ITP001 Axioms: ITP135^5.ax
%------------------------------------------------------------------------------
% File : ITP135^5 : TPTP v9.0.0. 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 : 287 ( 171 unt; 22 typ; 0 def)
% Number of atoms : 4534 ( 332 equ; 0 cnn)
% Maximal formula atoms : 55 ( 15 avg)
% Number of connectives : 5538 ( 47 ~; 8 |; 19 &;5391 @)
% ( 34 <=>; 39 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg;5391 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 4 ( 4 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 69 usr; 65 con; 0-2 aty)
% Number of variables : 438 ( 0 ^ 438 !; 0 ?; 438 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Ecomplex_2EIM,type,
c_2Ecomplex_2EIM: $i ).
thf(mem_c_2Ecomplex_2EIM,axiom,
mem @ c_2Ecomplex_2EIM @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Ecomplex_2EIM,type,
fo__c_2Ecomplex_2EIM: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Ecomplex_2EIM,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Ecomplex_2EIM @ X0 ) )
= ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Ecomplex_2ERE,type,
c_2Ecomplex_2ERE: $i ).
thf(mem_c_2Ecomplex_2ERE,axiom,
mem @ c_2Ecomplex_2ERE @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Ecomplex_2ERE,type,
fo__c_2Ecomplex_2ERE: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Ecomplex_2ERE,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Ecomplex_2ERE @ X0 ) )
= ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Ecomplex_2Earg,type,
c_2Ecomplex_2Earg: $i ).
thf(mem_c_2Ecomplex_2Earg,axiom,
mem @ c_2Ecomplex_2Earg @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Ecomplex_2Earg,type,
fo__c_2Ecomplex_2Earg: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Ecomplex_2Earg,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Ecomplex_2Earg @ X0 ) )
= ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__add,type,
c_2Ecomplex_2Ecomplex__add: $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__div,type,
c_2Ecomplex_2Ecomplex__div: $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__exp,type,
c_2Ecomplex_2Ecomplex__exp: $i ).
thf(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 ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__inv,type,
c_2Ecomplex_2Ecomplex__inv: $i ).
thf(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 ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__mul,type,
c_2Ecomplex_2Ecomplex__mul: $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__neg,type,
c_2Ecomplex_2Ecomplex__neg: $i ).
thf(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 ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__of__num,type,
c_2Ecomplex_2Ecomplex__of__num: $i ).
thf(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 ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__of__real,type,
c_2Ecomplex_2Ecomplex__of__real: $i ).
thf(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 ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__pow,type,
c_2Ecomplex_2Ecomplex__pow: $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__scalar__lmul,type,
c_2Ecomplex_2Ecomplex__scalar__lmul: $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__scalar__rmul,type,
c_2Ecomplex_2Ecomplex__scalar__rmul: $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ecomplex_2Ecomplex__sub,type,
c_2Ecomplex_2Ecomplex__sub: $i ).
thf(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 ) ) ) ).
thf(tp_c_2Ecomplex_2Econj,type,
c_2Ecomplex_2Econj: $i ).
thf(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 ) ) ).
thf(tp_c_2Ecomplex_2Ei,type,
c_2Ecomplex_2Ei: $i ).
thf(mem_c_2Ecomplex_2Ei,axiom,
mem @ c_2Ecomplex_2Ei @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Ecomplex_2Emodu,type,
c_2Ecomplex_2Emodu: $i ).
thf(mem_c_2Ecomplex_2Emodu,axiom,
mem @ c_2Ecomplex_2Emodu @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Ecomplex_2Emodu,type,
fo__c_2Ecomplex_2Emodu: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Ecomplex_2Emodu,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Ecomplex_2Emodu @ X0 ) )
= ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__LEMMA1,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3d: tp__ty_2Erealax_2Ereal] :
( ( surj__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 @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V3d ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__LEMMA2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2ERE,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Epair_2EFST @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(ax_thm_2Ecomplex_2EIM,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Epair_2ESND @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_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 @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__RE__IM__EQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z = V1w )
<=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__of__real,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ERE__COMPLEX__OF__REAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2ERE @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= V0x ) ).
