ITP001 Axioms: ITP135+5.ax
%------------------------------------------------------------------------------
% File : ITP135+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : complex+2.ax [Gau20]
% : HL4135+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 261 ( 29 unt; 0 def)
% Number of atoms : 794 ( 328 equ)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 580 ( 47 ~; 8 |; 19 &)
% ( 34 <=>; 472 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 54 ( 54 usr; 47 con; 0-2 aty)
% Number of variables : 434 ( 434 !; 0 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Ecomplex_2EIM,axiom,
mem(c_2Ecomplex_2EIM,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
fof(mem_c_2Ecomplex_2ERE,axiom,
mem(c_2Ecomplex_2ERE,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
fof(mem_c_2Ecomplex_2Earg,axiom,
mem(c_2Ecomplex_2Earg,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
fof(mem_c_2Ecomplex_2Ecomplex__add,axiom,
mem(c_2Ecomplex_2Ecomplex__add,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ).
fof(mem_c_2Ecomplex_2Ecomplex__div,axiom,
mem(c_2Ecomplex_2Ecomplex__div,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ).
fof(mem_c_2Ecomplex_2Ecomplex__exp,axiom,
mem(c_2Ecomplex_2Ecomplex__exp,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Ecomplex_2Ecomplex__inv,axiom,
mem(c_2Ecomplex_2Ecomplex__inv,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Ecomplex_2Ecomplex__mul,axiom,
mem(c_2Ecomplex_2Ecomplex__mul,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ).
fof(mem_c_2Ecomplex_2Ecomplex__neg,axiom,
mem(c_2Ecomplex_2Ecomplex__neg,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Ecomplex_2Ecomplex__of__num,axiom,
mem(c_2Ecomplex_2Ecomplex__of__num,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Ecomplex_2Ecomplex__of__real,axiom,
mem(c_2Ecomplex_2Ecomplex__of__real,arr(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Ecomplex_2Ecomplex__pow,axiom,
mem(c_2Ecomplex_2Ecomplex__pow,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ).
fof(mem_c_2Ecomplex_2Ecomplex__scalar__lmul,axiom,
mem(c_2Ecomplex_2Ecomplex__scalar__lmul,arr(ty_2Erealax_2Ereal,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ).
fof(mem_c_2Ecomplex_2Ecomplex__scalar__rmul,axiom,
mem(c_2Ecomplex_2Ecomplex__scalar__rmul,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ).
fof(mem_c_2Ecomplex_2Ecomplex__sub,axiom,
mem(c_2Ecomplex_2Ecomplex__sub,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ).
fof(mem_c_2Ecomplex_2Econj,axiom,
mem(c_2Ecomplex_2Econj,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Ecomplex_2Ei,axiom,
mem(c_2Ecomplex_2Ei,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
fof(mem_c_2Ecomplex_2Emodu,axiom,
mem(c_2Ecomplex_2Emodu,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__LEMMA1,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ! [V3d] :
( mem(V3d,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__mul,V0a),V2c)),ap(ap(c_2Erealax_2Ereal__mul,V1b),V3d))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0a),V3d)),ap(ap(c_2Erealax_2Ereal__mul,V1b),V2c))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,V0a),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,V1b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,V2c),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,V3d),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__LEMMA2,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Etransc_2Esqrt,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,V0x),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,V1y),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))))) ) ) ).
fof(ax_thm_2Ecomplex_2ERE,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2ERE,V0z) = ap(c_2Epair_2EFST(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0z) ) ).
fof(ax_thm_2Ecomplex_2EIM,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2EIM,V0z) = ap(c_2Epair_2ESND(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0z) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2EIM,V0z)) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__RE__IM__EQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z = V1w
<=> ( ap(c_2Ecomplex_2ERE,V0z) = ap(c_2Ecomplex_2ERE,V1w)
& ap(c_2Ecomplex_2EIM,V0z) = ap(c_2Ecomplex_2EIM,V1w) ) ) ) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__of__real,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2Ecomplex__of__real,V0x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ).
fof(conj_thm_2Ecomplex_2ERE__COMPLEX__OF__REAL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2ERE,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)) = V0x ) ).
fof(conj_thm_2Ecomplex_2EIM__COMPLEX__OF__REAL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2EIM,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__of__num,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Ecomplex_2Ecomplex__of__num,V0n) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(c_2Ereal_2Ereal__of__num,V0n)) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__0,axiom,
ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__1,axiom,
ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__10,axiom,
ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__0__THM,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ).
