) (e. (` F (cv k)) (CC))) (E. x (br (opr (+) (seq1) F) (~~>) (cv x)))) (= (sum_ k (NN0) (` F (cv k))) (opr (` F (0)) (+) (sum_ k (NN) (` F (cv k))))))) stmt (isumclt ((k x) (F k) (F x) (M x)) ((e. F (V))) (-> (/\ (e. M (ZZ)) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (e. (sum_ k (` (ZZ>) M) (` F (cv k))) (CC)))) stmt (isumreclt ((j k) (j x) (F j) (k x) (F k) (F x) (M j) (M k) (M x)) ((e. F (V))) (-> (/\/\ (e. M (ZZ)) (A.e. k (` (ZZ>) M) (e. (` F (cv k)) (RR))) (E. x (br (opr (<,> M (+)) (seq) F) (~~>) (cv x)))) (e. (sum_ k (` (ZZ>) M) (` F (cv k))) (RR)))) stmt (expcnvlem1 ((ph x) (x y) (A x) (A y)) ((e. A (RR)) (-> (e. (cv y) (NN)) (-> (br A (<) (cv y)) ph))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) ph)))) stmt (expcnvlem2 () ((/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B)) (e. C (NN))) (-> (br (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) C) (br (opr A (^) C) (<) B))) stmt (expcnvlem3 () ((/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (-> (e. C (NN)) (-> (br (opr (opr (opr (1) (/) B) (-) (1)) (/) (opr (opr (1) (/) A) (-) (1))) (<) C) (br (opr A (^) C) (<) B)))) stmt (expcnvlem4 ((x y) (A x) (A y) (B x) (B y)) ((/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B))))) stmt (expcnvlem5 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\/\ (e. A (RR)) (br (0) (<) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B)))))) stmt (expcnvlem6 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (/\/\ (e. A (RR)) (br (0) (<_) A) (br A (<) (1))) (/\ (e. B (RR)) (br (0) (<) B))) (E.e. x (NN) (A.e. y (NN) (-> (br (cv x) (<_) (cv y)) (br (opr A (^) (cv y)) (<) B)))))) stmt (expcnv ((j k) (j x) (A j) (k x) (A k) (A x) (F j) (F x) (F k)) ((e. F (V))) (-> (/\/\ (e. A (CC)) (A.e. k (NN) (= (` F (cv k)) (opr A (^) (cv k)))) (br (` (abs) A) (<) (1))) (br F (~~>) (0)))) stmt (geoser ((j k) (j m) (j y) (A j) (k m) (k y) (A k) (m y) (A m) (A y) (F j) (F m) (N j)) ((= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k)))))) (e. A (CC))) (-> (/\ (e. N (NN0)) (=/= A (1))) (= (` (opr (+) (seq0) F) N) (opr (opr (1) (-) (opr A (^) (opr N (+) (1)))) (/) (opr (1) (-) A))))) stmt (geolimilem ((k n) (k y) (A k) (n y) (A n) (A y) (F n) (G n)) ((= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k)))))) (e. A (CC)) (br (` (abs) A) (<) (1)) (Fn G (NN0)) (-> (e. (cv n) (NN0)) (= (` G (cv n)) (opr (-u (opr A (x.) (` F (cv n)))) (/) (opr (1) (-) A))))) (br (opr (+) (seq0) F) (~~>) (opr (1) (/) (opr (1) (-) A)))) stmt (geolimi ((j k) (j n) (j y) (j z) (A j) (k n) (k y) (k z) (A k) (n y) (n z) (A n) (y z) (A y) (A z) (F j) (F n) (F z)) ((= F ({<,>|} k y (/\ (e. (cv k) (NN0)) (= (cv y) (opr A (^) (cv k)))))) (e. A (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq0) F) (~~>) (opr (1) (/) (opr (1) (-) A)))) stmt (geolim1i ((k y) (A k) (A y)) ((= F ({<,>|} k y (/\ (e. (cv k) (NN)) (= (cv y) (opr A (^) (cv k)))))) (e. A (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq1) F) (~~>) (opr A (/) (opr (1) (-) A)))) stmt (geolim1 ((k y) (A k) (A y)) ((= F ({<,>|} k y (/\ (e. (cv k) (NN)) (= (cv y) (opr A (^) (cv k))))))) (-> (/\ (e. A (CC)) (br (` (abs) A) (<) (1))) (br (opr (+) (seq1) F) (~~>) (opr A (/) (opr (1) (-) A))))) stmt (geosum ((k y) (A k) (A y) (N y) (N k)) ((e. A (CC))) (-> (/\ (e. N (NN0)) (=/= A (1))) (= (sum_ k (opr (0) (...) N) (opr A (^) (cv k))) (opr (opr (1) (-) (opr A (^) (opr N (+) (1)))) (/) (opr (1) (-) A))))) stmt (cvgratlem1ALT ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z) (C y) (F x) (F y) (F z)) ((:--> F (NN) (RR)) (e. A (RR)) (e. B (NN)) (e. C (NN))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) (br (` F (opr B (+) C)) (<_) (opr (opr A (^) C) (x.) (` F B))))) stmt (cvgratlem2ALT ((A x) (B x) (F x)) ((:--> F (NN) (RR)) (e. A (RR)) (e. B (NN)) (e. C (NN))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))) (-> (br B (<) C) (br (` F C) (<_) (opr (opr (` F B) (/) (opr A (^) B)) (x.) (opr A (^) C)))))) stmt (cvgratlem3ALT ((A x) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (C y) (C z) (G x)) ((:--> F (NN) (CC)) (= G ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (` (abs) (` F (cv y))))))) (e. A (RR)) (e. B (NN)) (e. C (NN))) (-> (/\ (br (0) (<) A) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x)))))))) (-> (br B (<) C) (br (` (abs) (` F C)) (<_) (opr (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) (opr A (^) C)))))) stmt (cvgratlem1 ((x y) (x z) (y z) (A x) (A y) (A z) (B x) (B y) (B z) (C y) (F x) (F y) (F z)) ((:--> F (NN) (RR))) (-> (/\ (e. C (NN)) (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x))))))))) (br (` F (opr B (+) C)) (<_) (opr (opr A (^) C) (x.) (` F B))))) stmt (cvgratlem2 ((A x) (B x) (F x)) ((:--> F (NN) (RR))) (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` F (opr (cv x) (+) (1))) (<) (opr A (x.) (` F (cv x)))))))) (-> (/\ (e. C (NN)) (br B (<) C)) (br (` F C) (<_) (opr (opr (` F B) (/) (opr A (^) B)) (x.) (opr A (^) C)))))) stmt (cvgratlem3 ((A x) (x y) (x z) (B x) (y z) (B y) (B z) (F x) (F y) (F z) (C y) (C z) (G x)) ((:--> F (NN) (CC)) (= G ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (` (abs) (` F (cv y)))))))) (-> (/\ (/\ (e. A (RR)) (br (0) (<) A)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (-> (/\ (e. C (NN)) (br B (<) C)) (br (` (abs) (` F C)) (<_) (opr (opr (` (abs) (` F B)) (/) (opr A (^) B)) (x.) (opr A (^) C)))))) stmt (cvgratlem4 ((A x) (B x) (F x)) ((:--> F (NN) (CC))) (-> (/\ (e. A (RR)) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (br (0) (<) A))) stmt (cvgratlem5 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (u x) (B x) (u y) (B y) (u v) (B v) (B u) (F x) (F y) (F v) (F u) (G y) (G v)) ((:--> F (NN) (CC)) (= G ({<,>|} z w (/\ (e. (cv z) (NN)) (= (cv w) (opr A (^) (cv z))))))) (-> (/\ (/\ (e. A (RR)) (br A (<) (1))) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) F) (~~>) (cv y))))) stmt (cvgrat ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (F x) (F y)) ((:--> F (NN) (CC))) (-> (/\ (/\ (e. A (RR)) (br A (<) (1))) (/\ (e. B (NN)) (A.e. x (NN) (-> (br B (<_) (cv x)) (br (` (abs) (` F (opr (cv x) (+) (1)))) (<) (opr A (x.) (` (abs) (` F (cv x))))))))) (E. y (br (opr (+) (seq1) F) (~~>) (cv y))))) stmt (fsum0diaglem1 () () (-> (/\ (e. N A) (e. K (opr (0) (...) N))) (C_ (opr (0) (...) (opr N (-) K)) (opr (0) (...) N)))) stmt (fsum0diaglem2 ((k m) (k n) (A k) (m n) (A m) (A n) (j m) (j n) (B j) (B m) (B n) (j k)) () (-> (e. (cv n) (NN0)) (-> (-> (/\ (A.e. j (opr (0) (...) (cv n)) (e. A (CC))) (A.e. k (opr (0) (...) (cv n)) (e. B (CC)))) (= (sum_ j (opr (0) (...) (cv n)) (sum_ k (opr (0) (...) (opr (cv n) (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) (cv n)) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B)))))) (-> (/\ (A.e. j (opr (0) (...) (opr (cv n) (+) (1))) (e. A (CC))) (A.e. k (opr (0) (...) (opr (cv n) (+) (1))) (e. B (CC)))) (= (sum_ j (opr (0) (...) (opr (cv n) (+) (1))) (sum_ k (opr (0) (...) (opr (opr (cv n) (+) (1)) (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) (opr (cv n) (+) (1))) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))))))))) stmt (fsum0diag ((k m) (k n) (A k) (m n) (A m) (A n) (j m) (j n) (B j) (B m) (B n) (j k) (N j) (N k) (N m) (N n)) () (-> (/\/\ (e. N (NN0)) (A.e. j (opr (0) (...) N) (e. A (CC))) (A.e. k (opr (0) (...) N) (e. B (CC)))) (= (sum_ j (opr (0) (...) N) (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr A (x.) B))) (sum_ k (opr (0) (...) N) (sum_ j (opr (0) (...) (cv k)) (opr A (x.) ([_/]_ (opr (cv k) (-) (cv j)) k B))))))) stmt (fsum0diag2 ((k m) (k n) (k x) (A k) (m n) (m x) (A m) (n x) (A n) (A x) (j m) (j n) (j x) (B j) (B m) (B n) (B x) (j k) (N j) (N k) (N m) (N n) (N x)) () (-> (/\/\ (e. N (NN0)) (A.e. j (opr (0) (...) N) (e. A (CC))) (A.e. k (opr (0) (...) N) (e. B (CC)))) (= (sum_ j (opr (0) (...) N) (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr A (x.) B))) (sum_ j (opr (0) (...) N) (sum_ k (opr (0) (...) (cv j)) (opr ([_/]_ (opr (cv j) (-) (cv k)) j A) (x.) B)))))) stmt (efcltlem1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (F x) (F w)) ((= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y))))))) (e. A (CC)) (=/= A (0))) (E. x (br (opr (+) (seq1) F) (~~>) (cv x)))) stmt (efcltlem2 ((y z) (A y) (A z)) ((= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y)))))))) (-> (= A (0)) (br (opr (+) (seq1) F) (~~>) (0)))) stmt (efcltlem3 ((x y) (x z) (A x) (y z) (A y) (A z) (F x)) ((= F ({<,>|} y z (/\ (e. (cv y) (NN)) (= (cv z) (opr (opr A (^) (cv y)) (/) (` (!) (cv y)))))))) (-> (e. A (CC)) (E. x (br (opr (+) (seq1) F) (~~>) (cv x))))) stmt (efcltlem2a ((j k) (j m) (j y) (A j) (k m) (k y) (A k) (m y) (A m) (A y) (F k) (F m)) ((= F ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))))) (-> (= A (0)) (br (opr (+) (seq0) F) (~~>) (1)))) stmt (efcltlem4 ((j k) (j m) (j w) (j x) (j y) (A j) (k m) (k w) (k x) (k y) (A k) (m w) (m x) (m y) (A m) (w x) (w y) (A w) (x y) (A x) (A y) (F k) (F m) (F w) (F x)) ((= F ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))))) (-> (e. A (CC)) (E. x (br (opr (+) (seq0) F) (~~>) (cv x))))) stmt (dfef2lem ((j k) (j x) (j y) (A j) (k x) (k y) (A k) (x y) (A x) (A y)) ((e. A (CC))) (= (sum_ k (NN0) (opr (opr A (^) (cv k)) (/) (` (!) (cv k)))) (opr (1) (+) (sum_ k (NN) (` ({<,>|} j y (/\ (e. (cv j) (NN)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))) (cv k)))))) stmt (efvalt ((x y) (k x) (A x) (k y) (A y) (A k)) () (-> (e. A (CC)) (= (` (exp) A) (sum_ k (NN0) (opr (opr A (^) (cv k)) (/) (` (!) (cv k))))))) stmt (eval () () (= (e) (sum_ k (NN0) (opr (opr (1) (^) (cv k)) (/) (` (!) (cv k)))))) stmt (efclt ((j k) (j x) (j y) (A j) (k x) (k y) (A k) (x y) (A x) (A y)) () (-> (e. A (CC)) (e. (` (exp) A) (CC)))) stmt (efcvg ((A j) (A k) (A x) (A y) (j k) (j x) (j y) (k x) (k y) (x y) (F k) (F x)) ((= F ({<,>|} j y (/\ (e. (cv j) (NN0)) (= (cv y) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))))) (-> (e. A (CC)) (br (opr (+) (seq0) F) (~~>) (` (exp) A)))) stmt (efcvgfsum ((k n) (k x) (k y) (k z) (A k) (n x) (n y) (n z) (A n) (x y) (x z) (A x) (y z) (A y) (A z)) ((= F ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ k (opr (0) (...) (cv n)) (opr (opr A (^) (cv k)) (/) (` (!) (cv k))))))))) (-> (e. A (CC)) (br F (~~>) (` (exp) A)))) stmt (reefcl ((j k) (j x) (j y) (A j) (k x) (k y) (A k) (x y) (A x) (A y)) ((e. A (RR))) (e. (` (exp) A) (RR))) stmt (erelem1 ((x y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (/\ (:--> F (NN) (RR)) (:--> G (NN) (RR)))) stmt (erelem2 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (br (opr (+) (seq1) F) (~~>) (2))) stmt (erelem3 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (-> (e. (cv z) (NN)) (/\ (br (0) (<_) (` G (cv z))) (br (` G (cv z)) (<_) (` F (cv z)))))) stmt (erelem4 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (br (opr (+) (seq1) G) (~~>) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) stmt (erelem5 ((x y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (e. (sup (ran (opr (+) (seq1) G)) (RR) (<)) (RR))) stmt (erelem6 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (= (e) (opr (1) (+) (sup (ran (opr (+) (seq1) G)) (RR) (<))))) stmt (erelem7 ((x y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (e. (e) (RR))) stmt (ele3lem ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (br (e) (<_) (3))) stmt (ege2le3lem1 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (br (` (opr (+) (seq1) G) (1)) (<_) (sup (ran (opr (+) (seq1) G)) (RR) (<)))) stmt (ege2lem2 ((x y)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (br (2) (<_) (e))) stmt (ege2le3lem2 ((x y) (x z) (y z) (F z) (G z)) ((= F ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (2) (x.) (opr (opr (1) (/) (2)) (^) (cv x))))))) (= G ({<,>|} x y (/\ (e. (cv x) (NN)) (= (cv y) (opr (1) (/) (` (!) (cv x)))))))) (/\ (br (2) (<_) (e)) (br (e) (<_) (3)))) stmt (ere () () (e. (e) (RR))) stmt (ereALT ((x y)) () (e. (e) (RR))) stmt (ege2 ((x y)) () (br (2) (<_) (e))) stmt (ele3 ((x y)) () (br (e) (<_) (3))) stmt (ege2le3 ((x y)) () (/\ (br (2) (<_) (e)) (br (e) (<_) (3)))) stmt (ef0 ((j y)) () (= (` (exp) (0)) (1))) stmt (efcj ((j k) (j m) (j y) (A j) (k m) (k y) (A k) (m y) (A m) (A y)) ((e. A (CC))) (= (` (exp) (` (*) A)) (` (*) (` (exp) A)))) stmt (efcjt () () (-> (e. A (CC)) (= (` (exp) (` (*) A)) (` (*) (` (exp) A))))) stmt (efaddlem1 () ((e. N (NN))) (-> (e. (cv j) (opr (1) (...) N)) (e. N (` (ZZ>) (opr (opr N (-) (cv j)) (+) (1)))))) stmt (efaddlem2 () ((e. N (NN))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (e. (cv k) (NN)))) stmt (efaddlem3 ((j k) (N j) (N k)) ((e. N (NN)) (e. A (CC)) (e. B (CC))) (e. (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) (CC))) stmt (efaddlem4 ((j k) (N j) (N k)) ((e. N (NN)) (e. A (CC)) (e. B (CC))) (e. (` (abs) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))))) (RR))) stmt (efaddlem5 ((j k) (A j) (A k) (B j) (B k) (N j) (N k)) ((e. N (NN)) (e. A (CC)) (e. B (CC))) (= (sum_ j (opr (0) (...) N) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))) (sum_ j (opr (0) (...) N) (sum_ k (opr (0) (...) (opr N (-) (cv j))) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))))) stmt (efaddlem6 ((j k) (A j) (A k) (B j) (B k) (N j) (N k)) ((e. N (NN)) (e. A (CC)) (e. B (CC))) (= (opr (opr (sum_ j (opr (0) (...) N) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) N) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) N) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j))))) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))))) stmt (efaddlem7 () ((e. A (CC)) (e. B (CC)) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (e. T (NN))) stmt (efaddlem8 () ((e. N (NN)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (e. S (NN))) stmt (efaddlem9 () ((e. N (NN))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (/\ (/\ (e. (cv j) (NN0)) (br (cv j) (<_) N)) (/\ (e. (cv k) (NN0)) (br (cv k) (<_) N))))) stmt (efaddlem10 () ((e. N (NN)) (e. A (CC))) (-> (/\ (e. (cv j) (NN0)) (br (cv j) (<_) N)) (/\ (/\ (e. (` (abs) (opr A (^) (cv j))) (RR)) (e. (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N) (RR))) (/\ (br (0) (<_) (` (abs) (opr A (^) (cv j)))) (br (` (abs) (opr A (^) (cv j))) (<_) (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N)))))) stmt (efaddlem11 () ((e. N (NN)) (e. A (CC)) (e. B (CC))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (` (abs) (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k)))) (<_) (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N) (x.) (opr (` (floor) (opr (` (abs) B) (+) (1))) (^) N))))) stmt (efaddlem12 () ((e. N (NN)) (e. A (CC)) (e. B (CC)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (br (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (^) N) (x.) (opr (` (floor) (opr (` (abs) B) (+) (1))) (^) N)) (<_) (opr T (^) S))) stmt (efaddlem13 () ((e. N (NN)) (e. A (CC)) (e. B (CC)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (` (abs) (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k)))) (<_) (opr T (^) S)))) stmt (efaddlem14 () ((e. N (NN))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (opr N (+) (1)) (<_) (opr (cv j) (+) (cv k))))) stmt (efaddlem15 () ((e. N (NN)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2))))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (` (!) S) (<_) (opr (` (!) (cv j)) (x.) (` (!) (cv k)))))) stmt (efaddlem16 ((j k) (N j) (N k) (C j) (C k)) ((e. N (NN)) (e. C (RR)) (br (0) (<_) C)) (br (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) C)) (<_) (opr (opr N (^) (2)) (x.) C))) stmt (efaddlem17 ((j k) (N j) (N k)) ((e. N (NN)) (e. A (CC)) (e. B (CC)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (-> (/\ (e. (cv j) (opr (1) (...) N)) (e. (cv k) (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N))) (br (` (abs) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))) (<_) (opr (opr T (^) S) (/) (` (!) S))))) stmt (efaddlem18 ((j k) (N j) (N k)) ((e. N (NN)) (e. A (CC)) (e. B (CC)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (e. (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr T (^) S) (/) (` (!) S)))) (RR))) stmt (efaddlem19 ((j k) (N j) (N k)) ((e. N (NN)) (e. A (CC)) (e. B (CC)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (br (` (abs) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))))) (<_) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr T (^) S) (/) (` (!) S)))))) stmt (efaddlem20 () ((e. N (NN)) (e. A (CC)) (e. B (CC)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2)))) (br (opr (opr N (^) (2)) (x.) (opr (opr T (^) S) (/) (` (!) S))) (<_) (opr (opr (opr S (^) (2)) (x.) (opr (opr (4) (x.) T) (^) S)) (/) (` (!) S)))) stmt (efaddlem21 () ((e. A (CC)) (e. B (CC)) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))) (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3))))))) (e. R (NN))) stmt (efaddlem22 ((j k) (N j) (N k) (S j) (S k) (T j) (T k)) ((e. N (NN)) (e. A (CC)) (e. B (CC)) (= S (` (floor) (opr (opr N (+) (1)) (/) (2)))) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))) (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3))))))) (br (` (abs) (sum_ j (opr (1) (...) N) (sum_ k (opr (opr (opr N (-) (cv j)) (+) (1)) (...) N) (opr (opr (opr A (^) (cv j)) (x.) (opr B (^) (cv k))) (/) (opr (` (!) (cv j)) (x.) (` (!) (cv k))))))) (<) (opr (opr (2) (x.) R) (/) N))) stmt (efaddlem23 ((j k) (A j) (A k) (B j) (B k) (N j) (N k) (T j) (T k)) ((e. N (NN)) (e. A (CC)) (e. B (CC)) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))) (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3))))))) (-> (/\ (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N)) (br (` (abs) (opr (opr (sum_ j (opr (0) (...) N) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) N) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) N) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))))) (<) (cv x)))) stmt (efaddlem24 ((j k) (A j) (A k) (B j) (B k) (N j) (N k) (T j) (T k) (j x) (k x)) ((e. A (CC)) (e. B (CC)) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))) (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3))))))) (-> (/\/\ (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (e. N (NN)) (br (opr (` (floor) (opr (opr (2) (x.) R) (/) (cv x))) (+) (1)) (<_) N)) (br (` (abs) (opr (opr (sum_ j (opr (0) (...) N) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) N) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) N) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))))) (<) (cv x)))) stmt (efaddlem25 ((j k) (j m) (A j) (k m) (A k) (A m) (B j) (B k) (B m) (m n) (R m) (R n) (T j) (T k) (m x) (j n) (k n) (n x) (j x) (k x)) ((e. A (CC)) (e. B (CC)) (= T (opr (opr (` (floor) (opr (` (abs) A) (+) (1))) (x.) (` (floor) (opr (` (abs) B) (+) (1)))) (^) (2))) (= R (opr (2) (x.) (opr (opr (2) (^) (opr (3) (^) (2))) (x.) (opr (opr (4) (x.) T) (^) (opr (opr (4) (x.) T) (+) (3))))))) (-> (/\ (e. (cv x) (RR)) (br (0) (<) (cv x))) (E.e. m (NN) (A.e. n (NN) (-> (br (cv m) (<_) (cv n)) (br (` (abs) (opr (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k))))) (-) (sum_ j (opr (0) (...) (cv n)) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))))) (<) (cv x))))))) stmt (efaddlem26 ((j n) (j y) (A j) (n y) (A n) (A y) (k n) (k y) (B k) (B n) (B y) (F m) (G m) (H m) (j k) (j m) (k m) (m n) (m y)) ((e. A (CC)) (e. B (CC)) (= F ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))))))) (= G ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))))))) (= H ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))))))))) (br H (~~>) (opr (` (exp) A) (x.) (` (exp) B)))) stmt (efaddlem27 ((f j) (f k) (f m) (f n) (f x) (f y) (A f) (j k) (j m) (j n) (j x) (j y) (A j) (k m) (k n) (k x) (k y) (A k) (m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y) (B f) (B j) (B k) (B m) (B n) (B x) (B y) (F f) (F m) (F x) (H f) (H m) (H x)) ((e. A (CC)) (e. B (CC)) (= F ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (sum_ j (opr (0) (...) (cv n)) (opr (opr (opr A (+) B) (^) (cv j)) (/) (` (!) (cv j)))))))) (= H ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))))))))) (br H (~~>) (` (exp) (opr A (+) B)))) stmt (efaddlem28 ((j k) (j n) (j y) (A j) (k n) (k y) (A k) (n y) (A n) (A y) (B j) (B k) (B n) (B y)) ((e. A (CC)) (e. B (CC)) (= H ({<,>|} n y (/\ (e. (cv n) (NN0)) (= (cv y) (opr (sum_ j (opr (0) (...) (cv n)) (opr (opr A (^) (cv j)) (/) (` (!) (cv j)))) (x.) (sum_ k (opr (0) (...) (cv n)) (opr (opr B (^) (cv k)) (/) (` (!) (cv k)))))))))) (= (` (exp) (opr A (+) B)) (opr (` (exp) A) (x.) (` (exp) B)))) stmt (efadd ((j k) (j n) (j y) (A j) (k n) (k y) (A k) (n y) (A n) (A y) (B j) (B k) (B n) (B y)) ((e. A (CC)) (e. B (CC))) (= (` (exp) (opr A (+) B)) (opr (` (exp) A) (x.) (` (exp) B)))) stmt (efaddt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (exp) (opr A (+) B)) (opr (` (exp) A) (x.) (` (exp) B))))) stmt (efcant () () (-> (e. A (CC)) (= (opr (` (exp) A) (x.) (` (exp) (-u A))) (1)))) stmt (efne0t () () (-> (e. A (CC)) (=/= (` (exp) A) (0)))) stmt (efnn0valtlem ((x y) (N x) (N y)) () (-> (e. N (NN)) (= (` (exp) N) (` (opr (x.) (seq1) (X. (NN) ({} (e)))) N)))) stmt (efnn0valt () () (-> (e. N (NN0)) (= (` (exp) N) (opr (e) (^) N)))) stmt (sinvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (sin) A) (opr (opr (` (exp) (opr (i) (x.) A)) (-) (` (exp) (opr (-u (i)) (x.) A))) (/) (opr (2) (x.) (i)))))) stmt (cosvalt ((x y) (A x) (A y)) () (-> (e. A (CC)) (= (` (cos) A) (opr (opr (` (exp) (opr (i) (x.) A)) (+) (` (exp) (opr (-u (i)) (x.) A))) (/) (2))))) stmt (sinclt () () (-> (e. A (CC)) (e. (` (sin) A) (CC)))) stmt (cosclt () () (-> (e. A (CC)) (e. (` (cos) A) (CC)))) stmt (resinvalt () () (-> (e. A (RR)) (= (` (sin) A) (` (Im) (` (exp) (opr (i) (x.) A)))))) stmt (recosvalt () () (-> (e. A (RR)) (= (` (cos) A) (` (Re) (` (exp) (opr (i) (x.) A)))))) stmt (resinclt () () (-> (e. A (RR)) (e. (` (sin) A) (RR)))) stmt (recosclt () () (-> (e. A (RR)) (e. (` (cos) A) (RR)))) stmt (sinnegt () () (-> (e. A (CC)) (= (` (sin) (-u A)) (-u (` (sin) A))))) stmt (cosnegt () () (-> (e. A (CC)) (= (` (cos) (-u A)) (` (cos) A)))) stmt (sin0 () () (= (` (sin) (0)) (0))) stmt (cos0 () () (= (` (cos) (0)) (1))) stmt (efivalt () () (-> (e. A (CC)) (= (` (exp) (opr (i) (x.) A)) (opr (` (cos) A) (+) (opr (i) (x.) (` (sin) A)))))) stmt (efmivalt () () (-> (e. A (CC)) (= (` (exp) (opr (-u (i)) (x.) A)) (opr (` (cos) A) (-) (opr (i) (x.) (` (sin) A)))))) stmt (efeult () () (-> (e. A (CC)) (= (` (exp) A) (opr (` (exp) (` (Re) A)) (x.) (opr (` (cos) (` (Im) A)) (+) (opr (i) (x.) (` (sin) (` (Im) A)))))))) stmt (sinadd () ((e. A (CC)) (e. B (CC))) (= (` (sin) (opr A (+) B)) (opr (opr (` (sin) A) (x.) (` (cos) B)) (+) (opr (` (cos) A) (x.) (` (sin) B))))) stmt (cosadd () ((e. A (CC)) (e. B (CC))) (= (` (cos) (opr A (+) B)) (opr (opr (` (cos) A) (x.) (` (cos) B)) (-) (opr (` (sin) A) (x.) (` (sin) B))))) stmt (cosaddt () () (-> (/\ (e. A (CC)) (e. B (CC))) (= (` (cos) (opr A (+) B)) (opr (opr (` (cos) A) (x.) (` (cos) B)) (-) (opr (` (sin) A) (x.) (` (sin) B)))))) stmt (sincossqt () () (-> (e. A (CC)) (= (opr (opr (` (sin) A) (^) (2)) (+) (opr (` (cos) A) (^) (2))) (1)))) stmt (sinbndt () () (-> (e. A (RR)) (/\ (br (-u (1)) (<_) (` (sin) A)) (br (` (sin) A) (<_) (1))))) stmt (nn0ennn ((x y)) () (br (NN0) (~~) (NN))) stmt (nnenom ((x y) (x z) (y z)) () (br (NN) (~~) (om))) stmt (xpnnen ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) () (br (X. (NN) (NN)) (~~) (NN))) stmt (xpomen () () (br (X. (om) (om)) (~~) (om))) stmt (znnenlem () () (-> (/\ (/\ (br (0) (<_) (cv x)) (-. (br (0) (<_) (cv y)))) (/\ (e. (cv x) (ZZ)) (e. (cv y) (ZZ)))) (<-> (= (cv y) (cv x)) (= (opr (2) (x.) (cv x)) (opr (opr (-u (2)) (x.) (cv y)) (+) (1)))))) stmt (znnen ((x y)) () (br (ZZ) (~~) (NN))) stmt (qnnen ((x y) (x z) (w x) (y z) (w y) (w z)) () (br (QQ) (~~) (NN))) stmt (unbenlem ((m n) (m z) (m w) (m v) (m u) (m t) (A m) (n z) (n w) (n v) (n u) (n t) (A n) (w z) (v z) (u z) (t z) (A z) (v w) (u w) (t w) (A w) (u v) (t v) (A v) (t u) (A u) (A t) (G m) (G n) (G v) (G u) (G t)) ((= G (|` (rec ({<,>|} z w (= (cv w) (opr (cv z) (+) (1)))) (1)) (om)))) (-> (/\ (C_ A (NN)) (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n))))) (br A (~~) (om)))) stmt (unben ((m n) (m x) (m y) (A m) (n x) (n y) (A n) (x y) (A x) (A y)) () (-> (/\ (C_ A (NN)) (A.e. m (NN) (E.e. n A (br (cv m) (<) (cv n))))) (br A (~~) (NN)))) stmt (infpnlem1 ((N j) (M j) (K j)) ((= K (opr (` (!) N) (+) (1)))) (-> (/\ (e. N (NN)) (e. M (NN))) (-> (/\ (/\ (br (1) (<) M) (e. (opr K (/) M) (NN))) (A.e. j (NN) (-> (/\ (br (1) (<) (cv j)) (e. (opr K (/) (cv j)) (NN))) (br M (<_) (cv j))))) (/\ (br N (<) M) (A.e. j (NN) (-> (e. (opr M (/) (cv j)) (NN)) (\/ (= (cv j) (1)) (= (cv j) M)))))))) stmt (infpnlem2 ((j k) (N j) (N k) (K j) (K k)) ((= K (opr (` (!) N) (+) (1)))) (-> (e. N (NN)) (E.e. j (NN) (/\ (br N (<) (cv j)) (A.e. k (NN) (-> (e. (opr (cv j) (/) (cv k)) (NN)) (\/ (= (cv k) (1)) (= (cv k) (cv j))))))))) stmt (infpn ((j k) (N j) (N k)) () (-> (e. N (NN)) (E.e. j (NN) (/\ (br N (<) (cv j)) (A.e. k (NN) (-> (e. (opr (cv j) (/) (cv k)) (NN)) (\/ (= (cv k) (1)) (= (cv k) (cv j))))))))) stmt (infpn2 ((j k) (j n) (j m) (k n) (k m) (m n) (S j) (S k)) ((= S ({e.