ITP001 Axioms: ITP135_7.ax
%------------------------------------------------------------------------------
% File : ITP135_7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : complex.ax [Gau19]
% : HL4135_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 430 ( 222 unt; 120 typ; 0 def)
% Number of atoms : 431 ( 385 equ)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 171 ( 50 ~; 11 |; 21 &)
% ( 47 <=>; 42 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 83 ( 56 >; 27 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 116 ( 116 usr; 51 con; 0-4 aty)
% Number of variables : 577 ( 548 !; 1 ?; 577 :)
% ( 28 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Earithmetic_2E_2A_2E0,type,
c_2Earithmetic_2E_2A_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2A_2E2,type,
c_2Earithmetic_2E_2A_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2E_2B_2E0,type,
c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2B_2E2,type,
c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Epair_2E_2C_2E0,type,
c_2Epair_2E_2C_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).
tff(c_2Epair_2E_2C_2E2,type,
c_2Epair_2E_2C_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Earithmetic_2E_2D_2E0,type,
c_2Earithmetic_2E_2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2D_2E2,type,
c_2Earithmetic_2E_2D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Ereal_2E_2F_2E0,type,
c_2Ereal_2E_2F_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2E_2F_2E2,type,
c_2Ereal_2E_2F_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Enum_2E0_2E0,type,
c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Earithmetic_2EBIT1_2E0,type,
c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT1_2E1,type,
c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EBIT2_2E0,type,
c_2Earithmetic_2EBIT2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT2_2E1,type,
c_2Earithmetic_2EBIT2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ECOND_2E0,type,
c_2Ebool_2ECOND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Ebool_2ECOND_2E3,type,
c_2Ebool_2ECOND_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Epair_2EFST_2E0,type,
c_2Epair_2EFST_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a) ).
tff(c_2Epair_2EFST_2E1,type,
c_2Epair_2EFST_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27a ) ).
tff(c_2Ecomplex_2EIM_2E0,type,
c_2Ecomplex_2EIM_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2EIM_2E1,type,
c_2Ecomplex_2EIM_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Earithmetic_2ENUMERAL_2E0,type,
c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2ENUMERAL_2E1,type,
c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ecomplex_2ERE_2E0,type,
c_2Ecomplex_2ERE_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2ERE_2E1,type,
c_2Ecomplex_2ERE_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Epair_2ESND_2E0,type,
c_2Epair_2ESND_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b) ).
tff(c_2Epair_2ESND_2E1,type,
c_2Epair_2ESND_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27b ) ).
tff(c_2Enum_2ESUC_2E0,type,
c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Enum_2ESUC_2E1,type,
c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2EZERO_2E0,type,
c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Eabs_2E0,type,
c_2Ereal_2Eabs_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Eabs_2E1,type,
c_2Ereal_2Eabs_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Eacs_2E0,type,
c_2Etransc_2Eacs_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Eacs_2E1,type,
c_2Etransc_2Eacs_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ecomplex_2Earg_2E0,type,
c_2Ecomplex_2Earg_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Earg_2E1,type,
c_2Ecomplex_2Earg_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Easn_2E0,type,
c_2Etransc_2Easn_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Easn_2E1,type,
c_2Etransc_2Easn_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ecomplex_2Ecomplex__add_2E0,type,
c_2Ecomplex_2Ecomplex__add_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))) ).
tff(c_2Ecomplex_2Ecomplex__add_2E2,type,
c_2Ecomplex_2Ecomplex__add_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__div_2E0,type,
c_2Ecomplex_2Ecomplex__div_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))) ).
tff(c_2Ecomplex_2Ecomplex__div_2E2,type,
c_2Ecomplex_2Ecomplex__div_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__exp_2E0,type,
c_2Ecomplex_2Ecomplex__exp_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ecomplex_2Ecomplex__exp_2E1,type,
c_2Ecomplex_2Ecomplex__exp_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__inv_2E0,type,
c_2Ecomplex_2Ecomplex__inv_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ecomplex_2Ecomplex__inv_2E1,type,
c_2Ecomplex_2Ecomplex__inv_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__mul_2E0,type,
c_2Ecomplex_2Ecomplex__mul_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))) ).
tff(c_2Ecomplex_2Ecomplex__mul_2E2,type,
c_2Ecomplex_2Ecomplex__mul_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__neg_2E0,type,
c_2Ecomplex_2Ecomplex__neg_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ecomplex_2Ecomplex__neg_2E1,type,
c_2Ecomplex_2Ecomplex__neg_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__of__num_2E0,type,
c_2Ecomplex_2Ecomplex__of__num_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ecomplex_2Ecomplex__of__num_2E1,type,
c_2Ecomplex_2Ecomplex__of__num_2E1: tyop_2Enum_2Enum > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__of__real_2E0,type,
c_2Ecomplex_2Ecomplex__of__real_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ecomplex_2Ecomplex__of__real_2E1,type,
c_2Ecomplex_2Ecomplex__of__real_2E1: tyop_2Erealax_2Ereal > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__pow_2E0,type,
c_2Ecomplex_2Ecomplex__pow_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))) ).