fof(ax_thm_2Ecomplex_2Ei,axiom,
c_2Ecomplex_2Ei = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ).
fof(ax_thm_2Ecomplex_2Ecomplex__add,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2ERE,V1w))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2EIM,V1w))) ) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__neg,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,V0z) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,ap(c_2Ecomplex_2ERE,V0z))),ap(c_2Erealax_2Ereal__neg,ap(c_2Ecomplex_2EIM,V0z))) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__mul,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2ERE,V1w))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2EIM,V1w)))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2EIM,V1w))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2ERE,V1w)))) ) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__inv,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__inv,V0z) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))))),ap(ap(c_2Ereal_2E_2F,ap(c_2Erealax_2Ereal__neg,ap(c_2Ecomplex_2EIM,V0z))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))))) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__sub,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) ) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__div,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(c_2Ecomplex_2Ecomplex__inv,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__COMM,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__add,V1w),V0z) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__ASSOC,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),ap(ap(c_2Ecomplex_2Ecomplex__add,V1w),V2v)) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)),V2v) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RID,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__LID,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)),V0z) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RINV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),ap(c_2Ecomplex_2Ecomplex__neg,V0z)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__LINV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__COMM,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V0z) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__ASSOC,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V2v)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)),V2v) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RID,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LID,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0z) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RINV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(c_2Ecomplex_2Ecomplex__inv,V0z)) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LINV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__inv,V0z)),V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__LDISTRIB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(ap(c_2Ecomplex_2Ecomplex__add,V1w),V2v)) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RDISTRIB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)),V2v) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V2v)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V2v)
<=> V1w = V2v ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__RADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V2v) = ap(ap(c_2Ecomplex_2Ecomplex__add,V1w),V2v)
<=> V0z = V1w ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RID__UNIQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w) = V0z
<=> V1w = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__LID__UNIQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w) = V1w
<=> V0z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEGNEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(c_2Ecomplex_2Ecomplex__neg,V0z)) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__EQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(c_2Ecomplex_2Ecomplex__neg,V0z) = V1w
<=> V0z = ap(c_2Ecomplex_2Ecomplex__neg,V1w) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__NEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(c_2Ecomplex_2Ecomplex__neg,V0z) = ap(c_2Ecomplex_2Ecomplex__neg,V1w)
<=> V0z = V1w ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__RNEG__UNIQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> V1w = ap(c_2Ecomplex_2Ecomplex__neg,V0z) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__LNEG__UNIQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> V0z = ap(c_2Ecomplex_2Ecomplex__neg,V1w) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__ADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RZERO,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LZERO,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)),V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__LMUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),V1w) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__RMUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__MUL2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ENTIRE,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> ( V0z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
| V1w = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__0,axiom,
ap(c_2Ecomplex_2Ecomplex__neg,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__EQ0,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(c_2Ecomplex_2Ecomplex__neg,V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> V0z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__REFL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__RZERO,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__LZERO,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)),V0z) = ap(c_2Ecomplex_2Ecomplex__neg,V0z) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__LNEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),V1w) = ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__NEG2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__sub,V1w),V0z) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__SUB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__sub,V1w),V0z) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__RNEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__ADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)),V1w) = V0z ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__ADD2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,V1w),ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)) = V0z ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__SUB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)),V0z) = V1w ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__SUB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)),V0z) = ap(c_2Ecomplex_2Ecomplex__neg,V1w) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__SUB2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)) = V1w ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__SUB2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)) = ap(c_2Ecomplex_2Ecomplex__neg,V1w) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD2__SUB2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2u] :
( mem(V2u,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3v] :
( mem(V3v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)),ap(ap(c_2Ecomplex_2Ecomplex__add,V2u),V3v)) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V2u)),ap(ap(c_2Ecomplex_2Ecomplex__sub,V1w),V3v)) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__TRIANGLE,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)),ap(ap(c_2Ecomplex_2Ecomplex__sub,V1w),V2v)) = ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V2v) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__0,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> V0z = V1w ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__SUB__LADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z = ap(ap(c_2Ecomplex_2Ecomplex__sub,V1w),V2v)
<=> ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V2v) = V1w ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__SUB__RADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w) = V2v
<=> V0z = ap(ap(c_2Ecomplex_2Ecomplex__add,V2v),V1w) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RNEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) = ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LNEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),V1w) = ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__LDISTRIB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(ap(c_2Ecomplex_2Ecomplex__sub,V1w),V2v)) = ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__RDISTRIB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)),V2v) = ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V2v)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIFFSQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)),ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V0z)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LMUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v)