thf(conj_thm_2Ecomplex_2EIM__COMPLEX__OF__REAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2EIM @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__of__num,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__0,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__1,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__10,axiom,
( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__0__THM,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ei,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ c_2Ecomplex_2Ei )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__add,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__neg,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__mul,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( 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 @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__inv,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__sub,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__div,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__COMM,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__ASSOC,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RID,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__LID,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RINV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__LINV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__COMM,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__ASSOC,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RID,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LID,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_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 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RINV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LINV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__LDISTRIB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RDISTRIB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
<=> ( V1w = V2v ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__RADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
<=> ( V0z = V1w ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RID__UNIQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= V0z )
<=> ( V1w
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__LID__UNIQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= V1w )
<=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEGNEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__EQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= V1w )
<=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__NEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
<=> ( V0z = V1w ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__RNEG__UNIQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V1w
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__LNEG__UNIQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__ADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RZERO,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LZERO,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__LMUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__RMUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__MUL2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ENTIRE,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( V1w
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__0,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__EQ0,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__REFL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__RZERO,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__LZERO,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__LNEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__NEG2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__SUB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__RNEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__ADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__ADD2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__SUB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= V1w ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__SUB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__SUB2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= V1w ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__SUB2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD2__SUB2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2u: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V3v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2u ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2u ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3v ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__TRIANGLE,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__0,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0z = V1w ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__SUB__LADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
<=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) )
= V1w ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__SUB__RADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= V2v )
<=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RNEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LNEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__LDISTRIB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__RDISTRIB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIFFSQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LMUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
<=> ( ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( V1w = V2v ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__RMUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
<=> ( ( V2v
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( V0z = V1w ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LMUL2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( V1w = V2v )
<=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__RMUL__IMP,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) )
=> ( V1w = V2v ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LMUL__IMP,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) )
=> ( V1w = V2v ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__INV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__MUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1w
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INVINV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= V0z ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__LINV__UNIQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__RINV__UNIQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( V1w
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__0,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV1,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__INV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__NEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__EQ__0,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__NZ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__INJ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
<=> ( V0z = V1w ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__LDIV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__RDIV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__DIV2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__1OVER,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV1,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__ADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__SUB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__RMUL__CANCEL,axiom,
! [V0v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0v
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0v ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__LMUL__CANCEL,axiom,
! [V0v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0v
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0v ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0v ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__DENOM__CANCEL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__INNER__CANCEL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__OUTER__CANCEL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__MUL2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1w
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ! [V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RAT,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2u: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V3v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V1w
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V3v
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2u ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3v ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2u ) ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3v ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__RAT,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2u: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V3v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V1w
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V3v
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2u ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3v ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3v ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2u ) ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3v ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__LZERO,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__REFL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__INV2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1w
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__RDIV__EQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V2v
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) )
<=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) )
= V1w ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__LDIV__EQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2v: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V2v
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) )
= V1w )
<=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2v ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__EQ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( V0x = V1y ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__INV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__SUB,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__REAL__DIV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__NUM__EQ,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( V0m = V1n ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__NUM__ADD,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__OF__NUM__MUL,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__scalar__lmul,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__scalar__rmul,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1k: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__NEG,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ADD,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__SUB,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__SCALAR__LMUL2,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V3w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__LMUL__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__RMUL__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ZERO,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ZERO__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ONE,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_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 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__NEG1,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_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 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DOUBLE,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_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_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ENTIRE,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( V0k