tff(c_2Ecomplex_2Ecomplex__pow_2E2,type,
c_2Ecomplex_2Ecomplex__pow_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Enum_2Enum ) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__scalar__lmul_2E0,type,
c_2Ecomplex_2Ecomplex__scalar__lmul_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))) ).
tff(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2,type,
c_2Ecomplex_2Ecomplex__scalar__lmul_2E2: ( tyop_2Erealax_2Ereal * tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__scalar__rmul_2E0,type,
c_2Ecomplex_2Ecomplex__scalar__rmul_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))) ).
tff(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2,type,
c_2Ecomplex_2Ecomplex__scalar__rmul_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal ) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Ecomplex__sub_2E0,type,
c_2Ecomplex_2Ecomplex__sub_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))) ).
tff(c_2Ecomplex_2Ecomplex__sub_2E2,type,
c_2Ecomplex_2Ecomplex__sub_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Econj_2E0,type,
c_2Ecomplex_2Econj_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ecomplex_2Econj_2E1,type,
c_2Ecomplex_2Econj_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Ecos_2E0,type,
c_2Etransc_2Ecos_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Ecos_2E1,type,
c_2Etransc_2Ecos_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Eexp_2E0,type,
c_2Etransc_2Eexp_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Eexp_2E1,type,
c_2Etransc_2Eexp_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ecomplex_2Ei_2E0,type,
c_2Ecomplex_2Ei_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Einv_2E0,type,
c_2Erealax_2Einv_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Einv_2E1,type,
c_2Erealax_2Einv_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ecomplex_2Emodu_2E0,type,
c_2Ecomplex_2Emodu_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Ecomplex_2Emodu_2E1,type,
c_2Ecomplex_2Emodu_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Epi_2E0,type,
c_2Etransc_2Epi_2E0: tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Epow_2E0,type,
c_2Ereal_2Epow_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Epow_2E2,type,
c_2Ereal_2Epow_2E2: ( tyop_2Erealax_2Ereal * tyop_2Enum_2Enum ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__add_2E0,type,
c_2Erealax_2Ereal__add_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__add_2E2,type,
c_2Erealax_2Ereal__add_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__lt_2E0,type,
c_2Erealax_2Ereal__lt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Erealax_2Ereal__lt_2E2,type,
c_2Erealax_2Ereal__lt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Ereal__lte_2E0,type,
c_2Ereal_2Ereal__lte_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__lte_2E2,type,
c_2Ereal_2Ereal__lte_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Erealax_2Ereal__mul_2E0,type,
c_2Erealax_2Ereal__mul_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__mul_2E2,type,
c_2Erealax_2Ereal__mul_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__neg_2E0,type,
c_2Erealax_2Ereal__neg_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Ereal__neg_2E1,type,
c_2Erealax_2Ereal__neg_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__of__num_2E0,type,
c_2Ereal_2Ereal__of__num_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Ereal__of__num_2E1,type,
c_2Ereal_2Ereal__of__num_2E1: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__sub_2E0,type,
c_2Ereal_2Ereal__sub_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Ereal__sub_2E2,type,
c_2Ereal_2Ereal__sub_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Esin_2E0,type,
c_2Etransc_2Esin_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Esin_2E1,type,
c_2Etransc_2Esin_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Esqrt_2E0,type,
c_2Etransc_2Esqrt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Esqrt_2E1,type,
c_2Etransc_2Esqrt_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2A_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2A_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Epair_2E_2C_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2D_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2E_2F_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2E_2F_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT2_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ebool_2ECOND_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Epair_2EFST_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Epair_2EFST_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2EIM_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2EIM_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Ecomplex_2EIM_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2ERE_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2ERE_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Ecomplex_2ERE_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Epair_2ESND_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Epair_2ESND_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ereal_2Eabs_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Eabs_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Eacs_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eacs_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Eacs_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2Earg_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Earg_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Ecomplex_2Earg_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Easn_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Easn_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Easn_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecomplex_2Ecomplex__add_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecomplex_2Ecomplex__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecomplex_2Ecomplex__div_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__div_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecomplex_2Ecomplex__div_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2Ecomplex__exp_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__exp_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecomplex_2Ecomplex__exp_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2Ecomplex__inv_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__inv_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecomplex_2Ecomplex__inv_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecomplex_2Ecomplex__mul_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecomplex_2Ecomplex__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2Ecomplex__neg_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecomplex_2Ecomplex__neg_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2Ecomplex__of__num_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__of__num_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecomplex_2Ecomplex__of__num_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2Ecomplex__of__real_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__of__real_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecomplex_2Ecomplex__of__real_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecomplex_2Ecomplex__pow_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecomplex_2Ecomplex__pow_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecomplex_2Ecomplex__scalar__lmul_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecomplex_2Ecomplex__scalar__lmul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecomplex_2Ecomplex__scalar__rmul_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecomplex_2Ecomplex__scalar__rmul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecomplex_2Ecomplex__sub_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecomplex_2Ecomplex__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2Econj_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecomplex_2Econj_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Ecos_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Eexp_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Eexp_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Einv_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Einv_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecomplex_2Emodu_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Emodu_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Ecomplex_2Emodu_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Epow_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),c_2Ereal_2Epow_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__lte_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__mul_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__neg_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Ereal__of__num_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__sub_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Ereal__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Esin_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Esin_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Esqrt_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esqrt_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Esqrt_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Ecomplex_2ERE,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2ERE_2E1(V0z_2E0) = c_2Epair_2EFST_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0z_2E0) ) ).