<=> ( V0z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
| V1w = V2v ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__RMUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V2v)
<=> ( V2v = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
| V0z = V1w ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LMUL2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ( V1w = V2v
<=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v) ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__RMUL__IMP,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
& ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V0z) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V2v),V0z) )
=> V1w = V2v ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LMUL__IMP,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
& ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v) )
=> V1w = V2v ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__INV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Ecomplex__neg,V0z)) = ap(c_2Ecomplex_2Ecomplex__neg,ap(c_2Ecomplex_2Ecomplex__inv,V0z)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__MUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
& V1w != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) )
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__inv,V0z)),ap(c_2Ecomplex_2Ecomplex__inv,V1w)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INVINV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Ecomplex__inv,V0z)) = V0z ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__LINV__UNIQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))
=> V0z = ap(c_2Ecomplex_2Ecomplex__inv,V1w) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__RINV__UNIQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))
=> V1w = ap(c_2Ecomplex_2Ecomplex__inv,V0z) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__0,axiom,
ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV1,axiom,
ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__INV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Ecomplex__inv,V0z)) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__NEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Ecomplex__neg,V0z)) = ap(c_2Ecomplex_2Ecomplex__neg,ap(c_2Ecomplex_2Ecomplex__inv,V0z)) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__EQ__0,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(c_2Ecomplex_2Ecomplex__inv,V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> V0z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__NZ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Ecomplex_2Ecomplex__inv,V0z) != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__INJ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(c_2Ecomplex_2Ecomplex__inv,V0z) = ap(c_2Ecomplex_2Ecomplex__inv,V1w)
<=> V0z = V1w ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__LDIV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),V1w) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__RDIV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__DIV2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__div,ap(c_2Ecomplex_2Ecomplex__neg,V0z)),ap(c_2Ecomplex_2Ecomplex__neg,V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__1OVER,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__inv,V0z) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0z) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV1,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__ADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V2v)),ap(ap(c_2Ecomplex_2Ecomplex__div,V1w),V2v)) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)),V2v) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__SUB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V2v)),ap(ap(c_2Ecomplex_2Ecomplex__div,V1w),V2v)) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)),V2v) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__RMUL__CANCEL,axiom,
! [V0v] :
( mem(V0v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0v != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__mul,V1z),V0v)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V2w),V0v)) = ap(ap(c_2Ecomplex_2Ecomplex__div,V1z),V2w) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__LMUL__CANCEL,axiom,
! [V0v] :
( mem(V0v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0v != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0v),V1z)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V0v),V2w)) = ap(ap(c_2Ecomplex_2Ecomplex__div,V1z),V2w) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__DENOM__CANCEL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__div,V1w),V0z)),ap(ap(c_2Ecomplex_2Ecomplex__div,V2v),V0z)) = ap(ap(c_2Ecomplex_2Ecomplex__div,V1w),V2v) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__INNER__CANCEL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__div,V1w),V0z)),ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V2v)) = ap(ap(c_2Ecomplex_2Ecomplex__div,V1w),V2v) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__OUTER__CANCEL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w)),ap(ap(c_2Ecomplex_2Ecomplex__div,V2v),V0z)) = ap(ap(c_2Ecomplex_2Ecomplex__div,V2v),V1w) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__MUL2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
& V1w != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) )
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__div,V2v),V1w) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RAT,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2u] :
( mem(V2u,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3v] :
( mem(V3v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V1w != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
& V3v != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) )
=> ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w)),ap(ap(c_2Ecomplex_2Ecomplex__div,V2u),V3v)) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V3v)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V2u))),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V3v)) ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__RAT,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2u] :
( mem(V2u,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3v] :
( mem(V3v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V1w != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
& V3v != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) )
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w)),ap(ap(c_2Ecomplex_2Ecomplex__div,V2u),V3v)) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V3v)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V2u))),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V3v)) ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__LZERO,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__div,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)),V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__REFL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__INV2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
& V1w != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) )
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(c_2Ecomplex_2Ecomplex__inv,V0z)),ap(c_2Ecomplex_2Ecomplex__inv,V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__sub,V1w),V0z)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__RDIV__EQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V2v != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ( V0z = ap(ap(c_2Ecomplex_2Ecomplex__div,V1w),V2v)
<=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V2v) = V1w ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LDIV__EQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2v] :
( mem(V2v,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V2v != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ( ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V2v) = V1w
<=> V0z = ap(ap(c_2Ecomplex_2Ecomplex__mul,V1w),V2v) ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__EQ,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ap(c_2Ecomplex_2Ecomplex__of__real,V0x) = ap(c_2Ecomplex_2Ecomplex__of__real,V1y)