|} n (NN) (/\ (br (1) (<) (cv n)) (A.e. m (NN) (-> (e. (opr (cv n) (/) (cv m)) (NN)) (\/ (= (cv m) (1)) (= (cv m) (cv n))))))))) (br S (~~) (NN))) stmt (ruclem1 () ((e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br A (<) (opr (opr (opr (2) (x.) A) (+) B) (/) (3))))) stmt (ruclem2 () ((e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (opr (opr (opr (2) (x.) A) (+) B) (/) (3)) (<) (opr (opr A (+) (opr (2) (x.) B)) (/) (3))))) stmt (ruclem3 () ((e. A (RR)) (e. B (RR))) (<-> (br A (<) B) (br (opr (opr A (+) (opr (2) (x.) B)) (/) (3)) (<) B))) stmt (ruclem4 () () (-> (/\ (= A C) (= B D)) (= (if (/\ (br (` (1st) A) (<) B) (br B (<) (` (2nd) A))) (<,> (opr (opr (opr (2) (x.) B) (+) (` (2nd) A)) (/) (3)) (opr (opr B (+) (opr (2) (x.) (` (2nd) A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) A)) (+) (` (2nd) A)) (/) (3)) (opr (opr (` (1st) A) (+) (opr (2) (x.) (` (2nd) A))) (/) (3)))) (if (/\ (br (` (1st) C) (<) D) (br D (<) (` (2nd) C))) (<,> (opr (opr (opr (2) (x.) D) (+) (` (2nd) C)) (/) (3)) (opr (opr D (+) (opr (2) (x.) (` (2nd) C))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) C)) (+) (` (2nd) C)) (/) (3)) (opr (opr (` (1st) C) (+) (opr (2) (x.) (` (2nd) C))) (/) (3))))))) stmt (ruclem5 () ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (e. C (V))) stmt (ruclem6 () ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (= (|` C (\ (NN) ({} (1)))) (|` F (\ (NN) ({} (1)))))) stmt (ruclem7 () ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1))))))) (= (` C (1)) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) stmt (ruclem8 () ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (e. A (NN))) (= (` C (opr A (+) (1))) (` F (opr A (+) (1))))) stmt (ruclem9 ((x y) (x z) (y z) (A z)) ((= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) A))))) (e. D (V))) stmt (ruclem10 () ((e. A (NN)) (e. D (V)) (e. C (V)) (= G (o. (1st) (opr D (seq1) C)))) (= (` (1st) (` (opr D (seq1) C) A)) (` G A))) stmt (ruclem11 () ((e. A (NN)) (e. D (V)) (e. C (V)) (= H (o. (2nd) (opr D (seq1) C)))) (= (` (2nd) (` (opr D (seq1) C) A)) (` H A))) stmt (ruclem12 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v)) ((= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (= D ({<<,>,>|} w v u (/\ (/\ (e. (cv w) (X. (RR) (RR))) (e. (cv v) (RR))) (= (cv u) (if (/\ (br (` (1st) (cv w)) (<) (cv v)) (br (cv v) (<) (` (2nd) (cv w)))) (<,> (opr (opr (opr (2) (x.) (cv v)) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (cv v) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv w))) (+) (` (2nd) (cv w))) (/) (3)) (opr (opr (` (1st) (cv w)) (+) (opr (2) (x.) (` (2nd) (cv w)))) (/) (3))))))))) stmt (ruclem13 ((x y) (x z) (y z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))))))))) (:--> (opr D (seq1) C) (NN) (X. (RR) (RR)))) stmt (ruclem14 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C)))) (= (` (opr D (seq1) C) (1)) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) stmt (ruclem15 ((x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A w) (A v) (A u) (C w) (C v) (C u) (D w) (D v) (D u) (G w) (G v) (G u) (H w) (H v) (H u) (F z) (F w) (F v) (F u)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (= (` (opr D (seq1) C) (opr A (+) (1))) (if (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (<,> (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3)) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3)) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3)))))) stmt (ruclem16 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C)))) (= (` G (1)) (opr (` F (1)) (+) (1)))) stmt (ruclem17 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C)))) (:--> G (NN) (RR))) stmt (ruclem18 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (= (` G (opr A (+) (1))) (opr (opr (opr (2) (x.) (` F (opr A (+) (1)))) (+) (` H A)) (/) (3))))) stmt (ruclem19 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A))) (= (` H (opr A (+) (1))) (opr (opr (` F (opr A (+) (1))) (+) (opr (2) (x.) (` H A))) (/) (3))))) stmt (ruclem20 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (-. (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A)))) (= (` G (opr A (+) (1))) (opr (opr (opr (2) (x.) (` G A)) (+) (` H A)) (/) (3))))) stmt (ruclem21 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (-. (/\ (br (` G A) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H A)))) (= (` H (opr A (+) (1))) (opr (opr (` G A) (+) (opr (2) (x.) (` H A))) (/) (3))))) stmt (ruclem22 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (e. (` G A) (RR))) stmt (ruclem23 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (e. (` H A) (RR))) stmt (ruclem24 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-> (br (` G A) (<) (` H A)) (br (` G (opr A (+) (1))) (<) (` H (opr A (+) (1)))))) stmt (ruclem25 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (br (` G A) (<) (` H A))) stmt (ruclem26 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (br (` G A) (<) (` G (opr A (+) (1))))) stmt (ruclem27 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (br (` H (opr A (+) (1))) (<) (` H A))) stmt (ruclem28 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-. (/\ (br (` G (opr A (+) (1))) (<) (` F (opr A (+) (1)))) (br (` F (opr A (+) (1))) (<) (` H (opr A (+) (1))))))) stmt (ruclem29 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN))) (-. (/\ (br (` G A) (<) (` F A)) (br (` F A) (<) (` H A))))) stmt (ruclem30 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN)) (e. B (NN))) (-> (br (` G A) (<) (` G (opr A (+) B))) (br (` G A) (<) (` G (opr A (+) (opr B (+) (1))))))) stmt (ruclem31 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN)) (e. B (NN))) (-> (br (` H (opr A (+) B)) (<) (` H B)) (br (` H (opr (opr A (+) (1)) (+) B)) (<) (` H B)))) stmt (ruclem32 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (A w) (A v) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v) (B w) (B v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (e. A (NN)) (e. B (NN))) (br (` G A) (<) (` H B))) stmt (ruclem33 ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C)))) (/\/\ (C_ (ran G) (RR)) (-. (= (ran G) ({/}))) (E.e. w (RR) (A.e. v (ran G) (br (cv v) (<_) (cv w)))))) stmt (ruclem34 ((S w) (x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w) (C w) (C v) (D w) (D v) (G w) (G v) (H w) (H v) (F z) (F w) (F v)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (= S (sup (ran G) (RR) (<)))) (e. S (RR))) stmt (ruclem35 ((S w) (x y) (x z) (w x) (v x) (u x) (y z) (w y) (v y) (u y) (w z) (v z) (u z) (v w) (u w) (u v) (A w) (A v) (A u) (C w) (C v) (C u) (D w) (D v) (D u) (G w) (G v) (G u) (H w) (H v) (H u) (F z) (F w) (F v) (F u)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (= S (sup (ran G) (RR) (<))) (e. A (NN))) (/\ (br (` G A) (<) S) (br S (<) (` H A)))) stmt (ruclem36 ((x y) (x z) (y z) (F z)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (= S (sup (ran G) (RR) (<))) (e. A (NN))) (-. (= (` F A) S))) stmt (ruclem37 ((S w) (x y) (x z) (w x) (y z) (w y) (w z) (C w) (D w) (G w) (H w) (F z) (F w)) ((:--> F (NN) (RR)) (= C (u. ({} (<,> (1) (<,> (opr (` F (1)) (+) (1)) (opr (` F (1)) (+) (2))))) (|` F (\ (NN) ({} (1)))))) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (X. (RR) (RR))) (e. (cv y) (RR))) (= (cv z) (if (/\ (br (` (1st) (cv x)) (<) (cv y)) (br (cv y) (<) (` (2nd) (cv x)))) (<,> (opr (opr (opr (2) (x.) (cv y)) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (cv y) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3))) (<,> (opr (opr (opr (2) (x.) (` (1st) (cv x))) (+) (` (2nd) (cv x))) (/) (3)) (opr (opr (` (1st) (cv x)) (+) (opr (2) (x.) (` (2nd) (cv x)))) (/) (3)))))))) (= G (o. (1st) (opr D (seq1) C))) (= H (o. (2nd) (opr D (seq1) C))) (= S (sup (ran G) (RR) (<)))) (-. (:-onto-> F (NN) (RR)))) stmt (ruclem38 ((F z) (x y) (x z) (y z)) ((:--> F (NN) (RR))) (-. (:-onto-> F (NN) (RR)))) stmt (ruclem39 () () (-. (:-onto-> F (NN) (RR)))) stmt (ruc () () (br (NN) (~<) (RR))) stmt (resdomq () () (br (QQ) (~<) (RR))) stmt (aleph1re () () (br (` (aleph) (1o)) (~<_) (RR))) stmt (infxpidmlem1 () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) (cv x)) (br (cv x) (~~) (X. (cv x) (cv x)))) (-> (/\ (br (cv x) (~~) A) (br (cv x) (~~) B)) (br (cv x) (~~) (u. A B))))) stmt (infxpidmlem2 ((f x) (t x) (A x) (f t) (A f) (A t) (B x) (B f) (B t) (H x)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. B (V))) (<-> (e. B H) (\/ (= B ({/})) (E. x (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> B (X. (cv x) (cv x)) (cv x))))))) stmt (infxpidmlem3 ((D x) (f x) (t x) (A x) (f t) (A f) (A t) (B x) (B f) (B t) (H x)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. B (V)) (e. D (V))) (-> (/\ (/\ (br (om) (~<_) D) (C_ D A)) (:-1-1-onto-> B (X. D D) D)) (e. B H))) stmt (infxpidmlem4 ((f x) (g x) (t x) (A x) (f g) (f t) (A f) (g t) (A g) (A t) (H x) (H g)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (-> (e. (cv g) H) (= (dom (cv g)) (X. (ran (cv g)) (ran (cv g)))))) stmt (infxpidmlem5 ((y z) (f y) (g y) (t y) (A y) (f z) (g z) (t z) (A z) (f g) (f t) (A f) (g t) (A g) (A t) (C y) (C z) (C f) (C g) (C t) (H y) (H z) (H g)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t)))))))) (-> (/\ (C_ C H) (e. (cv g) C)) (-> (/\ (e. (cv y) (ran (cv g))) (e. (cv z) (ran (cv g)))) (e. (<,> (cv y) (cv z)) (dom (U. C)))))) stmt (infxpidmlem6 ((f y) (g y) (t y) (A y) (f g) (f t) (A f) (g t) (A g) (A t) (B y) (B f) (B g) (B t) (C y) (C f) (C g) (C t) (H y) (H g)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (= B (ran (U. C)))) (<-> (e. (cv y) B) (E.e. g C (e. (cv y) (ran (cv g)))))) stmt (infxpidmlem7 ((x y) (x z) (w x) (f x) (g x) (h x) (v x) (t x) (A x) (y z) (w y) (f y) (g y) (h y) (v y) (t y) (A y) (w z) (f z) (g z) (h z) (v z) (t z) (A z) (f w) (g w) (h w) (v w) (t w) (A w) (f g) (f h) (f v) (f t) (A f) (g h) (g v) (g t) (A g) (h v) (h t) (A h) (t v) (A v) (A t) (B x) (B y) (B z) (B w) (B v) (B f) (B g) (B h) (B t) (C x) (C y) (C z) (C w) (C v) (C f) (C g) (C h) (C t) (H x) (H y) (H z) (H w) (H v) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (= B (ran (U. C)))) (-> (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) (:-1-1-onto-> (U. C) (X. B B) B))) stmt (infxpidmlem8 ((x y) (f x) (g x) (h x) (t x) (A x) (f y) (g y) (h y) (t y) (A y) (f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (B x) (B y) (B f) (B g) (B h) (B t) (C x) (C y) (C f) (C g) (C h) (C t) (H x) (H y) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (= B (ran (U. C))) (e. C (V))) (-> (/\ (C_ C H) (A.e. g C (A.e. h C (\/ (C_ (cv g) (cv h)) (C_ (cv h) (cv g)))))) (e. (U. C) H))) stmt (infxpidmlem9 ((f z) (g z) (h z) (t z) (A z) (f g) (f h) (f t) (A f) (g h) (g t) (A g) (h t) (A h) (A t) (H z) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. A (V))) (E.e. g H (A.e. h H (-. (C: (cv g) (cv h)))))) stmt (infxpidmlem10 ((f y) (g y) (h y) (v y) (t y) (A y) (f g) (f h) (f v) (f t) (A f) (g h) (g v) (g t) (A g) (h v) (h t) (A h) (t v) (A v) (A t) (H y) (H v) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. A (V))) (-> (A.e. h H (-. (C: (cv g) (cv h)))) (-> (br (om) (~<_) A) (-. (= (cv g) ({/})))))) stmt (infxpidmlem11 ((x y) (f x) (g x) (h x) (u x) (t x) (A x) (f y) (g y) (h y) (u y) (t y) (A y) (f g) (f h) (f u) (f t) (A f) (g h) (g u) (g t) (A g) (h u) (h t) (A h) (t u) (A u) (A t) (H x) (H y) (H u) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. A (V))) (-> (/\ (/\ (/\ (/\ (br (om) (~<_) (cv x)) (C_ (cv x) A)) (:-1-1-onto-> (cv g) (X. (cv x) (cv x)) (cv x))) (/\ (br (cv x) (~~) (cv y)) (C_ (cv y) (\ A (cv x))))) (:-1-1-onto-> (cv u) (u. (X. (cv x) (cv y)) (u. (X. (cv y) (cv x)) (X. (cv y) (cv y)))) (cv y))) (E.e. h H (C: (cv g) (cv h))))) stmt (infxpidmlem12 ((x y) (f x) (g x) (h x) (u x) (t x) (A x) (f y) (g y) (h y) (u y) (t y) (A y) (f g) (f h) (f u) (f t) (A f) (g h) (g u) (g t) (A g) (h u) (h t) (A h) (t u) (A u) (A t) (H x) (H y) (H u) (H g) (H h)) ((= H ({|} f (\/ (= (cv f) ({/})) (E. t (/\ (/\ (br (om) (~<_) (cv t)) (C_ (cv t) A)) (:-1-1-onto-> (cv f) (X. (cv t) (cv t)) (cv t))))))) (e. A (V))) (-> (br (om) (~<_) A) (br (X. A A) (~~) A))) stmt (infxpidm ((f x) (A f) (A x)) ((e. A (V))) (-> (br (om) (~<_) A) (br (X. A A) (~~) A))) stmt (infunabs () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (u. A B) (~~) A))) stmt (infcdaabs () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (opr A (+c) B) (~~) A))) stmt (infcda () ((e. A (V)) (e. B (V))) (-> (br (om) (~<_) A) (br (opr A (+c) B) (~~) (u. A B)))) stmt (infdif () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<) A)) (br (\ A B) (~~) A))) stmt (infxpabs () ((e. A (V)) (e. B (V))) (-> (/\ (/\ (br (om) (~<_) A) (-. (= B ({/})))) (br B (~<_) A)) (br (X. A B) (~~) A))) stmt (infxpdom () ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (X. A B) (~<_) A))) stmt (infmap1 () ((e. A (V)) (e. B (V))) (-> (/\ (/\ (br (2o) (~<_) A) (br (om) (~<_) B)) (br A (~<_) B)) (br (opr A (^m) B) (~~) (opr (2o) (^m) B)))) stmt (iunctb ((A x)) ((e. A (V)) (e. B (V))) (-> (/\ (br A (~<_) (om)) (A.e. x A (br B (~<_) (om)))) (br (U_ x A B) (~<_) (om)))) stmt (unictb ((A x)) ((e. A (V))) (-> (/\ (br A (~<_) (om)) (A.e. x A (br (cv x) (~<_) (om)))) (br (U. A) (~<_) (om)))) stmt (unctb ((A x) (B x)) () (-> (/\ (br A (~<_) (om)) (br B (~<_) (om))) (br (u. A B) (~<_) (om)))) stmt (aleph1irr () () (br (` (aleph) (1o)) (~<_) (\ (RR) (QQ)))) stmt (infmap2lem1 ((x z) (w x) (v x) (f x) (A x) (w z) (v z) (f z) (A z) (v w) (f w) (A w) (f v) (A v) (A f) (B x) (B z) (B w) (B f) (B v) (R v) (R f)) ((e. A (V)) (e. B (V)) (= R ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z)))))) (-> (/\ (C_ (cv f) R) (Fn (cv f) (dom R))) (-> (e. (cv v) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B)))) (/\ (C_ (cv v) A) (:-onto-> (` (cv f) (cv v)) B (cv v)))))) stmt (infmap2lem2 ((x z) (w x) (v x) (u x) (f x) (A x) (w z) (v z) (u z) (f z) (A z) (v w) (u w) (f w) (A w) (u v) (f v) (A v) (f u) (A u) (A f) (B x) (B z) (B w) (B f) (B v) (B u) (R u) (R v) (R f)) ((e. A (V)) (e. B (V)) (= R ({<,>|} z w (/\ (/\ (C_ (cv z) A) (br (cv z) (~~) B)) (:-onto-> (cv w) B (cv z)))))) (br ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B))) (~<_) (opr A (^m) B))) stmt (infmap2 ((x z) (w x) (A x) (w z) (A z) (A w) (B x) (B z) (B w)) ((e. A (V)) (e. B (V))) (-> (/\ (br (om) (~<_) A) (br B (~<_) A)) (br (opr A (^m) B) (~~) ({|} x (/\ (C_ (cv x) A) (br (cv x) (~~) B)))))) stmt (alephadd () () (br (opr (` (aleph) A) (+c) (` (aleph) B)) (~~) (u. (` (aleph) A) (` (aleph) B)))) stmt (alephexp1 () () (-> (/\ (/\ (e. A (On)) (e. B (On))) (C_ A B)) (br (opr (` (aleph) A) (^m) (` (aleph) B)) (~~) (opr (2o) (^m) (` (aleph) B))))) stmt (alephsuc3 ((A x)) () (-> (e. A (On)) (br (` (aleph) (suc A)) (~~) ({e.