tff(thm_2Ecomplex_2EIM,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2EIM_2E1(V0z_2E0) = c_2Epair_2ESND_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0z_2E0) ) ).
tff(thm_2Ecomplex_2Ecomplex__of__real,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ecomplex_2Ecomplex__of__num,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__of__num_2E1(V0n_2E0) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)) ) ).
tff(thm_2Ecomplex_2Ei,axiom,
c_2Ecomplex_2Ei_2E0 = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ).
tff(thm_2Ecomplex_2Ecomplex__add,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2ERE_2E1(V1w_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2EIM_2E1(V1w_2E0))) ) ).
tff(thm_2Ecomplex_2Ecomplex__neg,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(c_2Ecomplex_2ERE_2E1(V0z_2E0)),c_2Erealax_2Ereal__neg_2E1(c_2Ecomplex_2EIM_2E1(V0z_2E0))) ) ).
tff(thm_2Ecomplex_2Ecomplex__mul,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2ERE_2E1(V1w_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2EIM_2E1(V1w_2E0))),c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2EIM_2E1(V1w_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2ERE_2E1(V1w_2E0)))) ) ).
tff(thm_2Ecomplex_2Ecomplex__inv,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ecomplex_2EIM_2E1(V0z_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))) ) ).
tff(thm_2Ecomplex_2Ecomplex__sub,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2Ecomplex__div,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__inv_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2Ecomplex__scalar__lmul,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V0k_2E0,c_2Ecomplex_2ERE_2E1(V1z_2E0)),c_2Erealax_2Ereal__mul_2E2(V0k_2E0,c_2Ecomplex_2EIM_2E1(V1z_2E0))) ) ).
tff(thm_2Ecomplex_2Ecomplex__scalar__rmul,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V0z_2E0,V1k_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),V1k_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),V1k_2E0)) ) ).
tff(thm_2Ecomplex_2Econj,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(V0z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ecomplex_2EIM_2E1(V0z_2E0))) ) ).
tff(thm_2Ecomplex_2Emodu,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Emodu_2E1(V0z_2E0) = c_2Etransc_2Esqrt_2E1(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ecomplex_2Earg,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Earg_2E1(V0z_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ecomplex_2EIM_2E1(V0z_2E0)),c_2Etransc_2Eacs_2E1(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0))),c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Eacs_2E1(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)))),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Etransc_2Epi_2E0))) ) ).
tff(thm_2Ecomplex_2Ecomplex__pow__def,axiom,
( ! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,c_2Enum_2E0_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
& ! [V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(V1z_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(V1z_2E0,c_2Ecomplex_2Ecomplex__pow_2E2(V1z_2E0,V2n_2E0)) ) ) ).
tff(thm_2Ecomplex_2Ecomplex__exp,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Etransc_2Eexp_2E1(c_2Ecomplex_2ERE_2E1(V0z_2E0)),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(c_2Ecomplex_2EIM_2E1(V0z_2E0)),c_2Etransc_2Esin_2E1(c_2Ecomplex_2EIM_2E1(V0z_2E0)))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__LEMMA1,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2c_2E0),c_2Erealax_2Ereal__mul_2E2(V1b_2E0,V3d_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3d_2E0),c_2Erealax_2Ereal__mul_2E2(V1b_2E0,V2c_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(V0a_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V1b_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(V2c_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V3d_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__LEMMA2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Etransc_2Esqrt_2E1(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V1y_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))) ).
tff(thm_2Ecomplex_2ECOMPLEX,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2EIM_2E1(V0z_2E0)) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__RE__IM__EQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 = V1w_2E0 )
<=> ( ( c_2Ecomplex_2ERE_2E1(V0z_2E0) = c_2Ecomplex_2ERE_2E1(V1w_2E0) )
& ( c_2Ecomplex_2EIM_2E1(V0z_2E0) = c_2Ecomplex_2EIM_2E1(V1w_2E0) ) ) ) ).