<=> V0x = V1y ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__ADD,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__add,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)),ap(c_2Ecomplex_2Ecomplex__of__real,V1y)) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(c_2Erealax_2Ereal__neg,V0x)) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__MUL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)),ap(c_2Ecomplex_2Ecomplex__of__real,V1y)) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__INV,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(c_2Erealax_2Einv,V0x)) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__SUB,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)),ap(c_2Ecomplex_2Ecomplex__of__real,V1y)) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__DIV,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__div,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)),ap(c_2Ecomplex_2Ecomplex__of__real,V1y)) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(ap(c_2Ereal_2E_2F,V0x),V1y)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__NUM__EQ,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ap(c_2Ecomplex_2Ecomplex__of__num,V0m) = ap(c_2Ecomplex_2Ecomplex__of__num,V1n)
<=> V0m = V1n ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__NUM__ADD,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__add,ap(c_2Ecomplex_2Ecomplex__of__num,V0m)),ap(c_2Ecomplex_2Ecomplex__of__num,V1n)) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(ap(c_2Earithmetic_2E_2B,V0m),V1n)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__OF__NUM__MUL,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__of__num,V0m)),ap(c_2Ecomplex_2Ecomplex__of__num,V1n)) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(ap(c_2Earithmetic_2E_2A,V0m),V1n)) ) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__scalar__lmul,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,V0k),ap(c_2Ecomplex_2ERE,V1z))),ap(ap(c_2Erealax_2Ereal__mul,V0k),ap(c_2Ecomplex_2EIM,V1z))) ) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__scalar__rmul,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1k] :
( mem(V1k,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V0z),V1k) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2ERE,V0z)),V1k)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2EIM,V0z)),V1k)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V1l),V2z)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(ap(c_2Erealax_2Ereal__mul,V0k),V1l)),V2z) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__NEG,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Erealax_2Ereal__neg,V0k)),V1z) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),ap(c_2Ecomplex_2Ecomplex__neg,V1z)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Erealax_2Ereal__neg,V0k)),V1z) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ADD,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(ap(c_2Erealax_2Ereal__add,V0k),V1l)),V2z) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2z)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V1l),V2z)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__SUB,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(ap(c_2Ereal_2Ereal__sub,V0k),V1l)),V2z) = ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2z)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V1l),V2z)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),ap(ap(c_2Ecomplex_2Ecomplex__add,V1z),V2w)) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2w)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),ap(ap(c_2Ecomplex_2Ecomplex__sub,V1z),V2w)) = ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2w)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__SCALAR__LMUL2,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3w] :
( mem(V3w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2z)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V1l),V3w)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(ap(c_2Erealax_2Ereal__mul,V0k),V1l)),ap(ap(c_2Ecomplex_2Ecomplex__mul,V2z),V3w)) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__LMUL__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z)),V2w) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1z),V2w)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__RMUL__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V1z),ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2w)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),ap(ap(c_2Ecomplex_2Ecomplex__mul,V1z),V2w)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ZERO,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0z) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ZERO__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ONE,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0z) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__NEG1,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0z) = ap(c_2Ecomplex_2Ecomplex__neg,V0z) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DOUBLE,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V0z) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V0z) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ENTIRE,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> ( V0k = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
| V1z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EQ__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2w)
<=> ( V0k = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
| V1z = V2w ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__EQ,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2z) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V1l),V2z)
<=> ( V0k = V1l
| V2z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__EQ1,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z) = V1z
<=> ( V0k = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))
| V1z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V0k != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
& V1z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) )
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Erealax_2Einv,V0k)),ap(c_2Ecomplex_2Ecomplex__inv,V1z)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__DIV2,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3w] :
( mem(V3w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( V1l != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
& V3w != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) )
=> ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V2z)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V1l),V3w)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(ap(c_2Ereal_2E_2F,V0k),V1l)),ap(ap(c_2Ecomplex_2Ecomplex__div,V2z),V3w)) ) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__MUL__COMM,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V1z),V0k) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2z),V0k)),V1l) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2z),ap(ap(c_2Erealax_2Ereal__mul,V0k),V1l)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__NEG,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__neg,ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V1z),V0k)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V1z),ap(c_2Erealax_2Ereal__neg,V0k)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__NEG__SCALAR__RMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,ap(c_2Ecomplex_2Ecomplex__neg,V1z)),V0k) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V1z),ap(c_2Erealax_2Ereal__neg,V0k)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ADD,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2z),ap(ap(c_2Erealax_2Ereal__add,V0k),V1l)) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2z),V0k)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2z),V1l)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__RSCALAR__RMUL__SUB,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2z),ap(ap(c_2Ereal_2Ereal__sub,V0k),V1l)) = ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2z),V0k)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2z),V1l)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RSCALAR__RMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,ap(ap(c_2Ecomplex_2Ecomplex__add,V1z),V2w)),V0k) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V1z),V0k)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2w),V0k)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SUB__SCALAR__RMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,ap(ap(c_2Ecomplex_2Ecomplex__sub,V1z),V2w)),V0k) = ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V1z),V0k)),ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V2w),V0k)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ZERO,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V0z),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__ZERO__SCALAR__RMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)),V0k) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ONE,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V0z),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__NEG1,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,V0z),ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) = ap(c_2Ecomplex_2Ecomplex__neg,V0z) ) ).