|} x (On) (br (cv x) (~~) (` (aleph) A)))))) stmt (alephexp2 ((A x)) () (-> (e. A (On)) (br (opr (2o) (^m) (` (aleph) A)) (~~) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A))))))) stmt (gch-kn ((A x)) () (-> (e. A (On)) (<-> (br (` (aleph) (suc A)) (~~) ({|} x (/\ (C_ (cv x) (` (aleph) A)) (br (cv x) (~~) (` (aleph) A))))) (br (` (aleph) (suc A)) (~~) (opr (2o) (^m) (` (aleph) A)))))) stmt (istopg ((x y) (x z) (J x) (y z) (J y) (J z)) () (-> (e. J A) (<-> (e. J (Top)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A.e. x J (A.e. y J (e. (i^i (cv x) (cv y)) J))))))) stmt (istop2g ((x y) (J x) (J y)) () (-> (e. J A) (<-> (e. J (Top)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A. x (-> (/\/\ (C_ (cv x) J) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) J))))))) stmt (uniopnt ((x y) (A x) (A y) (J x) (J y)) () (-> (/\ (e. J (Top)) (C_ A J)) (e. (U. A) J))) stmt (iunopnt ((x y) (A x) (A y) (B y) (J x) (J y)) () (-> (/\ (e. J (Top)) (A.e. x A (e. B J))) (e. (U_ x A B) J))) stmt (inopnt ((x y) (A x) (A y) (B y) (J x) (J y)) () (-> (/\/\ (e. J (Top)) (e. A J) (e. B J)) (e. (i^i A B) J))) stmt (0opnt () () (-> (e. J (Top)) (e. ({/}) J))) stmt (1open () ((= X (U. J))) (-> (e. J (Top)) (e. X J))) stmt (eltopss () ((= X (U. J))) (-> (/\ (e. J (Top)) (e. A J)) (C_ A X))) stmt (unitopt () () (-> (e. J (Top)) (e. (U. J) J))) stmt (eltopsp ((x y) (J x) (J y)) () (<-> (e. (<,> (U. J) J) (TopSp)) (e. J (Top)))) stmt (tpsex ((x y) (A x) (A y) (J x) (J y)) () (-> (e. (<,> A J) (TopSp)) (/\ (e. A (V)) (e. J (V))))) stmt (istps ((x y) (A x) (A y) (J x) (J y)) () (<-> (e. (<,> A J) (TopSp)) (/\ (e. J (Top)) (= A (U. J))))) stmt (istps2 () () (<-> (e. (<,> A J) (TopSp)) (/\ (/\ (e. J (Top)) (C_ J (P~ A))) (/\ (e. ({/}) J) (e. A J))))) stmt (istps3 ((x y) (A x) (A y) (J x) (J y)) () (<-> (e. (<,> A J) (TopSp)) (/\ (/\/\ (C_ J (P~ A)) (e. ({/}) J) (e. A J)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A.e. x J (A.e. y J (e. (i^i (cv x) (cv y)) J))))))) stmt (istps4 ((x y) (A x) (A y) (J x) (J y)) () (<-> (e. (<,> A J) (TopSp)) (/\ (/\/\ (C_ J (P~ A)) (e. ({/}) J) (e. A J)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A. x (-> (/\/\ (C_ (cv x) J) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) J))))))) stmt (istps5 ((x y) (A x) (A y) (J x) (J y)) () (<-> (e. (<,> A J) (TopSp)) (/\ (/\/\ (A.e. x J (C_ (cv x) A)) (e. ({/}) J) (e. A J)) (/\ (A. x (-> (C_ (cv x) J) (e. (U. (cv x)) J))) (A. x (-> (/\/\ (C_ (cv x) J) (-. (= (cv x) ({/}))) (E.e. y (om) (br (cv x) (~~) (cv y)))) (e. (|^| (cv x)) J))))))) stmt (isbasisg ((x y) (x z) (B x) (y z) (B y) (B z)) () (-> (e. B C) (<-> (e. B (Bases)) (A.e. x B (A.e. y B (C_ (i^i (cv x) (cv y)) (U. (i^i B (P~ (i^i (cv x) (cv y))))))))))) stmt (isbasis2g ((w x) (w y) (w z) (B w) (x y) (x z) (B x) (y z) (B y) (B z)) () (-> (e. B C) (<-> (e. B (Bases)) (A.e. x B (A.e. y B (A.e. z (i^i (cv x) (cv y)) (E.e. w B (/\ (e. (cv z) (cv w)) (C_ (cv w) (i^i (cv x) (cv y))))))))))) stmt (isbasis3g ((w x) (w y) (w z) (B w) (x y) (x z) (B x) (y z) (B y) (B z)) () (-> (e. B C) (<-> (e. B (Bases)) (/\/\ (A.e. x B (C_ (cv x) (U. B))) (A.e. x (U. B) (E.e. y B (e. (cv x) (cv y)))) (A.e. x B (A.e. y B (A.e. z (i^i (cv x) (cv y)) (E.e. w B (/\ (e. (cv z) (cv w)) (C_ (cv w) (i^i (cv x) (cv y)))))))))))) stmt (basis1t ((x y) (B x) (B y) (C x) (C y) (D y) (D x)) () (-> (/\/\ (e. B (Bases)) (e. 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B)) (A.e. x A (E.e. y B (/\ (e. (cv x) (cv y)) (C_ (cv y) A)))))))) stmt (tg1t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. B (Bases)) (e. A (` (topGen) B))) (C_ A (U. B)))) stmt (tg2t ((x y) (A x) (A y) (B x) (B y) (C x) (C y)) () (-> (/\/\ (e. B (Bases)) (e. A (` (topGen) B)) (e. C A)) (E.e. x B (/\ (e. C (cv x)) (C_ (cv x) A))))) stmt (bastgt ((B x)) () (-> (e. B (Bases)) (C_ B (` (topGen) B)))) stmt (unitgt ((x y) (B x) (B y)) () (-> (e. B (Bases)) (= (U. (` (topGen) B)) (U. B)))) stmt (0eltgt () () (-> (e. B (Bases)) (e. ({/}) (` (topGen) B)))) stmt (tgclt ((v w) (v x) (v y) (v z) (w x) (w y) (w z) (x y) (x z) (y z) (t u) (t v) (t w) (t x) (t y) (t z) (B t) (u v) (u w) (u x) (u y) (u z) (B u) (B v) (B w) (B x) (B y) (B z)) () (-> (e. B (Bases)) (e. (` (topGen) B) (Top)))) stmt (tgval3t ((w x) (w y) (w z) (x y) (x z) (y z) (B w) (B x) (B y) (B z)) () (-> (e. B (Bases)) (= (` (topGen) B) ({|} x (E. y (/\ (C_ (cv y) B) (= (cv x) (U. 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X (` (Cls) J)))) stmt (ntrval ((x y) (x z) (J x) (y z) (J y) (J z) (S x) (S y) (S z) (X x) (X y) (X z)) ((= X (U. J))) (-> (/\ (e. J (Top)) (C_ S X)) (= (` (` (int) J) S) (U. ({e.|} x J (C_ (cv x) S)))))) stmt (clsval ((x y) (x z) (J x) (y z) (J y) (J z) (S x) (S y) (S z) (X x) (X y) (X z)) ((= X (U. J))) (-> (/\ (e. J (Top)) (C_ S X)) (= (` (` (cls) J) S) (|^| ({e.|} x (` (Cls) J) (C_ S (cv x))))))) stmt (clsval2 ((x y) (x z) (J x) (y z) (J y) (J z) (S x) (S y) (S z) (X x) (X y) (X z)) ((= X (U. J))) (-> (/\ (e. J (Top)) (C_ S X)) (= (` (` (cls) J) S) (\ X (` (` (int) J) (\ X S)))))) stmt (ntropn ((J x) (S x) (X x)) ((= X (U. J))) (-> (/\ (e. J (Top)) (C_ S X)) (e. (` (` (int) J) S) J))) stmt (iinclsd ((A x) (J x)) () (-> (/\/\ (e. J (Top)) (=/= A ({/})) (A.e. x A (e. B (` (Cls) J)))) (e. (|^|_ x A B) (` (Cls) J)))) stmt (intclsd ((A x) (J x)) () (-> (/\/\ (e. J (Top)) (=/= A ({/})) (C_ A (` (Cls) J))) (e. (|^| A) (` (Cls) J)))) stmt (isopen3 ((J x) (S x) (X x)) ((= X (U. J))) (-> (/\ (e. J (Top)) (C_ S X)) (<-> (e. S J) (= (` (` (int) J) S) S)))) stmt (ntrtop () ((= X (U. J))) (-> (e. J (Top)) (= (` (` (int) J) X) X))) stmt (isnei ((J u) (J w) (J j) (J z) (J g) (J y) (u w) (j u) (u z) (g u) (u y) (j w) (w z) (g w) (w y) (j z) (g j) (j y) (g z) (y z) (g y) (X u) (X w) (X j) (X z) (X g) (X y) (A g) (A w) (A z) (B g) (B w) (B z)) ((= X (U. J)) (e. A (V)) (e. B (V)) (e. J (Top))) (<-> (br A (` (nei) J) B) (/\ (C_ A X) (E.e. g J (/\ (C_ B (cv g)) (C_ (cv g) A)))))) stmt (scnei ((A x) (B x) (C x) (J x)) ((e. A (V)) (e. B (V)) (e. C (V)) (e. J (Top))) (-> (/\ (br A (` (nei) J) B) (C_ C B)) (br A (` (nei) J) C))) stmt (neibel ((A g) (B g) (J g)) ((e. J (Top)) (e. A (V)) (e. B (V))) (-> (br A (` (nei) J) ({} B)) (e. B A))) stmt (intnei ((J g) (J j) (J k) (g j) (g k) (j k) (A g) (A j) (A k) (B g) (B j) (B k) (C g) (C j) (C k)) ((e. J (Top)) (e. A (V)) (e. B (V)) (e. C (V))) (-> (/\ (br A (` (nei) J) B) (br C (` (nei) J) B)) (br (i^i A C) (` (nei) J) B))) stmt (opneit ((J g) (A g)) ((e. J (Top)) (e. A (V))) (<-> (br A (` (nei) J) A) (e. A J))) stmt (onp ((A g) (B g) (J g)) ((e. J (Top)) (e. A J) (e. B (V))) (-> (C_ B A) (br A (` (nei) J) B))) stmt (ssneip ((A g) (B g) (C g) (J g) (X g)) ((= X (U. J)) (e. A (V)) (e. B (V)) (e. C (V)) (e. J (Top))) (-> (/\/\ (C_ A B) (C_ B X) (br A (` (nei) J) C)) (br B (` (nei) J) C))) stmt (enonei ((J x) (J y) (x y) (A x) (A y) (B x) (B y)) ((e. J (Top)) (e. A (V)) (e. B (V))) (-> (br B (` (nei) J) A) (E. x (/\ (br (cv x) (` (nei) J) A) (A. y (-> (C_ (cv y) (cv x)) (br B (` (nei) J) (cv y)))))))) stmt (ucnp () ((= X (U. J)) (e. J (Top)) (e. A (V))) (-> (C_ A X) (br X (` (nei) J) A))) stmt (cnfval ((f j) (f k) (f y) (f z) (J f) (j k) (j y) (j z) (J j) (k y) (k z) (J k) (y z) (J y) (J z) (K f) (K j) (K k) (K y) (K z) (X f) (X j) (X k) (X y) (X z) (Y f) (Y k) (Y y) (Y z)) ((= X (U. J)) (= Y (U. K))) (-> (/\ (e. J (Top)) (e. K (Top))) (= (opr J (Cn) K) ({e.|} f (opr Y (^m) X) (A.e. y K (e. (" (`' (cv f)) (cv y)) J)))))) stmt (cnpfval ((f j) (f k) (f v) (f w) (f x) (f y) (f z) (J f) (j k) (j v) (j w) (j x) (j y) (j z) (J j) (k v) (k w) (k x) (k y) (k z) (J k) (v w) (v x) (v y) (v z) (J v) (w x) (w y) (w z) (J w) (x y) (x z) (J x) (y z) (J y) (J z) (K f) (K j) (K k) (K w) (K x) (K y) (K z) (X f) (X j) (X k) (X x) (X y) (X z) (Y f) (Y k) (Y y) (Y z)) ((= X (U. J)) (= Y (U. K))) (-> (/\ (e. J (Top)) (e. K (Top))) (= (opr J (CnP) K) ({<,>|} x y (/\ (e. (cv x) X) (= (cv y) ({e.|} f (opr Y (^m) X) (A.e. w K (-> (e. (` (cv f) (cv x)) (cv w)) (E.e. v J (/\ (e. (cv x) (cv v)) (C_ (" (cv f) (cv v)) (cv w))))))))))))) stmt (iscn ((f y) (F f) (F y) (J f) (J y) (K y) (K f) (X f) (X y) (Y f) (Y y)) ((= X (U. J)) (= Y (U. K))) (-> (/\ (e. J (Top)) (e. K (Top))) (<-> (e. F (opr J (Cn) K)) (/\ (:--> F X Y) (A.e. y K (e. (" (`' F) (cv y)) J)))))) stmt (cnpval ((f x) (f y) (x y) (f v) (f w) (J f) (v w) (v x) (v y) (J v) (w x) (w y) (J w) (J x) (J y) (K y) (K w) (K v) (K f) (P f) (P v) (P w) (P x) (P y) (X f) (X v) (X w) (X y) (Y f) (Y v) (Y w) (Y y)) ((= X (U. J)) (= Y (U. K))) (-> (/\/\ (e. J (Top)) (e. K (Top)) (e. P X)) (= (` (opr J (CnP) K) P) ({e.|} f (opr Y (^m) X) (A.e. y K (-> (e. (` (cv f) P) (cv y)) (E.e. x J (/\ (e. P (cv x)) (C_ (" (cv f) (cv x)) (cv y)))))))))) stmt (iscnp ((f x) (f y) (F f) (x y) (F x) (F y) (J f) (J x) (J y) (K y) (K f) (P f) (P x) (P y) (X f) (X y) (Y f) (Y y)) ((= X (U. J)) (= Y (U. K))) (-> (/\/\ (e. J (Top)) (e. K (Top)) (e. P X)) (<-> (e. F (` (opr J (CnP) K) P)) (/\ (:--> F X Y) (A.e. y K (-> (e. (` F P) (cv y)) (E.e. x J (/\ (e. P (cv x)) (C_ (" F (cv x)) (cv y)))))))))) stmt (msrel ((x y) (x z) (w x) (v x) (y z) (w y) (v y) (w z) (v z) (v w)) () (Rel (Met))) stmt (ismsg ((v w) (v x) (v y) (v z) (X v) (w x) (w y) (w z) (X w) (x y) (x z) (X x) (y z) (X y) (X z) (D v) (D w) (D x) (D y) (D z)) () (-> (e. D A) (<-> (e. (<,> X D) (Met)) (/\ (:--> D (X. X X) (RR)) (A.e. x X (A.e. y X (/\ (<-> (= (opr (cv x) D (cv y)) (0)) (= (cv x) (cv y))) (A.e. z X (br (opr (cv x) D (cv y)) (<_) (opr (opr (cv z) D (cv x)) (+) (opr (cv z) D (cv y)))))))))))) stmt (msflem ((x y) (x z) (y z) (v w) (v x) (v y) (v z) (D v) (w x) (w y) (w z) (D w) (D x) (D y) (D z) (M v) (M w) (X v) (X w) (X x) (X y) (X z)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (e. M (Met)) (/\ (:--> D (X. 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M (Met)) (e. A X)) (= (opr A D A) (0)))) stmt (mssym () ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (/\/\ (e. M (Met)) (e. A X) (e. B X)) (= (opr A D B) (opr B D A)))) stmt (mstri () ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (/\ (/\ (e. M (Met)) (e. A X)) (/\ (e. B X) (e. C X))) (br (opr A D B) (<_) (opr (opr A D C) (+) (opr C D B))))) stmt (msge0 () ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (/\/\ (e. M (Met)) (e. A X) (e. B X)) (br (0) (<_) (opr A D B)))) stmt (msgt0 () ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (/\/\ (e. M (Met)) (e. A X) (e. B X)) (<-> (=/= A B) (br (0) (<) (opr A D B))))) stmt (blfval ((t u) (t z) (D t) (u z) (D u) (D z) (t w) (t x) (t y) (M t) (u w) (u x) (u y) (M u) (w x) (w y) (w z) (M w) (x y) (x z) (M x) (y z) (M y) (M z) (X t) (X u) (X w) (X x) (X y) (X z)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (e. M (Met)) (= (` (ball) M) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) X) (e. (cv y) (RR))) (/\ (br (0) (<) (cv y)) (= (cv z) ({e.|} w X (br (opr (cv x) D (cv w)) (<) (cv y)))))))))) stmt (blfval2 ((D z) (w x) (w y) (w z) (M w) (x y) (x z) (M x) (y z) (M y) (M z) (X w) (X x) (X y) (X z) (v y)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (e. M (Met)) (= (` (ball) M) ({<<,>,>|} x y z (/\ (/\ (e. (cv x) X) (e. (cv y) ({e.|} v (RR) (br (0) (<) (cv v))))) (= (cv z) ({e.|} w X (br (opr (cv x) D (cv w)) (<) (cv y))))))))) stmt (blval ((x y) (v w) (v x) (v y) (v z) (D v) (w x) (w y) (w z) (D w) (x z) (D x) (y z) (D y) (D z) (M v) (M w) (M x) (M y) (M z) (P w) (P x) (P y) (P z) (R v) (R w) (R x) (R y) (R z) (X v) (X w) (X x) (X y) (X z)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R))) (= (opr P (` (ball) M) R) ({e.|} x X (br (opr P D (cv x)) (<) R))))) stmt (elbl ((A x) (D x) (M x) (P x) (R x) (X x)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (/\ (/\ (e. M (Met)) (e. P X)) (/\ (e. R (RR)) (br (0) (<) R))) (<-> (e. 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A) (` (open) M)))) stmt (opnin ((f g) (f u) (f v) (f x) (f y) (f z) (A f) (g u) (g v) (g x) (g y) (g z) (A g) (u v) (u x) (u y) (u z) (A u) (v x) (v y) (v z) (A v) (x y) (x z) (A x) (y z) (A y) (A z) (f w) (B f) (g w) (B g) (u w) (B u) (v w) (B v) (w x) (w y) (B w) (B x) (B y) (M f) (M g) (M u) (M v) (w z) (M w) (M x) (M y) (M z)) () (-> (/\/\ (e. M (Met)) (e. A (` (open) M)) (e. B (` (open) M))) (e. (i^i A B) (` (open) M)))) stmt (opntop ((x y) (M x) (M y)) () (-> (e. M (Met)) (e. (` (open) M) (Top)))) stmt (0ms ((x y) (x z) (y z)) () (e. (<,> ({/}) ({/})) (Met))) stmt (ccms ((D u) (D v) (D w) (u v) (u w) (v w) (x y) (x z) (u x) (v x) (w x) (y z) (u y) (v y) (w y) (u z) (v z) (w z)) ((= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) (CC)) (e. (cv y) (CC))) (= (cv z) (` (abs) (opr (cv x) (-) (cv y)))))))) (e. (<,> (CC) D) (Met))) stmt (dscms ((X x) (X y) (X z) (x y) (x z) (y z) (X u) (X v) (X w) (u v) (u w) (v w) (D u) (D v) (D w) (u x) (u y) (u z) (v x) (v y) (v z) (w x) (w y) (w z)) ((e. X (V)) (= D ({<<,>,>|} x y z (/\ (/\ (e. (cv x) X) (e. (cv y) X)) (= (cv z) (if (= (cv x) (cv y)) (0) (1))))))) (e. (<,> X D) (Met))) stmt (lmfval ((w z) (D w) (D z) (f j) (f k) (f w) (f x) (f y) (f z) (M f) (j k) (j w) (j x) (j y) (j z) (M j) (k w) (k x) (k y) (k z) (M k) (w x) (w y) (M w) (x y) (x z) (M x) (y z) (M y) (M z) (X f) (X w) (X y) (X z)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (e. M (Met)) (= (` (~~>m) M) ({<,>|} f y (/\/\ (C_ (cv f) (X. (CC) X)) (e. (cv y) X) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` (cv f) (cv k)) X) (br (opr (` (cv f) (cv k)) D (cv y)) (<) (cv x))))))))))))) stmt (caufval ((w z) (D w) (D z) (f j) (f k) (f m) (f w) (f x) (f z) (M f) (j k) (j m) (j w) (j x) (j z) (M j) (k m) (k w) (k x) (k z) (M k) (m w) (m x) (m z) (M m) (w x) (M w) (x z) (M x) (M z) (X f) (X w) (X z)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (e. M (Met)) (= (` (cau) M) ({|} f (/\ (C_ (cv f) (X. (CC) X)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (A.e. m (ZZ) (-> (/\ (br (cv j) (<_) (cv k)) (br (cv j) (<_) (cv m))) (/\/\ (e. (` (cv f) (cv k)) X) (e. (` (cv f) (cv m)) X) (br (opr (` (cv f) (cv k)) D (` (cv f) (cv m))) (<) (cv x)))))))))))))) stmt (lmbr ((w z) (D w) (D z) (j k) (j w) (j x) (j z) (F j) (k w) (k x) (k z) (F k) (w x) (F w) (x z) (F x) (F z) (M j) (M k) (M w) (M x) (M z) (P j) (P k) (P w) (P x) (P z) (X w) (X z)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (/\/\ (e. M (Met)) (C_ F (X. (CC) X)) (e. P X)) (<-> (br F (` (~~>m) M) P) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (-> (br (cv j) (<_) (cv k)) (/\ (e. (` F (cv k)) X) (br (opr (` F (cv k)) D P) (<) (cv x))))))))))) stmt (iscau ((D f) (f j) (f k) (f m) (f x) (F f) (j k) (j m) (j x) (F j) (k m) (k x) (F k) (m x) (F m) (F x) (M f) (M j) (M k) (M m) (M x) (X f)) ((= X (` (1st) M)) (= D (` (2nd) M))) (-> (/\ (e. M (Met)) (C_ F (X. (CC) X))) (<-> (e. F (` (cau) M)) (A.e. x (RR) (-> (br (0) (<) (cv x)) (E.e. j (ZZ) (A.e. k (ZZ) (A.e. m (ZZ) (-> (/\ (br (cv j) (<_) (cv k)) (br (cv j) (<_) (cv m))) (/\/\ (e. (` F (cv k)) X) (e. (` F (cv m)) X) (br (opr (` F (cv k)) D (` F (cv m))) (<) (cv x)))))))))))) stmt (iscms ((f x) (f y) (M f) (x y) (M x) (M y) (X x) (X y)) ((= X (` (1st) M))) (<-> (e. M (Cm)) (/\ (e. M (Met)) (A.e. f (` (cau) M) (E.e. x X (br (cv f) (` (~~>m) M) (cv x))))))) stmt (avril1 ((F x)) () (-. (/\ (br A (P~ (RR)) (` (i) (1))) (br F ({/}) (opr (0) (x.) (1)))))) stmt (hvmulcl () ((e. A (CC)) (e. B (H~))) (e. (opr A (.s) B) (H~))) stmt (hvsubvalt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (H~)) (e. B (H~))) (= (opr A (-v) B) (opr A (+v) (opr (-u (1)) (.s) B))))) stmt (hvsubclt () () (-> (/\ (e. A (H~)) (e. B (H~))) (e. (opr A (-v) B) (H~)))) stmt (hvaddcl () ((e. A (H~)) (e. B (H~))) (e. (opr A (+v) B) (H~))) stmt (hvcom () ((e. A (H~)) (e. B (H~))) (= (opr A (+v) B) (opr B (+v) A))) stmt (hvsubval () ((e. A (H~)) (e. B (H~))) (= (opr A (-v) B) (opr A (+v) (opr (-u (1)) (.s) B)))) stmt (hvsubcl () ((e. A (H~)) (e. B (H~))) (e. (opr A (-v) B) (H~))) stmt (hvaddid2t () () (-> (e. A (H~)) (= (opr (0v) (+v) A) A))) stmt (hvmul0t () () (-> (e. A (CC)) (= (opr A (.s) (0v)) (0v)))) stmt (hvmul0ort () () (-> (/\ (e. A (CC)) (e. B (H~))) (<-> (= (opr A (.s) B) (0v)) (\/ (= A (0)) (= B (0v)))))) stmt (hvsubidt () () (-> (e. A (H~)) (= (opr A (-v) A) (0v)))) stmt (hvnegidt () () (-> (e. A (H~)) (= (opr A (+v) (opr (-u (1)) (.s) A)) (0v)))) stmt (hv2negt () () (-> (e. A (H~)) (= (opr (0v) (-v) A) (opr (-u (1)) (.s) A)))) stmt (hvaddid2 () ((e. A (H~))) (= (opr (0v) (+v) A) A)) stmt (hvnegid () ((e. A (H~))) (= (opr A (+v) (opr (-u (1)) (.s) A)) (0v))) stmt (hv2neg () ((e. A (H~))) (= (opr (0v) (-v) A) (opr (-u (1)) (.s) A))) stmt (hvm1negt () () (-> (/\ (e. A (CC)) (e. 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G (NN)))) (-> (/\ (br N (<_) D) (br N (<_) G)) (br (` (norm) (opr (` F D) (-v) (` F G))) (<) (` (sqr) (opr (opr (4) (x.) (opr (opr (2) (x.) R) (+) (1))) (/) N)))))) stmt (projlem28 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H u) (H v) (H x) (H z) (D x) (D y) (D z) (D w) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (ph x) (ph y) (ph z)) ((e. A (H~)) (e. H (CH)) (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))) (= R (-u (sup S (RR) (<)))) (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))) (e. D (RR))) (-> ph (-> (br (0) (<) D) (E.e. z (NN) (A.e. x (NN) (A.e. y (NN) (-> (/\ (br (cv z) (<_) (cv x)) (br (cv z) (<_) (cv y))) (br (` (norm) (opr (` F (cv x)) (-v) (` F (cv y)))) (<) D)))))))) stmt (projlem29 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (g u) (H u) (g v) (H v) (g x) (g z) (H g) (H x) (H z) (g y) (g w) (F g) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (g ph) (ph x) (ph y) (ph z)) ((e. A (H~)) (e. H (CH)) (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))) (= R (-u (sup S (RR) (<)))) (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (-> ph (e. F (Cauchy)))) stmt (projlem30 ((u v) (u z) (u w) (A u) (v z) (v w) (A v) (w z) (A z) (A w) (H u) (H v) (H z) (F w) (F z) (R z) (R w) (ph z)) ((e. A (H~)) (e. H (CH)) (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))) (= R (-u (sup S (RR) (<)))) (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w))))))))) (-> ph (E.e. z H (br F (~~>v) (cv z))))) stmt (projlem31 ((u v) (u x) (u y) (u z) (u w) (A u) (v x) (v y) (v z) (v w) (A v) (x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (H u) (H v) (H x) (H z) (F x) (F y) (F w) (F z) (R x) (R y) (R z) (R w) (ph x) (ph y) (ph z)) ((e. A (H~)) (e. H (CH)) (= S ({e.|} u (RR) (E.e. v H (= (cv u) (-u (` (norm) (opr (cv v) (-v) A))))))) (= R (-u (sup S (RR) (<)))) (<-> ph (/\ (:--> F (NN) H) (A.e. w (NN) (/\ (br (opr R (-) (opr (1) (/) (cv w))) (<) (` (norm) (opr (` F (cv w)) (-v) A))) (br (` (norm) (opr (` F (cv w)) (-v) A)) (<) (opr R (+) (opr (1) (/) (cv w)))))))) (e. F (V))) (-> ph (E.e. x H (A.e. y H (br (` (norm) (opr (cv x) (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A))))))) stmt (projlem ((x y) (f x) (u x) (v x) (w x) (H x) (f y) (u y) (v y) (w y) (H y) (f u) (f v) (f w) (H f) (u v) (u w) (H u) (v w) (H v) (H w) (A x) (A y) (A f) (A u) (A v) (A w)) ((e. A (H~)) (e. H (CH))) (E.e. x H (A.e. y H (br (` (norm) (opr (cv x) (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))))) stmt (pjthlem1 () ((e. A (H~)) (e. B (H~)) (e. D (H~)) (e. S (CC)) (= C (opr A (-v) B))) (<-> (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) (br (opr (` (norm) C) (^) (2)) (<_) (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2))))) stmt (pjthlem2 () ((e. D (H~)) (= R (opr (1) (/) (opr D (.i) D)))) (-> (-. (= D (0v))) (e. R (RR)))) stmt (pjthlem3 () ((e. D (H~)) (= R (opr (1) (/) (opr D (.i) D)))) (-> (-. (= D (0v))) (br (0) (<) R))) stmt (pjthlem4 () ((e. D (H~)) (= R (opr (1) (/) (opr D (.i) D))) (e. C (H~)) (= S (opr R (x.) (opr C (.i) D)))) (-> (-. (= D (0v))) (e. S (CC)))) stmt (pjthlem5 () ((e. D (H~)) (e. C (H~)) (e. S (CC))) (= (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)) (opr (opr (opr (` (norm) C) (^) (2)) (-) (` (*) (opr S (x.) (opr D (.i) C)))) (+) (opr (opr (opr S (x.) (` (*) S)) (x.) (opr D (.i) D)) (-) (opr S (x.) (opr D (.i) C)))))) stmt (pjthlem6 () ((e. D (H~)) (= R (opr (1) (/) (opr D (.i) D))) (e. C (H~)) (= S (opr R (x.) (opr C (.i) D)))) (-> (-. (= D (0v))) (= (opr S (x.) (opr D (.i) C)) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2)))))) stmt (pjthlem7 () ((e. D (H~)) (= R (opr (1) (/) (opr D (.i) D))) (e. C (H~)) (= S (opr R (x.) (opr C (.i) D)))) (-> (-. (= D (0v))) (= (opr (opr S (x.) (` (*) S)) (x.) (opr D (.i) D)) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2)))))) stmt (pjthlem8 () ((e. D (H~)) (= R (opr (1) (/) (opr D (.i) D))) (e. C (H~)) (= S (opr R (x.) (opr C (.i) D)))) (-> (-. (= D (0v))) (= (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)) (opr (opr (` (norm) C) (^) (2)) (-) (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))))))) stmt (pjthlem9 () ((e. A (H~)) (e. B (H~)) (e. D (H~)) (= R (opr (1) (/) (opr D (.i) D))) (= S (opr R (x.) (opr C (.i) D))) (= C (opr A (-v) B))) (-> (-. (= D (0v))) (<-> (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A))) (br (opr (` (norm) C) (^) (2)) (<_) (opr (` (norm) (opr C (-v) (opr S (.s) D))) (^) (2)))))) stmt (pjthlem10 () ((e. A (H~)) (e. B (H~)) (e. D (H~)) (= R (opr (1) (/) (opr D (.i) D))) (= S (opr R (x.) (opr C (.i) D))) (= C (opr A (-v) B))) (-> (/\ (-. (= D (0v))) (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A)))) (= (opr R (x.) (opr (` (abs) (opr C (.i) D)) (^) (2))) (0)))) stmt (pjthlem11 () ((e. A (H~)) (e. B (H~)) (e. D (H~)) (= R (opr (1) (/) (opr D (.i) D))) (= S (opr R (x.) (opr C (.i) D))) (= C (opr A (-v) B))) (-> (/\ (-. (= D (0v))) (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (opr B (+v) (opr S (.s) D)) (-v) A)))) (= (opr C (.i) D) (0)))) stmt (pjthlem12 ((A y) (B y) (D y) (S y) (H y)) ((e. A (H~)) (e. H (CH)) (e. B H) (e. D H) (= R (opr (1) (/) (opr D (.i) D))) (= S (opr R (x.) (opr C (.i) D))) (= C (opr A (-v) B))) (-> (A.e. y H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))) (-> (-. (= D (0v))) (= (opr C (.i) D) (0))))) stmt (pjthlem13 ((A y) (B y) (D y) (C y) (H y)) ((e. A (H~)) (e. H (CH)) (e. B H) (e. D H) (= C (opr A (-v) B))) (-> (A.e. y H (br (` (norm) (opr B (-v) A)) (<_) (` (norm) (opr (cv y) (-v) A)))) (= (opr C (.i) D) (0)))) stmt (pjthlem14 ((x z) (y z) (A z) (x y) (A x) (A y) (B z) (B x) (B y) (C z) (C x) (C y) (H z) (H x) (H y)) ((e. A (H~)) (e. H (CH)) (e. 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A (H~))) (E!e. x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y))))))) stmt (pjmvalt ((x y) (x z) (w x) (f x) (h x) (H x) (y z) (w y) (f y) (h y) (H y) (w z) (f z) (h z) (H z) (f w) (h w) (H w) (f h) (H f) (H h)) () (-> (e. H (CH)) (= (` (proj) H) ({<,>|} x y (/\ (e. (cv x) (H~)) (= (cv y) (U. ({e.|} z H (E.e. w (` (_|_) H) (= (cv x) (opr (cv z) (+v) (cv w)))))))))))) stmt (pjvalt ((x y) (x z) (w x) (H x) (y z) (w y) (H y) (w z) (H z) (H w) (A x) (A y) (A z) (A w)) () (-> (/\ (e. H (CH)) (e. A (H~))) (= (` (` (proj) H) A) (U. ({e.|} x H (E.e. y (` (_|_) H) (= A (opr (cv x) (+v) (cv y))))))))) stmt (axpjclt ((x y) (H x) (H y) (A x) (A y)) () (-> (/\ (e. H (CH)) (e. A (H~))) (e. (` (` (proj) H) A) H))) stmt (pjhclt () () (-> (/\ (e. H (CH)) (e. A (H~))) (e. (` (` (proj) H) A) (H~)))) stmt (omlsilem () ((e. G (SH)) (e. H (SH)) (C_ G H) (= (i^i H (` (_|_) G)) (0H)) (e. A H) (e. B G) (e. C (` (_|_) G))) (-> (= A (opr B (+v) C)) (e. A G))) stmt (omlsi ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) ((e. A (CH)) (e. B (SH)) (C_ A B) (= (i^i B (` (_|_) A)) (0H))) (= A B)) stmt (omls () ((e. A (CH)) (e. B (SH))) (-> (/\ (C_ A B) (= (i^i B (` (_|_) A)) (0H))) (= A B))) stmt (ococ () ((e. A (CH))) (= (` (_|_) (` (_|_) A)) A)) stmt (ococt () () (-> (e. A (CH)) (= (` (_|_) (` (_|_) A)) A))) stmt (dfch2 () () (= (CH) ({|} x (/\ (C_ (cv x) (H~)) (= (` (_|_) (` (_|_) (cv x))) (cv x)))))) stmt (pjcl () ((e. H (CH))) (-> (e. A (H~)) (e. (` (` (proj) H) A) H))) stmt (pjhcl () ((e. H (CH))) (-> (e. A (H~)) (e. (` (` (proj) H) A) (H~)))) stmt (pjcli () ((e. H (CH)) (e. A (H~))) (e. (` (` (proj) H) A) H)) stmt (pjhcli () ((e. H (CH)) (e. A (H~))) (e. (` (` (proj) H) A) (H~))) stmt (pjpj0 ((x y) (A x) (A y) (H x) (H y)) ((e. H (CH)) (e. A (H~))) (= A (opr (` (` (proj) H) A) (+v) (` (` (proj) (` (_|_) H)) A)))) stmt (axpjpjt () () (-> (/\ (e. H (CH)) (e. 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H (V))) (-> (/\ (/\ (/\ (:--> F (NN) A) (:--> G (NN) B)) (A.e. w (NN) (= (` H (cv w)) (opr (` F (cv w)) (+v) (` G (cv w)))))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) (-> (br H (~~>v) (cv z)) (br G (~~>v) (cv y))))) stmt (osumlem5 ((x y) (w x) (f x) (g x) (A x) (w y) (f y) (g y) (A y) (f w) (g w) (A w) (f g) (A f) (A g) (B x) (B y) (B w) (B f) (B g) (H x) (H y) (H w) (H f) (H g)) ((e. A (CH)) (e. B (CH))) (-> (:--> H (NN) (opr A (+H) B)) (E. f (E. g (/\ (/\ (:--> (cv f) (NN) A) (:--> (cv g) (NN) B)) (A.e. w (NN) (= (` H (cv w)) (opr (` (cv f) (cv w)) (+v) (` (cv g) (cv w)))))))))) stmt (osumlem6 ((f w) (g w) (A w) (f g) (A f) (A g) (B w) (B f) (B g) (H w) (H f) (H g) (w x) (f x) (g x) (w y) (f y) (g y) (w z) (f z) (g z)) ((e. A (CH)) (e. B (CH)) (C_ B (` (_|_) A)) (e. H (V))) (-> (/\ (/\ (:--> H (NN) (opr A (+H) B)) (br H (~~>v) (cv z))) (/\ (/\ (e. (cv x) A) (e. (cv y) (` (_|_) A))) (= (cv z) (opr (cv x) (+v) (cv y))))) (e. (cv y) B))) stmt (osumlem7 ((x y) (x z) (h x) (A x) (y z) (h y) (A y) (h z) (A z) (A h) (B x) (B y) (B z) (B h)) ((e. A (CH)) (e. B (CH)) (C_ B (` (_|_) A))) (e. (opr A (+H) B) (CH))) stmt (osumlem8 () ((e. A (CH)) (e. B (CH))) (-> (C_ B (` (_|_) A)) (e. (opr A (+H) B) (CH)))) stmt (osum () ((e. A (CH)) (e. B (CH))) (-> (C_ A (` (_|_) B)) (= (opr A (+H) B) (opr A (vH) B)))) stmt (osumcor () ((e. A (CH)) (e. B (CH))) (= (opr (i^i A B) (+H) (i^i A (` (_|_) B))) (opr (i^i A B) (vH) (i^i A (` (_|_) B))))) stmt (osumt () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (C_ A (` (_|_) B))) (= (opr A (+H) B) (opr A (vH) B)))) stmt (chsot () () (-> (e. A (CH)) (= (opr A (+H) (` (_|_) A)) (H~)))) stmt (osumcor2 () ((e. A (CH)) (e. B (CH))) (-> (br A (C_H) B) (= (opr A (+H) B) (opr A (vH) B)))) stmt (spansnj () ((e. A (CH)) (e. B (H~))) (= (opr A (+H) (` (span) ({} B))) (opr A (vH) (` (span) ({} B))))) stmt (spansnjt () () (-> (/\ (e. A (CH)) (e. 