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( V1z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EQ__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) )
<=> ( ( V0k
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( V1z = V2w ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__EQ,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) )
<=> ( ( V0k = V1l )
| ( V2z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__EQ1,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) )
= V1z )
<=> ( ( V0k
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
| ( V1z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V0k
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V1z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__DIV2,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V3w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( V1l
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( V3w
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3w ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__MUL__COMM,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__NEG,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__NEG__SCALAR__RMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ADD,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__RSCALAR__RMUL__SUB,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1l: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ADD__RSCALAR__RMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SUB__SCALAR__RMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ZERO,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__ZERO__SCALAR__RMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ONE,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__NEG1,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Econj,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_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 @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__REAL__REFL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__NUM__REFL,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__ADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__NEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__SUB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__MUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__INV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__DIV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__CONJ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECONJ__EQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= V1w )
<=> ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__EQ2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
<=> ( V0z = V1w ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RCONJ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LCONJ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECONJ__ZERO,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Econj @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Emodu,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esqrt @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Earg,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Etransc_2Eacs @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ 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 @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__POW2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ERE__IM__LE__MODU,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__POS,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__RCONJ1,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__LCONJ1,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__NEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__SUB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__CONJ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ c_2Ecomplex_2Econj @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__MUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__0,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__1,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__COMPLEX__INV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__DIV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V1w
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__SCALAR__LMUL,axiom,
! [V0k: tp__ty_2Erealax_2Ereal,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__REAL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ c_2Ecomplex_2Ecomplex__of__real @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__NUM,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__ZERO,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__NZ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__CASES,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
| ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ERE__DIV__MODU__BOUNDS,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__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 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EIM__DIV__MODU__BOUNDS,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__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 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ERE__DIV__MODU__ACS__BOUNDS,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Etransc_2Eacs @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Eacs @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EIM__DIV__MODU__ASN__BOUNDS,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Etransc_2Easn @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Etransc_2Easn @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ fo__c_2Etransc_2Epi ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ERE__DIV__MODU__ACS__COS,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Etransc_2Eacs @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EIM__DIV__MODU__ASN__SIN,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Etransc_2Easn @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EARG__COS,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EARG__SIN,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ERE__MODU__ARG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EIM__MODU__ARG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__TRIANGLE,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MODU__ARG__EQ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z = V1w )
<=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__UNIT,axiom,
! [A_27a: del,V0x: tp__ty_2Erealax_2Ereal,V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__MUL__ARG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_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 @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__INV__ARG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__DIV__ARG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_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 @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__pow__def,axiom,
( ! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2Ecomplex__pow__def__compute,axiom,
( ! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ! [V3z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V4n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3z ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3z ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V3z ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__NZ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POWINV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( V0z
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EMODU__COMPLEX__POW,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Emodu @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__ADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__1,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__ONE,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__MUL,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2w ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__INV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__DIV,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__L,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1k: tp__ty_2Erealax_2Ereal,V2z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__ZERO,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( V1z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__ZERO__EQ,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V1z
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__POW__POW,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EDE__MOIVRE__LEMMA,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( 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 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EDE__MOIVRE__THM,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Enum_2Enum @ 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 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecomplex_2Ecomplex__exp,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Etransc_2Eexp @ ( ap @ c_2Ecomplex_2ERE @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( ap @ c_2Ecomplex_2EIM @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EEXP__IMAGINARY,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ c_2Ecomplex_2Ei ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Etransc_2Ecos @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Etransc_2Esin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2EEULER__FORMULE,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Ecomplex_2Emodu @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__rmul @ c_2Ecomplex_2Ei ) @ ( ap @ c_2Ecomplex_2Earg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) )
= V0z ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__ADD,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__add @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__NEG,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__inv @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__SUB,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__sub @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__div @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__N,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__scalar__lmul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__N2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__pow @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__0,axiom,
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__NZ,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) )
!= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__ADD__MUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,V1w: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_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 @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V1w ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__NEG__MUL,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ecomplex_2Ecomplex__mul @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( ap @ c_2Ecomplex_2Ecomplex__neg @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Ecomplex_2ECOMPLEX__EXP__NEG__MUL2,axiom,
! [V0z: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_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 @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) ) @ ( ap @ c_2Ecomplex_2Ecomplex__exp @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0z ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( ap @ c_2Ecomplex_2Ecomplex__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
%------------------------------------------------------------------------------