fof(ax_thm_2Ecomplex_2Econj,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,V0z) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Erealax_2Ereal__neg,ap(c_2Ecomplex_2EIM,V0z))) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__REAL__REFL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2Econj,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)) = ap(c_2Ecomplex_2Ecomplex__of__real,V0x) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__NUM__REFL,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Ecomplex_2Econj,ap(c_2Ecomplex_2Ecomplex__of__num,V0n)) = ap(c_2Ecomplex_2Ecomplex__of__num,V0n) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__ADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__add,ap(c_2Ecomplex_2Econj,V0z)),ap(c_2Ecomplex_2Econj,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__NEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,ap(c_2Ecomplex_2Ecomplex__neg,V0z)) = ap(c_2Ecomplex_2Ecomplex__neg,ap(c_2Ecomplex_2Econj,V0z)) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__SUB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__sub,ap(c_2Ecomplex_2Econj,V0z)),ap(c_2Ecomplex_2Econj,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__MUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Econj,V0z)),ap(c_2Ecomplex_2Econj,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__INV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,ap(c_2Ecomplex_2Ecomplex__inv,V0z)) = ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Econj,V0z)) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__DIV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(c_2Ecomplex_2Econj,V0z)),ap(c_2Ecomplex_2Econj,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z)) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),ap(c_2Ecomplex_2Econj,V1z)) ) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__CONJ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Econj,ap(c_2Ecomplex_2Econj,V0z)) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECONJ__EQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(c_2Ecomplex_2Econj,V0z) = V1w
<=> V0z = ap(c_2Ecomplex_2Econj,V1w) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__EQ2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(c_2Ecomplex_2Econj,V0z) = ap(c_2Ecomplex_2Econj,V1w)
<=> V0z = V1w ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RCONJ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(c_2Ecomplex_2Econj,V0z)) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LCONJ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Econj,V0z)),V0z) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Ecomplex_2ECONJ__ZERO,axiom,
ap(c_2Ecomplex_2Econj,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ).
fof(ax_thm_2Ecomplex_2Emodu,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Emodu,V0z) = ap(c_2Etransc_2Esqrt,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) ) ).
fof(ax_thm_2Ecomplex_2Earg,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Earg,V0z) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ecomplex_2EIM,V0z))),ap(c_2Etransc_2Eacs,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z)))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Erealax_2Ereal__neg,ap(c_2Etransc_2Eacs,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z))))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),c_2Etransc_2Epi))) ) ).
fof(conj_thm_2Ecomplex_2EMODU__POW2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2Emodu,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ) ).
fof(conj_thm_2Ecomplex_2ERE__IM__LE__MODU,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Ecomplex_2ERE,V0z))),ap(c_2Ecomplex_2Emodu,V0z)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Ecomplex_2EIM,V0z))),ap(c_2Ecomplex_2Emodu,V0z))) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__POS,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ecomplex_2Emodu,V0z))) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RCONJ1,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),ap(c_2Ecomplex_2Econj,V0z)) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2Emodu,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LCONJ1,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Econj,V0z)),V0z) = ap(c_2Ecomplex_2Ecomplex__of__real,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2Emodu,V0z)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ) ).