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C (H~))) (-> (/\ (C: A B) (C_ B (opr A (vH) (` (span) ({} C))))) (= B (opr A (vH) (` (span) ({} C))))))) stmt (5oalem1 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. R (SH))) (-> (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (e. (cv z) C) (e. (opr (cv x) (-v) (cv z)) R))) (e. (cv v) (opr B (+H) (i^i A (opr C (+H) R)))))) stmt (5oalem2 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH))) (-> (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (= (opr (cv x) (+v) (cv y)) (opr (cv z) (+v) (cv w)))) (e. (opr (cv x) (-v) (cv z)) (i^i (opr A (+H) C) (opr B (+H) D))))) stmt (5oalem3 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g))))) (e. (opr (cv x) (-v) (cv z)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G)))))) stmt (5oalem4 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (e. (cv f) F) (e. (cv g) G))) (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv f) (+v) (cv g))) (= (opr (cv z) (+v) (cv w)) (opr (cv f) (+v) (cv g))))) (e. (opr (cv x) (-v) (cv z)) (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) F) (opr B (+H) G)) (+H) (i^i (opr C (+H) F) (opr D (+H) G))))))) stmt (5oalem5 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH)) (e. R (SH)) (e. S (SH))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (/\ (e. (cv z) C) (e. (cv w) D))) (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (/\ (e. (cv v) R) (e. (cv u) S)))) (/\ (/\ (= (opr (cv x) (+v) (cv y)) (opr (cv v) (+v) (cv u))) (= (opr (cv z) (+v) (cv w)) (opr (cv v) (+v) (cv u)))) (= (opr (cv f) (+v) (cv g)) (opr (cv v) (+v) (cv u))))) (e. (opr (cv x) (-v) (cv z)) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))))))))) stmt (5oalem6 () ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH)) (e. R (SH)) (e. S (SH))) (-> (/\ (/\ (/\ (/\ (e. (cv x) A) (e. (cv y) B)) (= (cv h) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. (cv w) D)) (= (cv h) (opr (cv z) (+v) (cv w))))) (/\ (/\ (/\ (e. (cv f) F) (e. (cv g) G)) (= (cv h) (opr (cv f) (+v) (cv g)))) (/\ (/\ (e. (cv v) R) (e. (cv u) S)) (= (cv h) (opr (cv v) (+v) (cv u)))))) (e. (cv h) (opr B (+H) (i^i A (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))))))))))) stmt (5oalem7 ((A h) (A f) (A g) (A u) (A v) (A w) (A x) (A y) (A z) (f h) (g h) (h u) (h v) (h w) (h x) (h y) (h z) (f g) (f u) (f v) (f w) (f x) (f y) (f z) (g u) (g v) (g w) (g x) (g y) (g z) (u v) (u w) (u x) (u y) (u z) (v w) (v x) (v y) (v z) (w x) (w y) (w z) (x y) (x z) (y z) (B h) (B f) (B g) (B u) (B v) (B w) (B x) (B y) (B z) (C h) (C f) (C g) (C u) (C v) (C w) (C x) (C y) (C z) (D h) (D f) (D g) (D u) (D v) (D w) (D x) (D y) (D z) (F h) (F f) (F g) (F u) (F v) (F w) (F x) (F y) (F z) (G h) (G f) (G g) (G u) (G v) (G w) (G x) (G y) (G z) (R h) (R f) (R g) (R u) (R v) (R w) (R x) (R y) (R z) (S h) (S f) (S g) (S u) (S v) (S w) (S x) (S y) (S z)) ((e. A (SH)) (e. B (SH)) (e. C (SH)) (e. D (SH)) (e. F (SH)) (e. G (SH)) (e. R (SH)) (e. S (SH))) (C_ (i^i (i^i (opr A (+H) B) (opr C (+H) D)) (i^i (opr F (+H) G) (opr R (+H) S))) (opr B (+H) (i^i A (opr C (+H) (i^i (i^i (i^i (opr A (+H) C) (opr B (+H) D)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr C (+H) R) (opr D (+H) S)))) (opr (i^i (i^i (opr A (+H) F) (opr B (+H) G)) (opr (i^i (opr A (+H) R) (opr B (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S)))) (+H) (i^i (i^i (opr C (+H) F) (opr D (+H) G)) (opr (i^i (opr C (+H) R) (opr D (+H) S)) (+H) (i^i (opr F (+H) R) (opr G (+H) S))))))))))) stmt (5oa () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (e. D (CH)) (e. F (CH)) (e. G (CH)) (e. R (CH)) (e. S (CH)) (C_ A (` (_|_) B)) (C_ C (` (_|_) D)) (C_ F (` (_|_) G)) (C_ R (` (_|_) S))) (C_ (i^i (i^i (opr A (vH) B) (opr C (vH) D)) (i^i (opr F (vH) G) (opr R (vH) S))) (opr B (vH) (i^i A (opr C (vH) (i^i (i^i (i^i (opr A (vH) C) (opr B (vH) D)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr C (vH) R) (opr D (vH) S)))) (opr (i^i (i^i (opr A (vH) F) (opr B (vH) G)) (opr (i^i (opr A (vH) R) (opr B (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S)))) (vH) (i^i (i^i (opr C (vH) F) (opr D (vH) G)) (opr (i^i (opr C (vH) R) (opr D (vH) S)) (vH) (i^i (opr F (vH) R) (opr G (vH) S))))))))))) stmt (3oalem1 ((x y) (x z) (w x) (v x) (B x) (y z) (w y) (v y) (B y) (w z) (v z) (B z) (v w) (B w) (B v) (C x) (C y) (C z) (C w) (C v) (R x) (R y) (R z) (R w) (R v) (S x) (S y) (S z) (S w) (S v)) ((e. B (CH)) (e. C (CH)) (e. R (CH)) (e. S (CH))) (-> (/\ (/\ (/\ (e. (cv x) B) (e. (cv y) R)) (= (cv v) (opr (cv x) (+v) (cv y)))) (/\ (/\ (e. (cv z) C) (e. 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(H~) (0H)) A) (0v)))) stmt (hoivalt () () (-> (e. A (H~)) (= (` (Iop) A) A))) stmt (hoico1t () () (-> (:--> T (H~) (H~)) (= (o. T (Iop)) T))) stmt (hoico2t () () (-> (:--> T (H~) (H~)) (= (o. (Iop) T) T))) stmt (hoaddclt ((x y) (S x) (S y) (T x) (T y)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (:--> (opr S (+op) T) (H~) (H~)))) stmt (homulclt ((x y) (A x) (A y) (T x) (T y)) () (-> (/\ (e. A (CC)) (:--> T (H~) (H~))) (:--> (opr A (.op) T) (H~) (H~)))) stmt (hoeqt ((T x) (U x)) () (-> (/\ (:--> T (H~) (H~)) (:--> U (H~) (H~))) (<-> (A.e. x (H~) (= (` T (cv x)) (` U (cv x)))) (= T U)))) stmt (hoeq ((S x) (T x)) ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (A.e. x (H~) (= (` S (cv x)) (` T (cv x)))) (= S T))) stmt (hoscl () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (e. A (H~)) (e. (` (opr S (+op) T) A) (H~)))) stmt (hodcl () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (e. A (H~)) (e. (` (opr S (-op) T) A) (H~)))) stmt (hoco () ((:--> S (H~) (H~)) (:--> T (H~) (H~))) (-> (e. 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(H~) (0H))) (= T U))) stmt (honpncan () ((:--> R (H~) (H~)) (:--> S (H~) (H~)) (:--> T (H~) (H~))) (= (opr (opr R (-op) S) (+op) (opr S (-op) T)) (opr R (-op) T))) stmt (ho01 ((x y) (T x) (T y)) ((:--> T (H~) (H~))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (0)))) (= T (X. (H~) (0H))))) stmt (ho02 ((x y) (T x) (T y)) ((:--> T (H~) (H~))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (0)))) (= T (X. (H~) (0H))))) stmt (hoeq1t ((x y) (S x) (S y) (T x) (T y)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (` S (cv x)) (.i) (cv y)) (opr (` T (cv x)) (.i) (cv y))))) (= S T)))) stmt (hoeq2t ((x y) (S x) (S y) (T x) (T y)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` S (cv y))) (opr (cv x) (.i) (` T (cv y)))))) (= S T)))) stmt (adjmo ((x y) (u v) (u x) (u y) (T u) (v x) (v y) (T v) (T x) (T y)) () (E* u (/\ (:--> (cv u) (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` (cv u) (cv x)) (.i) (cv y)))))))) stmt (adjsymt ((x y) (x z) (S x) (y z) (S y) (S z) (T x) (T y) (T z)) () (-> (/\ (:--> S (H~) (H~)) (:--> T (H~) (H~))) (<-> (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` S (cv y))) (opr (` T (cv x)) (.i) (cv y))))) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` S (cv x)) (.i) (cv y)))))))) stmt (eigre () ((e. A (H~)) (e. B (CC))) (-> (/\ (= (` T A) (opr B (.s) A)) (=/= A (0v))) (<-> (= (opr A (.i) (` T A)) (opr (` T A) (.i) A)) (e. B (RR))))) stmt (eigret () () (-> (/\ (/\ (e. A (H~)) (e. B (CC))) (/\ (= (` T A) (opr B (.s) A)) (=/= A (0v)))) (<-> (= (opr A (.i) (` T A)) (opr (` T A) (.i) A)) (e. B (RR))))) stmt (eigpos () ((e. A (H~)) (e. B (CC))) (-> (/\ (/\ (e. (opr A (.i) (` T A)) (RR)) (br (0) (<_) (opr A (.i) (` T A)))) (/\ (= (` T A) (opr B (.s) A)) (=/= A (0v)))) (/\ (e. B (RR)) (br (0) (<_) B)))) stmt (eigorth () ((e. A (H~)) (e. B (H~)) (e. C (CC)) (e. D (CC))) (-> (/\ (/\ (= (` T A) (opr C (.s) A)) (= (` T B) (opr D (.s) B))) (-. (= C (` (*) D)))) (<-> (= (opr A (.i) (` T B)) (opr (` T A) (.i) B)) (= (opr A (.i) B) (0))))) stmt (eigortht () () (-> (/\ (/\ (/\ (e. A (H~)) (e. B (H~))) (/\ (e. C (CC)) (e. D (CC)))) (/\ (/\ (= (` T A) (opr C (.s) A)) (= (` T B) (opr D (.s) B))) (-. (= C (` (*) D))))) (<-> (= (opr A (.i) (` T B)) (opr (` T A) (.i) B)) (= (opr A (.i) B) (0))))) stmt (nmopvalt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (:--> T (H~) (H~)) (= (` (normop) T) (sup ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) (RR*) (<))))) stmt (elcnopt ((t w) (t x) (t y) (t z) (T t) (w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) () (<-> (e. T (ConOp)) (/\ (:--> T (H~) (H~)) (A.e. x (H~) (A.e. y (RR) (-> (br (0) (<) (cv y)) (E.e. z (RR) (/\ (br (0) (<) (cv z)) (A.e. w (H~) (-> (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (cv z)) (br (` (norm) (opr (` T (cv w)) (-v) (` T (cv x)))) (<) (cv y)))))))))))) stmt (ellnopt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (<-> (e. T (LinOp)) (/\ (:--> T (H~) (H~)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (.s) (` T (cv y))) (+v) (` T (cv z)))))))))) stmt (lnopft ((x y) (x z) (T x) (y z) (T y) (T z)) () (-> (e. T (LinOp)) (:--> T (H~) (H~)))) stmt (elbdopt ((T t)) () (<-> (e. T (BndLinOp)) (/\ (e. T (LinOp)) (br (` (normop) T) (<) (+oo))))) stmt (bdoplnt () () (-> (e. T (BndLinOp)) (e. T (LinOp)))) stmt (bdopft () () (-> (e. T (BndLinOp)) (:--> T (H~) (H~)))) stmt (nmopsetret ((x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (C_ ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))) (RR)))) stmt (nmopsetn0 ((x y) (T x) (T y)) () (e. (` (norm) (` T (0v))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (norm) (` T (cv y))))))))) stmt (nmopxrt ((x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (e. (` (normop) T) (RR*)))) stmt (nmoprepnf ((x y) (T x) (T y)) () (-> (:--> T (H~) (H~)) (<-> (e. (` (normop) T) (RR)) (-. (= (` (normop) T) (+oo)))))) stmt (nmopgtmnf () () (-> (:--> T (H~) (H~)) (br (-oo) (<) (` (normop) T)))) stmt (nmopreltpnf () () (-> (:--> T (H~) (H~)) (<-> (e. (` (normop) T) (RR)) (br (` (normop) T) (<) (+oo))))) stmt (nmopret () () (-> (e. T (BndLinOp)) (e. (` (normop) T) (RR)))) stmt (elbdop2t () () (<-> (e. T (BndLinOp)) (/\ (e. T (LinOp)) (e. (` (normop) T) (RR))))) stmt (elunopt ((t x) (t y) (T t) (x y) (T x) (T y)) () (<-> (e. T (UniOp)) (/\ (:-onto-> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (` T (cv y))) (opr (cv x) (.i) (cv y)))))))) stmt (elhmopt ((t x) (t y) (T t) (x y) (T x) (T y)) () (<-> (e. T (HrmOp)) (/\ (:--> T (H~) (H~)) (A.e. x (H~) (A.e. y (H~) (= (opr (cv x) (.i) (` T (cv y))) (opr (` T (cv x)) (.i) (cv y)))))))) stmt (hmopft ((x y) (T x) (T y)) () (-> (e. T (HrmOp)) (:--> T (H~) (H~)))) stmt (hmopex () () (e. (HrmOp) (V))) stmt (nmfnvalt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (:--> T (H~) (CC)) (= (` (normfn) T) (sup ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) (RR*) (<))))) stmt (nmfnsetret ((x y) (T x) (T y)) () (-> (:--> T (H~) (CC)) (C_ ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))) (RR)))) stmt (nmfnsetn0 ((x y) (T x) (T y)) () (e. (` (abs) (` T (0v))) ({|} x (E.e. y (H~) (/\ (br (` (norm) (cv y)) (<_) (1)) (= (cv x) (` (abs) (` T (cv y))))))))) stmt (nmfnxrt ((x y) (T x) (T y)) () (-> (:--> T (H~) (CC)) (e. (` (normfn) T) (RR*)))) stmt (nmfnrepnf ((x y) (T x) (T y)) () (-> (:--> T (H~) (CC)) (<-> (e. (` (normfn) T) (RR)) (-. (= (` (normfn) T) (+oo)))))) stmt (nlfnvalt ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (:--> T (H~) (CC)) (= (` (null) T) ({e.|} x (H~) (= (` T (cv x)) (0)))))) stmt (elcnfnt ((t w) (t x) (t y) (t z) (T t) (w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) () (<-> (e. T (ConFn)) (/\ (:--> T (H~) (CC)) (A.e. x (H~) (A.e. y (RR) (-> (br (0) (<) (cv y)) (E.e. z (RR) (/\ (br (0) (<) (cv z)) (A.e. w (H~) (-> (br (` (norm) (opr (cv w) (-v) (cv x))) (<) (cv z)) (br (` (abs) (opr (` T (cv w)) (-) (` T (cv x)))) (<) (cv y)))))))))))) stmt (ellnfnt ((t x) (t y) (t z) (T t) (x y) (x z) (T x) (y z) (T y) (T z)) () (<-> (e. T (LinFn)) (/\ (:--> T (H~) (CC)) (A.e. x (CC) (A.e. y (H~) (A.e. z (H~) (= (` T (opr (opr (cv x) (.s) (cv y)) (+v) (cv z))) (opr (opr (cv x) (x.) (` T (cv y))) (+) (` T (cv z)))))))))) stmt (lnfnft ((x y) (x z) (T x) (y z) (T y) (T z)) () (-> (e. 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(` (` (eigval) T) A) (CC)))) stmt (eigvect () () (-> (/\ (:--> T (H~) (H~)) (e. A (` (eigvec) T))) (/\ (= (` T A) (opr (` (` (eigval) T) A) (.s) A)) (=/= A (0v))))) stmt (eighmret () () (-> (/\ (e. T (HrmOp)) (e. A (` (eigvec) T))) (e. (` (` (eigval) T) A) (RR)))) stmt (eighmortht () () (-> (/\ (/\ (e. T (HrmOp)) (e. A (` (eigvec) T))) (/\ (e. B (` (eigvec) T)) (-. (= (` (` (eigval) T) A) (` (` (eigval) T) B))))) (= (opr A (.i) B) (0)))) stmt (nmopneg ((x y) (T x) (T y)) ((:--> T (H~) (H~))) (= (` (normop) (opr (-u (1)) (.op) T)) (` (normop) T))) stmt (lnop0t () () (-> (e. T (LinOp)) (= (` T (0v)) (0v)))) stmt (lnopmult () () (-> (/\/\ (e. T (LinOp)) (e. A (CC)) (e. B (H~))) (= (` T (opr A (.s) B)) (opr A (.s) (` T B))))) stmt (lnopl () ((e. T (LinOp))) (-> (/\/\ (e. A (CC)) (e. B (H~)) (e. C (H~))) (= (` T (opr (opr A (.s) B) (+v) C)) (opr (opr A (.s) (` T B)) (+v) (` T C))))) stmt (lnopf () ((e. T (LinOp))) (:--> T (H~) (H~))) stmt (lnop0 () ((e. 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T (ConFn)) (E.e. x (RR) (A.e. y (H~) (br (` (abs) (` T (cv y))) (<_) (opr (cv x) (x.) (` (norm) (cv y))))))))) stmt (lnfncnbdt ((x y) (T x) (T y)) () (-> (e. T (LinFn)) (<-> (e. T (ConFn)) (e. (` (normfn) T) (RR))))) stmt (nlelsh ((x y) (T x) (T y)) ((e. T (LinFn))) (e. (` (null) T) (SH))) stmt (nlelch ((g h) (g m) (g n) (g v) (g x) (g y) (T g) (h m) (h n) (h v) (h x) (h y) (T h) (m n) (m v) (m x) (m y) (T m) (n v) (n x) (n y) (T n) (v x) (v y) (T v) (x y) (T x) (T y)) ((e. T (LinFn)) (e. T (ConFn))) (e. (` (null) T) (CH))) stmt (riesz3 ((u v) (u w) (T u) (v w) (T v) (T w)) ((e. T (LinFn)) (e. T (ConFn))) (E.e. w (H~) (A.e. v (H~) (= (` T (cv v)) (opr (cv v) (.i) (cv w)))))) stmt (riesz4 ((u v) (u w) (T u) (v w) (T v) (T w)) ((e. T (LinFn)) (e. T (ConFn))) (E!e. w (H~) (A.e. v (H~) (= (` T (cv v)) (opr (cv v) (.i) (cv w)))))) stmt (riesz4t ((v w) (T w) (T v)) () (-> (e. 