fof(conj_thm_2Ecomplex_2EMODU__NEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Emodu,ap(c_2Ecomplex_2Ecomplex__neg,V0z)) = ap(c_2Ecomplex_2Emodu,V0z) ) ).
fof(conj_thm_2Ecomplex_2EMODU__SUB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Emodu,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)) = ap(c_2Ecomplex_2Emodu,ap(ap(c_2Ecomplex_2Ecomplex__sub,V1w),V0z)) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__CONJ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Emodu,ap(c_2Ecomplex_2Econj,V0z)) = ap(c_2Ecomplex_2Emodu,V0z) ) ).
fof(conj_thm_2Ecomplex_2EMODU__MUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Emodu,ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V1w)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2Emodu,V0z)),ap(c_2Ecomplex_2Emodu,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__0,axiom,
ap(c_2Ecomplex_2Emodu,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ).
fof(conj_thm_2Ecomplex_2EMODU__1,axiom,
ap(c_2Ecomplex_2Emodu,ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Ecomplex_2EMODU__COMPLEX__INV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Ecomplex_2Emodu,ap(c_2Ecomplex_2Ecomplex__inv,V0z)) = ap(c_2Erealax_2Einv,ap(c_2Ecomplex_2Emodu,V0z)) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__DIV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V1w != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Ecomplex_2Emodu,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w)) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2Emodu,V0z)),ap(c_2Ecomplex_2Emodu,V1w)) ) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__SCALAR__LMUL,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Emodu,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V0k),V1z)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V0k)),ap(c_2Ecomplex_2Emodu,V1z)) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__REAL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2Emodu,ap(c_2Ecomplex_2Ecomplex__of__real,V0x)) = ap(c_2Ereal_2Eabs,V0x) ) ).
fof(conj_thm_2Ecomplex_2EMODU__NUM,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Ecomplex_2Emodu,ap(c_2Ecomplex_2Ecomplex__of__num,V0n)) = ap(c_2Ereal_2Ereal__of__num,V0n) ) ).
fof(conj_thm_2Ecomplex_2EMODU__ZERO,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> ap(c_2Ecomplex_2Emodu,V0z) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__NZ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ecomplex_2Emodu,V0z))) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__CASES,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
| p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ecomplex_2Emodu,V0z))) ) ) ).
fof(conj_thm_2Ecomplex_2ERE__DIV__MODU__BOUNDS,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) ).
fof(conj_thm_2Ecomplex_2EIM__DIV__MODU__BOUNDS,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2Emodu,V0z))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2Emodu,V0z))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) ).
fof(conj_thm_2Ecomplex_2ERE__DIV__MODU__ACS__BOUNDS,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Etransc_2Eacs,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z)))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Etransc_2Eacs,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z)))),c_2Etransc_2Epi)) ) ) ) ).
fof(conj_thm_2Ecomplex_2EIM__DIV__MODU__ASN__BOUNDS,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2E_2F,c_2Etransc_2Epi),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))))),ap(c_2Etransc_2Easn,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2Emodu,V0z)))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Etransc_2Easn,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2Emodu,V0z)))),ap(ap(c_2Ereal_2E_2F,c_2Etransc_2Epi),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))))) ) ) ) ).
fof(conj_thm_2Ecomplex_2ERE__DIV__MODU__ACS__COS,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Etransc_2Ecos,ap(c_2Etransc_2Eacs,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z)))) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z)) ) ) ).
fof(conj_thm_2Ecomplex_2EIM__DIV__MODU__ASN__SIN,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Etransc_2Esin,ap(c_2Etransc_2Easn,ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2Emodu,V0z)))) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2Emodu,V0z)) ) ) ).
fof(conj_thm_2Ecomplex_2EARG__COS,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Etransc_2Ecos,ap(c_2Ecomplex_2Earg,V0z)) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2ERE,V0z)),ap(c_2Ecomplex_2Emodu,V0z)) ) ) ).
fof(conj_thm_2Ecomplex_2EARG__SIN,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(c_2Etransc_2Esin,ap(c_2Ecomplex_2Earg,V0z)) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ecomplex_2EIM,V0z)),ap(c_2Ecomplex_2Emodu,V0z)) ) ) ).
fof(conj_thm_2Ecomplex_2ERE__MODU__ARG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2ERE,V0z) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2Emodu,V0z)),ap(c_2Etransc_2Ecos,ap(c_2Ecomplex_2Earg,V0z))) ) ).
fof(conj_thm_2Ecomplex_2EIM__MODU__ARG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2EIM,V0z) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ecomplex_2Emodu,V0z)),ap(c_2Etransc_2Esin,ap(c_2Ecomplex_2Earg,V0z))) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__TRIANGLE,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Ecomplex_2Emodu,V0z)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,ap(c_2Ecomplex_2Earg,V0z))),ap(c_2Etransc_2Esin,ap(c_2Ecomplex_2Earg,V0z)))) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MODU__ARG__EQ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z = V1w
<=> ( ap(c_2Ecomplex_2Emodu,V0z) = ap(c_2Ecomplex_2Emodu,V1w)
& ap(c_2Ecomplex_2Earg,V0z) = ap(c_2Ecomplex_2Earg,V1w) ) ) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__UNIT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ap(c_2Ecomplex_2Emodu,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,V0x)),ap(c_2Etransc_2Esin,V0x))) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__MUL__ARG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,V0x)),ap(c_2Etransc_2Esin,V0x))),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,V1y)),ap(c_2Etransc_2Esin,V1y))) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y))),ap(c_2Etransc_2Esin,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y))) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__INV__ARG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,V0x)),ap(c_2Etransc_2Esin,V0x))) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,ap(c_2Erealax_2Ereal__neg,V0x))),ap(c_2Etransc_2Esin,ap(c_2Erealax_2Ereal__neg,V0x))) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__DIV__ARG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,V0x)),ap(c_2Etransc_2Esin,V0x))),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,V1y)),ap(c_2Etransc_2Esin,V1y))) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y))),ap(c_2Etransc_2Esin,ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y))) ) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__pow__def,axiom,
( ! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),c_2Enum_2E0) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) )
& ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V1z),ap(c_2Enum_2ESUC,V2n)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V1z),ap(ap(c_2Ecomplex_2Ecomplex__pow,V1z),V2n)) ) ) ) ).
fof(conj_thm_2Ecomplex_2Ecomplex__pow__def__compute,axiom,
( ! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),c_2Enum_2E0) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) )
& ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V1z),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V2n))) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V1z),ap(ap(c_2Ecomplex_2Ecomplex__pow,V1z),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V2n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) )
& ! [V3z] :
( mem(V3z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V3z),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V4n))) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V3z),ap(ap(c_2Ecomplex_2Ecomplex__pow,V3z),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V4n)))) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__0,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)),ap(c_2Enum_2ESUC,V0n)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__NZ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),V1n) != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POWINV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( V0z != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(c_2Ecomplex_2Ecomplex__inv,ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),V1n)) = ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(c_2Ecomplex_2Ecomplex__inv,V0z)),V1n) ) ) ) ).
fof(conj_thm_2Ecomplex_2EMODU__COMPLEX__POW,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(c_2Ecomplex_2Emodu,ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),V1n)) = ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2Emodu,V0z)),V1n) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__ADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),ap(ap(c_2Earithmetic_2E_2B,V1m),V2n)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),V1m)),ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),V2n)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__1,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = ap(ap(c_2Ecomplex_2Ecomplex__mul,V0z),V0z) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__ONE,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0n) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__MUL,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2w] :
( mem(V2w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(ap(c_2Ecomplex_2Ecomplex__mul,V1z),V2w)),V0n) = ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(ap(c_2Ecomplex_2Ecomplex__pow,V1z),V0n)),ap(ap(c_2Ecomplex_2Ecomplex__pow,V2w),V0n)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__INV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(c_2Ecomplex_2Ecomplex__inv,V0z)),V1n) = ap(c_2Ecomplex_2Ecomplex__inv,ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),V1n)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__DIV,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(ap(c_2Ecomplex_2Ecomplex__div,V0z),V1w)),V2n) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),V2n)),ap(ap(c_2Ecomplex_2Ecomplex__pow,V1w),V2n)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__L,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1k] :
( mem(V1k,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,V1k),V2z)),V0n) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(ap(c_2Ereal_2Epow,V1k),V0n)),ap(ap(c_2Ecomplex_2Ecomplex__pow,V2z),V0n)) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__ZERO,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__pow,V1z),V0n) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
=> V1z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__ZERO__EQ,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1z] :
( mem(V1z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ecomplex_2Ecomplex__pow,V1z),ap(c_2Enum_2ESUC,V0n)) = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)
<=> V1z = ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__POW__POW,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),V1m)),V2n) = ap(ap(c_2Ecomplex_2Ecomplex__pow,V0z),ap(ap(c_2Earithmetic_2E_2A,V1m),V2n)) ) ) ) ).