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G (ConFn))))) stmt (cnlnadjlem3 ((g h) (g u) (g v) (g w) (g y) (h u) (h v) (h w) (h y) (u v) (u w) (u y) (v w) (v y) (w y) (B u) (F w) (G v) (G w) (T g) (T h) (T u) (T v) (T w) (T y)) ((e. T (LinOp)) (e. T (ConOp)) (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))) (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w))))))) (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B))))) (-> (e. (cv y) (H~)) (e. B (H~)))) stmt (cnlnadjlem4 ((g h) (g u) (g v) (g w) (g y) (A g) (h u) (h v) (h w) (h y) (A h) (u v) (u w) (u y) (A u) (v w) (v y) (A v) (w y) (A w) (A y) (B u) (F w) (G v) (G w) (T g) (T h) (T u) (T v) (T w) (T y)) ((e. T (LinOp)) (e. T (ConOp)) (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))) (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w))))))) (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B))))) (-> (e. A (H~)) (e. (` F A) (H~)))) stmt (cnlnadjlem5 ((f g) (f h) (f u) (f v) (f w) (f y) (f z) (A f) (g h) (g u) (g v) (g w) (g y) (g z) (A g) (h u) (h v) (h w) (h y) (h z) (A h) (u v) (u w) (u y) (u z) (A u) (v w) (v y) (v z) (A v) (w y) (w z) (A w) (y z) (A y) (A z) (B u) (C f) (F f) (F w) (F z) (G f) (G v) (G w) (G z) (T f) (T g) (T h) (T u) (T v) (T w) (T y) (T z)) ((e. T (LinOp)) (e. T (ConOp)) (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))) (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w))))))) (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B))))) (-> (/\ (e. A (H~)) (e. C (H~))) (= (opr (` T C) (.i) A) (opr C (.i) (` F A))))) stmt (cnlnadjlem6 ((f g) (f h) (f u) (f v) (f w) (f y) (f z) (g h) (g u) (g v) (g w) (g y) (g z) (h u) (h v) (h w) (h y) (h z) (u v) (u w) (u y) (u z) (v w) (v y) (v z) (w y) (w z) (y z) (B u) (f t) (f x) (F f) (t w) (t x) (t z) (F t) (w x) (F w) (x z) (F x) (F z) (G f) (v x) (G v) (G w) (G x) (G z) (T f) (g t) (g x) (T g) (h t) (h x) (T h) (t u) (t v) (t y) (T t) (u x) (T u) (T v) (T w) (x y) (T x) (T y) (T z)) ((e. T (LinOp)) (e. T (ConOp)) (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))) (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w))))))) (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B))))) (e. F (LinOp))) stmt (cnlnadjlem7 ((g h) (g u) (g v) (g w) (g y) (A g) (h u) (h v) (h w) (h y) (A h) (u v) (u w) (u y) (A u) (v w) (v y) (A v) (w y) (A w) (A y) (B u) (F w) (G v) (G w) (T g) (T h) (T u) (T v) (T w) (T y)) ((e. T (LinOp)) (e. T (ConOp)) (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))) (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w))))))) (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B))))) (-> (e. A (H~)) (br (` (norm) (` F A)) (<_) (opr (` (normop) T) (x.) (` (norm) A))))) stmt (cnlnadjlem8 ((g h) (g u) (g v) (g w) (g y) (g z) (h u) (h v) (h w) (h y) (h z) (u v) (u w) (u y) (u z) (v w) (v y) (v z) (w y) (w z) (y z) (B u) (w x) (F w) (x z) (F x) (F z) (v x) (G v) (G w) (G x) (G z) (g x) (T g) (h x) (T h) (u x) (T u) (T v) (T w) (x y) (T x) (T y) (T z)) ((e. T (LinOp)) (e. T (ConOp)) (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))) (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w))))))) (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B))))) (e. F (ConOp))) stmt (cnlnadjlem9 ((g h) (g u) (g v) (g w) (g y) (g z) (h u) (h v) (h w) (h y) (h z) (u v) (u w) (u y) (u z) (v w) (v y) (v z) (w y) (w z) (y z) (B u) (t w) (t x) (t z) (F t) (w x) (F w) (x z) (F x) (F z) (v x) (G v) (G w) (G x) (G z) (g t) (g x) (T g) (h t) (h x) (T h) (t u) (t v) (t y) (T t) (u x) (T u) (T v) (T w) (x y) (T x) (T y) (T z)) ((e. T (LinOp)) (e. T (ConOp)) (= G ({<,>|} g h (/\ (e. (cv g) (H~)) (= (cv h) (opr (` T (cv g)) (.i) (cv y)))))) (= B (U. ({e.|} w (H~) (A.e. v (H~) (= (opr (` T (cv v)) (.i) (cv y)) (opr (cv v) (.i) (cv w))))))) (= F ({<,>|} y u (/\ (e. (cv y) (H~)) (= (cv u) B))))) (E.e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. z (H~) (= (opr (` T (cv x)) (.i) (cv z)) (opr (cv x) (.i) (` (cv t) (cv z)))))))) stmt (cnlnadj ((f g) (f h) (f t) (f u) (f v) (f w) (f x) (f y) (f z) (T f) (g h) (g t) (g u) (g v) (g w) (g x) (g y) (g z) (T g) (h t) (h u) (h v) (h w) (h x) (h y) (h z) (T h) (t u) (t v) (t w) (t x) (t y) (t z) (T t) (u v) (u w) (u x) (u y) (u z) (T u) (v w) (v x) (v y) (v z) (T v) (w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) ((e. T (LinOp)) (e. T (ConOp))) (E.e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))))) stmt (cnlnadjeu ((t x) (t y) (T t) (x y) (T x) (T y)) ((e. T (LinOp)) (e. T (ConOp))) (E!e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y)))))))) stmt (cnlnadjeut ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (e. T (i^i (LinOp) (ConOp))) (E!e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y))))))))) stmt (cnlnadjt ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (e. T (i^i (LinOp) (ConOp))) (E.e. t (i^i (LinOp) (ConOp)) (A.e. x (H~) (A.e. y (H~) (= (opr (` T (cv x)) (.i) (cv y)) (opr (cv x) (.i) (` (cv t) (cv y))))))))) stmt (cnlnssadj ((t u) (t v) (t x) (t y) (t z) (u v) (u x) (u y) (u z) (v x) (v y) (v z) (x y) (x z) (y z)) () (C_ (i^i (LinOp) (ConOp)) (dom (adj)))) stmt (bdopssadj () () (C_ (BndLinOp) (dom (adj)))) stmt (bdopadjt () () (-> (e. T (BndLinOp)) (e. T (dom (adj))))) stmt (adjbdlnt ((t x) (t y) (T t) (x y) (T x) (T y)) () (-> (e. T (BndLinOp)) (e. (` (adj) T) (BndLinOp)))) stmt (adjbdlnb () () (<-> (e. T (BndLinOp)) (e. (` (adj) T) (BndLinOp)))) stmt (adjlnopt ((w x) (w y) (w z) (T w) (x y) (x z) (T x) (y z) (T y) (T z)) () (-> (e. T (dom (adj))) (e. 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(opr S (-op) T) (BndLinOp))) stmt (bdopco () ((e. S (BndLinOp)) (e. T (BndLinOp))) (e. (o. S T) (BndLinOp))) stmt (adjco ((x y) (S x) (S y) (T x) (T y)) ((e. S (BndLinOp)) (e. T (BndLinOp))) (= (` (adj) (o. S T)) (o. (` (adj) T) (` (adj) S)))) stmt (nmopcoadj ((T x)) ((e. T (BndLinOp))) (= (` (normop) (o. (` (adj) T) T)) (opr (` (normop) T) (^) (2)))) stmt (nmopcoadj2 () ((e. T (BndLinOp))) (= (` (normop) (o. T (` (adj) T))) (opr (` (normop) T) (^) (2)))) stmt (unierr () ((e. F (UniOp)) (e. G (UniOp)) (e. S (UniOp)) (e. T (UniOp))) (br (` (normop) (opr (o. F G) (-op) (o. S T))) (<_) (opr (` (normop) (opr F (-op) S)) (+) (` (normop) (opr G (-op) T))))) stmt (branmfnt ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w)) () (-> (e. A (H~)) (= (` (normfn) (` (bra) A)) (` (norm) A)))) stmt (brabnt () () (-> (e. A (H~)) (e. 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C (H~)) (e. D (H~)))) (= (o. (opr A (ketbra) B) (opr C (ketbra) D)) (opr (` (opr A (ketbra) B) C) (ketbra) D)))) stmt (leopg ((A x) (B x) (t u) (t x) (T t) (u x) (T u) (T x) (U t) (U u) (U x)) () (-> (/\ (e. T A) (e. U B)) (<-> (br T (<_op) U) (/\ (e. (opr U (-op) T) (HrmOp)) (A.e. x (H~) (br (0) (<_) (opr (` (opr U (-op) T) (cv x)) (.i) (cv x)))))))) stmt (leopt ((T x) (U x)) () (-> (/\ (e. T (HrmOp)) (e. U (HrmOp))) (<-> (br T (<_op) U) (A.e. x (H~) (br (0) (<_) (opr (` (opr U (-op) T) (cv x)) (.i) (cv x))))))) stmt (leop2t ((T x) (U x)) () (-> (/\ (e. T (HrmOp)) (e. U (HrmOp))) (<-> (br T (<_op) U) (A.e. x (H~) (br (opr (` T (cv x)) (.i) (cv x)) (<_) (opr (` U (cv x)) (.i) (cv x))))))) stmt (leop3t ((T x) (U x)) () (-> (/\ (e. T (HrmOp)) (e. U (HrmOp))) (<-> (br T (<_op) U) (br (0op) (<_op) (opr U (-op) T))))) stmt (leoppost ((T x)) () (-> (e. T (HrmOp)) (<-> (br (0op) (<_op) T) (A.e. x (H~) (br (0) (<_) (opr (` T (cv x)) (.i) (cv x))))))) stmt (leoprft ((T x)) () (-> (e. 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B (CH))) (-> ph (br (` S B) (<) (1)))) stmt (strlem6 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2)))))) (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1)))) (e. A (CH)) (e. B (CH))) (-> ph (-. (-> (= (` S A) (1)) (= (` S B) (1)))))) stmt (str ((x y) (u x) (f x) (A x) (u y) (f y) (A y) (f u) (A u) (A f) (B x) (B y) (B u) (B f)) ((e. A (CH)) (e. B (CH))) (-> (A.e. f (States) (-> (= (` (cv f) A) (1)) (= (` (cv f) B) (1)))) (C_ A B))) stmt (strb ((A f) (B f)) ((e. A (CH)) (e. B (CH))) (<-> (A.e. f (States) (-> (= (` (cv f) A) (1)) (= (` (cv f) B) (1)))) (C_ A B))) stmt (hstrlem2 ((x y) (C x) (C y) (u x) (u y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))))) (-> (e. C (CH)) (= (` S C) (` (` (proj) C) (cv u))))) stmt (hstrlem3a ((x z) (w x) (u x) (w z) (u z) (u w) (S z) (S w) (x y) (u y) (y z) (w y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u))))))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (e. S (CHStates)))) stmt (hstrlem3 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u)))))) (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1)))) (e. A (CH)) (e. B (CH))) (-> ph (e. S (CHStates)))) stmt (hstrlem4 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u)))))) (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1)))) (e. A (CH)) (e. B (CH))) (-> ph (= (` (norm) (` S A)) (1)))) stmt (hstrlem5 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u)))))) (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1)))) (e. A (CH)) (e. B (CH))) (-> ph (br (` (norm) (` S B)) (<) (1)))) stmt (hstrlem6 ((ph x) (x y) (u x) (u y) (A x) (A y) (B x) (B y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (` (` (proj) (cv x)) (cv u)))))) (<-> ph (/\ (e. (cv u) (\ A B)) (= (` (norm) (cv u)) (1)))) (e. A (CH)) (e. B (CH))) (-> ph (-. (-> (= (` (norm) (` S A)) (1)) (= (` (norm) (` S B)) (1)))))) stmt (hstr ((x y) (u x) (f x) (A x) (u y) (f y) (A y) (f u) (A u) (A f) (B x) (B y) (B u) (B f)) ((e. A (CH)) (e. B (CH))) (-> (A.e. f (CHStates) (-> (= (` (norm) (` (cv f) A)) (1)) (= (` (norm) (` (cv f) B)) (1)))) (C_ A B))) stmt (hstrb ((A f) (B f)) ((e. A (CH)) (e. B (CH))) (<-> (A.e. f (CHStates) (-> (= (` (norm) (` (cv f) A)) (1)) (= (` (norm) (` (cv f) B)) (1)))) (C_ A B))) stmt (large ((A f)) ((e. A (CH))) (<-> (-. (= A (0H))) (E.e. f (States) (= (` (cv f) A) (1))))) stmt (jplem1 () ((e. A (CH))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (e. (cv u) A) (= (opr (` (norm) (` (` (proj) A) (cv u))) (^) (2)) (1))))) stmt (jplem2 ((u x) (x y) (u y) (A x) (A y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2)))))) (e. A (CH))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (e. (cv u) A) (= (` S A) (1))))) stmt (jp ((u x) (x y) (u y) (A x) (A y) (B x) (B y)) ((= S ({<,>|} x y (/\ (e. (cv x) (CH)) (= (cv y) (opr (` (norm) (` (` (proj) (cv x)) (cv u))) (^) (2)))))) (e. A (CH)) (e. B (CH))) (-> (/\ (e. (cv u) (H~)) (= (` (norm) (cv u)) (1))) (<-> (/\ (= (` S A) (1)) (= (` S B) (1))) (= (` S (i^i A B)) (1))))) stmt (golem1 () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (= F (opr (` (_|_) A) (vH) (i^i A B))) (= G (opr (` (_|_) B) (vH) (i^i B C))) (= H (opr (` (_|_) C) (vH) (i^i C A))) (= D (opr (` (_|_) B) (vH) (i^i B A))) (= R (opr (` (_|_) C) (vH) (i^i C B))) (= S (opr (` (_|_) A) (vH) (i^i A C)))) (-> (e. (cv f) (States)) (= (opr (opr (` (cv f) F) (+) (` (cv f) G)) (+) (` (cv f) H)) (opr (opr (` (cv f) D) (+) (` (cv f) R)) (+) (` (cv f) S))))) stmt (golem2 () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (= F (opr (` (_|_) A) (vH) (i^i A B))) (= G (opr (` (_|_) B) (vH) (i^i B C))) (= H (opr (` (_|_) C) (vH) (i^i C A))) (= D (opr (` (_|_) B) (vH) (i^i B A))) (= R (opr (` (_|_) C) (vH) (i^i C B))) (= S (opr (` (_|_) A) (vH) (i^i A C)))) (-> (e. (cv f) (States)) (-> (= (` (cv f) (i^i (i^i F G) H)) (1)) (= (` (cv f) D) (1))))) stmt (goeq ((F f) (G f) (H f) (D f)) ((e. A (CH)) (e. B (CH)) (e. C (CH)) (= F (opr (` (_|_) A) (vH) (i^i A B))) (= G (opr (` (_|_) B) (vH) (i^i B C))) (= H (opr (` (_|_) C) (vH) (i^i C A))) (= D (opr (` (_|_) B) (vH) (i^i B A)))) (C_ (i^i (i^i F G) H) D)) stmt (stcltr1 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y))))))) (e. A (CH)) (e. B (CH))) (-> ph (-> (-> (= (` S A) (1)) (= (` S B) (1))) (C_ A B)))) stmt (stcltr2 ((x y) (A x) (A y) (S x) (S y)) ((<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y))))))) (e. A (CH))) (-> ph (-> (= (` S A) (1)) (= A (H~))))) stmt (stcltrlem1 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y))))))) (e. A (CH)) (e. B (CH))) (-> ph (-> (= (` S B) (1)) (= (` S (opr (` (_|_) A) (vH) (i^i A B))) (1))))) stmt (stcltrlem2 ((x y) (A x) (A y) (B x) (B y) (S x) (S y)) ((<-> ph (/\ (e. S (States)) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` S (cv x)) (1)) (= (` S (cv y)) (1))) (C_ (cv x) (cv y))))))) (e. A (CH)) (e. B (CH))) (-> ph (C_ B (opr (` (_|_) A) (vH) (i^i A B))))) stmt (stcltrth ((x y) (s x) (A x) (s y) (A y) (A s) (B x) (B y) (B s)) ((e. A (CH)) (e. B (CH)) (E.e. s (States) (A.e. x (CH) (A.e. y (CH) (-> (-> (= (` (cv s) (cv x)) (1)) (= (` (cv s) (cv y)) (1))) (C_ (cv x) (cv y))))))) (C_ B (opr (` (_|_) A) (vH) (i^i A B)))) stmt (cvbrt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A ( (/\ (e. A (CH)) (e. B (CH))) (<-> (br A ( (/\ (C: A (cv x)) (C_ (cv x) B)) (= (cv x) B))))))) stmt (cvcon3t ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A ( (/\ (e. A (CH)) (e. B (CH))) (-> (br A ( (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A ( (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A ( (/\ (C: A C) (C_ C B)) (= C B))))) stmt (cvnbtwn3t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A ( (/\ (C_ A C) (C: C B)) (= C A))))) stmt (cvnbtwn4t () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A ( (/\ (C_ A C) (C_ C B)) (\/ (= C A) (= C B)))))) stmt (cvnsymt () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (br A ( (e. A (CH)) (-. (br A ( (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (/\ (br A ( (/\ (e. A (CH)) (e. B (H~))) (-> (-. (C_ (` (span) ({} B)) A)) (br A ( (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)))))))) stmt (mdit ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (MH) B) (-> (C_ C B) (= (i^i (opr C (vH) A) B) (opr C (vH) (i^i A B))))))) stmt (mdbr2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (-> (C_ (cv x) B) (C_ (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)))))))) stmt (mdbr3 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (= (i^i (opr (i^i (cv x) B) (vH) A) B) (opr (i^i (cv x) B) (vH) (i^i A B))))))) stmt (mdbr4 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (C_ (i^i (opr (i^i (cv x) B) (vH) A) B) (opr (i^i (cv x) B) (vH) (i^i A B))))))) stmt (dmdbrt ((x y) (x z) (A x) (y z) (A y) (A z) (B x) (B y) (B z)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH*) B) (A.e. x (CH) (-> (C_ B (cv x)) (= (opr (i^i (cv x) A) (vH) B) (i^i (cv x) (opr A (vH) B)))))))) stmt (dmdmdt ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH*) B) (br (` (_|_) A) (MH) (` (_|_) B))))) stmt (dmdbrtOLD ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ B (cv x)) (= (opr (i^i (cv x) A) (vH) B) (i^i (cv x) (opr A (vH) B)))))))) stmt (mddmdt () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (br (` (_|_) A) (MH*) (` (_|_) B))))) stmt (dmdit ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br A (MH*) B) (-> (C_ B C) (= (opr (i^i C A) (vH) B) (i^i C (opr A (vH) B))))))) stmt (dmditOLD ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (-> (C_ B C) (= (opr (i^i C A) (vH) B) (i^i C (opr A (vH) B))))))) stmt (dmdbr2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ B (cv x)) (C_ (i^i (cv x) (opr A (vH) B)) (opr (i^i (cv x) A) (vH) B))))))) stmt (dmdi2 () () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (-> (C_ B C) (C_ (i^i C (opr A (vH) B)) (opr (i^i C A) (vH) B)))))) stmt (dmdbr3 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (= (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (i^i (opr (cv x) (vH) B) (opr A (vH) B))))))) stmt (dmdbr4 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)))))) stmt (dmdi4 ((A x) (B x) (C x)) () (-> (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (-> (br (` (_|_) A) (MH) (` (_|_) B)) (C_ (i^i (opr C (vH) B) (opr A (vH) B)) (opr (i^i (opr C (vH) B) A) (vH) B))))) stmt (dmdbr5 ((x y) (A x) (A y) (B x) (B y)) () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (CH) (-> (C_ (cv x) (opr A (vH) B)) (C_ (cv x) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))))))) stmt (mddmd ((x y) (A x) (A y)) () (-> (e. A (CH)) (<-> (A.e. x (CH) (br A (MH) (cv x))) (A.e. x (CH) (br (` (_|_) A) (MH) (` (_|_) (cv x))))))) stmt (mdsl0 ((A x) (B x) (C x) (D x)) () (-> (/\ (/\ (e. A (CH)) (e. B (CH))) (/\ (e. C (CH)) (e. D (CH)))) (-> (/\ (/\ (/\ (C_ C A) (C_ D B)) (= (i^i A B) (0H))) (br A (MH) B)) (br C (MH) D)))) stmt (ssmd1 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br A (MH) B)))) stmt (ssmd2 ((A x) (B x)) () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br B (MH) A)))) stmt (ssdmd1 () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br (` (_|_) A) (MH) (` (_|_) B))))) stmt (ssdmd2 () () (-> (/\ (e. A (CH)) (e. B (CH))) (-> (C_ A B) (br (` (_|_) B) (MH) (` (_|_) A))))) stmt (dmdsl3t () () (-> (/\ (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (/\/\ (br B (MH*) A) (C_ A C) (C_ C (opr A (vH) B)))) (= (opr (i^i C B) (vH) A) C))) stmt (mdsl3t () () (-> (/\ (/\/\ (e. A (CH)) (e. B (CH)) (e. C (CH))) (/\/\ (br A (MH) B) (C_ (i^i A B) C) (C_ C B))) (= (i^i (opr C (vH) A) B) C))) stmt (mdslle1 () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (e. D (CH))) (-> (/\/\ (br B (MH*) A) (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B))) (<-> (C_ C D) (C_ (i^i C B) (i^i D B))))) stmt (mdslj1 () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (e. D (CH))) (-> (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ A (i^i C D)) (C_ (opr C (vH) D) (opr A (vH) B)))) (= (i^i (opr C (vH) D) B) (opr (i^i C B) (vH) (i^i D B))))) stmt (mdslj2 () ((e. A (CH)) (e. B (CH)) (e. C (CH)) (e. D (CH))) (-> (/\ (/\ (br A (MH) B) (br B (MH*) A)) (/\ (C_ (i^i A B) (i^i C D)) (C_ (opr C (vH) D) B))) (= (opr (i^i C D) (vH) A) (i^i (opr C (vH) A) (opr D (vH) A))))) stmt (mdsl1 ((x y) (A x) (A y) (B x) (B y)) ((e. A (CH)) (e. B (CH))) (<-> (A.e. x (CH) (-> (/\ (C_ (i^i A B) (cv x)) (C_ (cv x) (opr A (vH) B))) (-> (C_ (cv x) B) (= (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B)))))) (br A (MH) B))) stmt (mdsl2 ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (A.e. x (CH) (-> (/\ (C_ (i^i A B) (cv x)) (C_ (cv x) B)) (C_ (i^i (opr (cv x) (vH) A) B) (opr (cv x) (vH) (i^i A B))))))) stmt (mdsl1OLD ((x y) (A x) (A y) (B x) (B y)) ((e. A (CH)) (e. 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(cv p) (Atoms)) (C_ (i^i B C) A)) (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (C_ (cv p) (opr A (vH) B)))) (-> (-. (= B (0H))) (E.e. r (Atoms) (E.e. q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B)))))))) stmt (mdsymlem3 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (/\ (/\ (e. (cv p) (Atoms)) (-. (C_ (i^i B C) A))) (C_ (cv p) (opr A (vH) B))) (-. (= A (0H)))) (E.e. r (Atoms) (E.e. q (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) stmt (mdsymlem4 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (e. (cv p) (Atoms)) (-> (/\ (br (` (_|_) B) (MH) (` (_|_) A)) (/\ (/\ (-. (= A (0H))) (-. (= B (0H)))) (C_ (cv p) (opr A (vH) B)))) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B)))))))) stmt (mdsymlem5 ((q r) (C q) (C r) (c p) (c q) (c r) (A c) (p q) (p r) (A p) (A q) (A r) (B c) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (e. (cv q) (Atoms)) (e. (cv r) (Atoms))) (-> (-. (= (cv q) (cv p))) (-> (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))) (-> (/\ (/\ (e. (cv c) (CH)) (C_ A (cv c))) (e. (cv p) (Atoms))) (-> (C_ (cv p) (cv c)) (C_ (cv p) (opr (i^i (cv c) B) (vH) A)))))))) stmt (mdsymlem6 ((q r) (C q) (C r) (c p) (c q) (c r) (A c) (p q) (p r) (A p) (A q) (A r) (B c) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (A.e. p (Atoms) (-> (C_ (cv p) (opr A (vH) B)) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B))))))) (br (` (_|_) B) (MH) (` (_|_) A)))) stmt (mdsymlem7 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (-. (= A (0H))) (-. (= B (0H)))) (<-> (br (` (_|_) B) (MH) (` (_|_) A)) (A.e. p (Atoms) (-> (C_ (cv p) (opr A (vH) B)) (E.e. q (Atoms) (E.e. r (Atoms) (/\ (C_ (cv p) (opr (cv q) (vH) (cv r))) (/\ (C_ (cv q) A) (C_ (cv r) B)))))))))) stmt (mdsymlem8 ((q r) (C q) (C r) (p q) (p r) (A p) (A q) (A r) (B p) (B q) (B r)) ((e. A (CH)) (e. B (CH)) (= C (opr A (vH) (cv p)))) (-> (/\ (-. (= A (0H))) (-. (= B (0H)))) (<-> (br (` (_|_) B) (MH) (` (_|_) A)) (br (` (_|_) A) (MH) (` (_|_) B))))) stmt (mdsym ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (br B (MH) A))) stmt (mdsymt () () (-> (/\ (e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (br B (MH) A)))) stmt (atdmd2 () () (-> (/\ (e. A (CH)) (e. B (Atoms))) (br (` (_|_) A) (MH) (` (_|_) B)))) stmt (sumdmdi ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (CH)) (e. B (CH))) (-> (= (opr A (+H) B) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B)))) stmt (cmmd () ((e. A (CH)) (e. B (CH))) (-> (br A (C_H) B) (br A (MH) B))) stmt (cmdmd () ((e. A (CH)) (e. B (CH))) (-> (br A (C_H) B) (br A (MH*) B))) stmt (sumdmdlem ((y z) (w y) (A y) (w z) (A z) (A w) (B y) (B z) (B w) (C y) (C z) (C w)) ((e. A (CH)) (e. B (CH))) (-> (/\ (e. C (H~)) (-. (e. C (opr A (+H) B)))) (= (i^i (opr B (+H) (` (span) ({} C))) A) (i^i B A)))) stmt (sumdmdlem2 ((x y) (A x) (A y) (B x) (B y)) ((e. A (CH)) (e. B (CH))) (-> (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))) (= (opr A (+H) B) (opr A (vH) B)))) stmt (sumdmd ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (= (opr A (+H) B) (opr A (vH) B)) (br (` (_|_) A) (MH) (` (_|_) B)))) stmt (dmdbr4at ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (C_ (i^i (opr (cv x) (vH) B) (opr A (vH) B)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))))) stmt (dmdbr5at ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (-> (C_ (cv x) (opr A (vH) B)) (C_ (cv x) (opr (i^i (opr (cv x) (vH) B) A) (vH) B)))))) stmt (dmdbr6at ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (= (i^i (opr A (vH) B) (cv x)) (i^i (opr (i^i (opr (cv x) (vH) B) A) (vH) B) (cv x)))))) stmt (dmdbr7at ((A x) (B x)) ((e. A (CH)) (e. B (CH))) (<-> (br (` (_|_) A) (MH) (` (_|_) B)) (A.e. x (Atoms) (C_ (i^i (opr A (vH) B) (cv x)) (opr (i^i (opr (cv x) (vH) B) A) (vH) B))))) stmt (mdoc1 () ((e. A (CH))) (br A (MH) (` (_|_) A))) stmt (mdoc2 () ((e. A (CH))) (br (` (_|_) A) (MH) A)) stmt (dmdoc1 () ((e. A (CH))) (br A (MH*) (` (_|_) A))) stmt (dmdoc2 () ((e. A (CH))) (br (` (_|_) A) (MH*) A)) stmt (mdcompl () ((e. A (CH)) (e. B (CH))) (<-> (br A (MH) B) (br (i^i A (` (_|_) (i^i A B))) (MH) (i^i B (` (_|_) (i^i A B)))))) stmt (dmdcompl () ((e. A (CH)) (e. B (CH))) (<-> (br A (MH*) B) (br (i^i A (` (_|_) (i^i A B))) (MH*) (i^i B (` (_|_) (i^i A B)))))) stmt (mddmdin0 ((x y) (A x) (A y) (B x) (B y)) ((e. A (CH)) (e. B (CH)) (A.e. x (CH) (A.e. y (CH) (-> (/\ (br (cv x) (MH*) (cv y)) (= (i^i (cv x) (cv y)) (0H))) (br (cv x) (MH) (cv y)))))) (-> (br A (MH*) B) (br A (MH) B))) stmt (cdjreu ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w)) ((e. A (SH)) (e. B (SH))) (-> (/\ (e. C (opr A (+H) B)) (= (i^i A B) (0H))) (E!e. x A (E.e. y B (= C (opr (cv x) (+v) (cv y))))))) stmt (cdj1 ((x y) (x z) (w x) (v x) (A x) (y z) (w y) (v y) (A y) (w z) (v z) (A z) (v w) (A w) (A v) (B x) (B y) (B z) (B w) (B v)) ((e. A (SH)) (e. B (SH))) (-> (E.e. w (RR) (/\ (br (0) (<) (cv w)) (A.e. y A (A.e. v B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv v))) (<_) (opr (cv w) (x.) (` (norm) (opr (cv y) (+v) (cv v))))))))) (E.e. x (RR) (/\ (br (0) (<) (cv x)) (A.e. y A (A.e. z B (-> (= (` (norm) (cv y)) (1)) (br (cv x) (<_) (` (norm) (opr (cv y) (-v) (cv z))))))))))) stmt (cdj3lem1 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w)) ((e. A (SH)) (e. B (SH))) (-> (E.e. x (RR) (/\ (br (0) (<) (cv x)) (A.e. y A (A.e. z B (br (opr (` (norm) (cv y)) (+) (` (norm) (cv z))) (<_) (opr (cv x) (x.) (` (norm) (opr (cv y) (+v) (cv z))))))))) (= (i^i A B) (0H)))) stmt (cdj3lem2 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w)) ((e. A (SH)) (e. B (SH)) (= S ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} z A (E.e. w B (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (-> (/\/\ (e. C A) (e. D B) (= (i^i A B) (0H))) (= (` S (opr C (+v) D)) C))) stmt (cdj3lem2a ((u v) (S v) (S u) (x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((e. A (SH)) (e. B (SH)) (= S ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} z A (E.e. w B (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (-> (/\ (e. C (opr A (+H) B)) (= (i^i A B) (0H))) (e. (` S C) A))) stmt (cdj3lem2b ((u v) (t v) (h v) (S v) (t u) (h u) (S u) (h t) (S t) (S h) (x y) (x z) (w x) (v x) (u x) (t x) (h x) (A x) (y z) (w y) (v y) (u y) (t y) (h y) (A y) (w z) (v z) (u z) (t z) (h z) (A z) (v w) (u w) (t w) (h w) (A w) (u v) (t v) (h v) (A v) (t u) (h u) (A u) (h t) (A t) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B h)) ((e. A (SH)) (e. B (SH)) (= S ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} z A (E.e. w B (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (-> (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. x A (A.e. y B (br (opr (` (norm) (cv x)) (+) (` (norm) (cv y))) (<_) (opr (cv v) (x.) (` (norm) (opr (cv x) (+v) (cv y))))))))) (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. u (opr A (+H) B) (br (` (norm) (` S (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u))))))))) stmt (cdj3lem3 ((x y) (x z) (w x) (A x) (y z) (w y) (A y) (w z) (A z) (A w) (B x) (B y) (B z) (B w) (C x) (C y) (C z) (C w) (D x) (D y) (D z) (D w)) ((e. A (SH)) (e. B (SH)) (= T ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} w B (E.e. z A (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (-> (/\/\ (e. C A) (e. D B) (= (i^i A B) (0H))) (= (` T (opr C (+v) D)) D))) stmt (cdj3lem3a ((u v) (T v) (T u) (x y) (x z) (w x) (v x) (u x) (A x) (y z) (w y) (v y) (u y) (A y) (w z) (v z) (u z) (A z) (v w) (u w) (A w) (u v) (A v) (A u) (B x) (B y) (B z) (B w) (B v) (B u) (C x) (C y) (C z) (C w) (C v) (C u)) ((e. A (SH)) (e. B (SH)) (= T ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} w B (E.e. z A (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (-> (/\ (e. C (opr A (+H) B)) (= (i^i A B) (0H))) (e. (` T C) B))) stmt (cdj3lem3b ((u v) (t v) (h v) (T v) (t u) (h u) (T u) (h t) (T t) (T h) (x y) (x z) (w x) (v x) (u x) (t x) (h x) (A x) (y z) (w y) (v y) (u y) (t y) (h y) (A y) (w z) (v z) (u z) (t z) (h z) (A z) (v w) (u w) (t w) (h w) (A w) (u v) (t v) (h v) (A v) (t u) (h u) (A u) (h t) (A t) (A h) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B h)) ((e. A (SH)) (e. B (SH)) (= T ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} w B (E.e. z A (= (cv x) (opr (cv z) (+v) (cv w))))))))))) (-> (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. x A (A.e. y B (br (opr (` (norm) (cv x)) (+) (` (norm) (cv y))) (<_) (opr (cv v) (x.) (` (norm) (opr (cv x) (+v) (cv y))))))))) (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. u (opr A (+H) B) (br (` (norm) (` T (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u))))))))) stmt (cdj3 ((x y) (x z) (w x) (v x) (u x) (t x) (h x) (f x) (g x) (A x) (y z) (w y) (v y) (u y) (t y) (h y) (f y) (g y) (A y) (w z) (v z) (u z) (t z) (h z) (f z) (g z) (A z) (v w) (u w) (t w) (h w) (f w) (g w) (A w) (u v) (t v) (h v) (f v) (g v) (A v) (t u) (h u) (f u) (g u) (A u) (h t) (f t) (g t) (A t) (f h) (g h) (A h) (f g) (A f) (A g) (B x) (B y) (B z) (B w) (B v) (B u) (B t) (B h) (B f) (B g) (S v) (S u) (S t) (S h) (S f) (S g) (T v) (T u) (T t) (T h) (T f) (T g)) ((e. A (SH)) (e. B (SH)) (= S ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} z A (E.e. w B (= (cv x) (opr (cv z) (+v) (cv w)))))))))) (= T ({<,>|} x y (/\ (e. (cv x) (opr A (+H) B)) (= (cv y) (U. ({e.|} w B (E.e. z A (= (cv x) (opr (cv z) (+v) (cv w)))))))))) (<-> ph (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. u (opr A (+H) B) (br (` (norm) (` S (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u)))))))) (<-> ps (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. u (opr A (+H) B) (br (` (norm) (` T (cv u))) (<_) (opr (cv v) (x.) (` (norm) (cv u))))))))) (<-> (E.e. v (RR) (/\ (br (0) (<) (cv v)) (A.e. x A (A.e. y B (br (opr (` (norm) (cv x)) (+) (` (norm) (cv y))) (<_) (opr (cv v) (x.) (` (norm) (opr (cv x) (+v) (cv y))))))))) (/\/\ (= (i^i A B) (0H)) ph ps))) # 2038331 Metamath steps converted to 323986 Ghilbert steps