fof(conj_thm_2Ecomplex_2EDE__MOIVRE__LEMMA,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,V0x)),ap(c_2Etransc_2Esin,V0x))),V1n) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V1n)),V0x))),ap(c_2Etransc_2Esin,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V1n)),V0x))) ) ) ).
fof(conj_thm_2Ecomplex_2EDE__MOIVRE__THM,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Ecomplex_2Emodu,V0z)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,ap(c_2Ecomplex_2Earg,V0z))),ap(c_2Etransc_2Esin,ap(c_2Ecomplex_2Earg,V0z))))),V1n) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(ap(c_2Ereal_2Epow,ap(c_2Ecomplex_2Emodu,V0z)),V1n)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V1n)),ap(c_2Ecomplex_2Earg,V0z)))),ap(c_2Etransc_2Esin,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V1n)),ap(c_2Ecomplex_2Earg,V0z))))) ) ) ).
fof(ax_thm_2Ecomplex_2Ecomplex__exp,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__exp,V0z) = ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Etransc_2Eexp,ap(c_2Ecomplex_2ERE,V0z))),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,ap(c_2Ecomplex_2EIM,V0z))),ap(c_2Etransc_2Esin,ap(c_2Ecomplex_2EIM,V0z)))) ) ).
fof(conj_thm_2Ecomplex_2EEXP__IMAGINARY,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ecomplex_2Ecomplex__exp,ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,c_2Ecomplex_2Ei),V0x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Etransc_2Ecos,V0x)),ap(c_2Etransc_2Esin,V0x)) ) ).
fof(conj_thm_2Ecomplex_2EEULER__FORMULE,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Ecomplex_2Emodu,V0z)),ap(c_2Ecomplex_2Ecomplex__exp,ap(ap(c_2Ecomplex_2Ecomplex__scalar__rmul,c_2Ecomplex_2Ei),ap(c_2Ecomplex_2Earg,V0z)))) = V0z ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__ADD,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__exp,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__exp,V0z)),ap(c_2Ecomplex_2Ecomplex__exp,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__NEG,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__exp,ap(c_2Ecomplex_2Ecomplex__neg,V0z)) = ap(c_2Ecomplex_2Ecomplex__inv,ap(c_2Ecomplex_2Ecomplex__exp,V0z)) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__SUB,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__exp,ap(ap(c_2Ecomplex_2Ecomplex__sub,V0z),V1w)) = ap(ap(c_2Ecomplex_2Ecomplex__div,ap(c_2Ecomplex_2Ecomplex__exp,V0z)),ap(c_2Ecomplex_2Ecomplex__exp,V1w)) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__N,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(c_2Ecomplex_2Ecomplex__exp,ap(ap(c_2Ecomplex_2Ecomplex__scalar__lmul,ap(c_2Ereal_2Ereal__of__num,V1n)),V0z)) = ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(c_2Ecomplex_2Ecomplex__exp,V0z)),V1n) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__N2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(c_2Ecomplex_2Ecomplex__exp,ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__of__num,V1n)),V0z)) = ap(ap(c_2Ecomplex_2Ecomplex__pow,ap(c_2Ecomplex_2Ecomplex__exp,V0z)),V1n) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__0,axiom,
ap(c_2Ecomplex_2Ecomplex__exp,ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0)) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__NZ,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(c_2Ecomplex_2Ecomplex__exp,V0z) != ap(c_2Ecomplex_2Ecomplex__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__ADD__MUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1w] :
( mem(V1w,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__exp,ap(ap(c_2Ecomplex_2Ecomplex__add,V0z),V1w))),ap(c_2Ecomplex_2Ecomplex__exp,ap(c_2Ecomplex_2Ecomplex__neg,V0z))) = ap(c_2Ecomplex_2Ecomplex__exp,V1w) ) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__NEG__MUL,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__exp,V0z)),ap(c_2Ecomplex_2Ecomplex__exp,ap(c_2Ecomplex_2Ecomplex__neg,V0z))) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ).
fof(conj_thm_2Ecomplex_2ECOMPLEX__EXP__NEG__MUL2,axiom,
! [V0z] :
( mem(V0z,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ecomplex_2Ecomplex__mul,ap(c_2Ecomplex_2Ecomplex__exp,ap(c_2Ecomplex_2Ecomplex__neg,V0z))),ap(c_2Ecomplex_2Ecomplex__exp,V0z)) = ap(c_2Ecomplex_2Ecomplex__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ).
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