tff(thm_2Ecomplex_2ERE__COMPLEX__OF__REAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2ERE_2E1(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Ecomplex_2EIM__COMPLEX__OF__REAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2EIM_2E1(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__0,axiom,
c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ).
tff(thm_2Ecomplex_2ECOMPLEX__1,axiom,
c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ).
tff(thm_2Ecomplex_2ECOMPLEX__10,axiom,
c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ecomplex_2ECOMPLEX__0__THM,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__COMM,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__add_2E2(V1w_2E0,V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__ASSOC,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__add_2E2(V1w_2E0,V2v_2E0)) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0),V2v_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__RID,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__LID,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0),V0z_2E0) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__RINV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__LINV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__COMM,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__ASSOC,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V2v_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0),V2v_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__RID,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__LID,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0z_2E0) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__RINV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__LINV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__LDISTRIB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__add_2E2(V1w_2E0,V2v_2E0)) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__RDISTRIB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0),V2v_2E0) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V2v_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__LADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V2v_2E0) )
<=> ( V1w_2E0 = V2v_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__RADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V2v_2E0) = c_2Ecomplex_2Ecomplex__add_2E2(V1w_2E0,V2v_2E0) )
<=> ( V0z_2E0 = V1w_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__RID__UNIQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0) = V0z_2E0 )
<=> ( V1w_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__LID__UNIQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0) = V1w_2E0 )
<=> ( V0z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEGNEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0)) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__EQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0) = V1w_2E0 )
<=> ( V0z_2E0 = c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__NEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0) = c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0) )
<=> ( V0z_2E0 = V1w_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__RNEG__UNIQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V1w_2E0 = c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__LNEG__UNIQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0z_2E0 = c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__ADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__RZERO,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__LZERO,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__LMUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),V1w_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__RMUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__MUL2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ENTIRE,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1w_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__0,axiom,
c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__EQ0,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__REFL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__RZERO,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__LZERO,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__LNEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),V1w_2E0) = c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__NEG2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) = c_2Ecomplex_2Ecomplex__sub_2E2(V1w_2E0,V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__SUB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__sub_2E2(V1w_2E0,V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__RNEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) = c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__ADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0),V1w_2E0) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__ADD2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(V1w_2E0,c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0)) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__SUB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0),V0z_2E0) = V1w_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__SUB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__SUB2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0)) = V1w_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__SUB2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD2__SUB2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2u_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0),c_2Ecomplex_2Ecomplex__add_2E2(V2u_2E0,V3v_2E0)) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V2u_2E0),c_2Ecomplex_2Ecomplex__sub_2E2(V1w_2E0,V3v_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__TRIANGLE,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0),c_2Ecomplex_2Ecomplex__sub_2E2(V1w_2E0,V2v_2E0)) = c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V2v_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__0,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0z_2E0 = V1w_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__SUB__LADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 = c_2Ecomplex_2Ecomplex__sub_2E2(V1w_2E0,V2v_2E0) )
<=> ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V2v_2E0) = V1w_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__SUB__RADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0) = V2v_2E0 )
<=> ( V0z_2E0 = c_2Ecomplex_2Ecomplex__add_2E2(V2v_2E0,V1w_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__RNEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) = c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__LNEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),V1w_2E0) = c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__LDISTRIB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__sub_2E2(V1w_2E0,V2v_2E0)) = c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__RDISTRIB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0),V2v_2E0) = c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V2v_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIFFSQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0),c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V0z_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__LMUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0) )
<=> ( ( V0z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1w_2E0 = V2v_2E0 ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__RMUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0) = c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V2v_2E0) )
<=> ( ( V2v_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V0z_2E0 = V1w_2E0 ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__LMUL2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( ( V1w_2E0 = V2v_2E0 )
<=> ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0) ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__RMUL__IMP,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V0z_2E0) = c_2Ecomplex_2Ecomplex__mul_2E2(V2v_2E0,V0z_2E0) ) )
=> ( V1w_2E0 = V2v_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__LMUL__IMP,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0) ) )
=> ( V1w_2E0 = V2v_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__INV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0)) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__MUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1w_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__inv_2E1(V1w_2E0)) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INVINV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0)) = V0z_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__LINV__UNIQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
=> ( V0z_2E0 = c_2Ecomplex_2Ecomplex__inv_2E1(V1w_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__RINV__UNIQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
=> ( V1w_2E0 = c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__0,axiom,
c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV1,axiom,
c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__INV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0)) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__NEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__EQ__0,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__NZ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0) != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__INJ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0) = c_2Ecomplex_2Ecomplex__inv_2E1(V1w_2E0) )
<=> ( V0z_2E0 = V1w_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__LDIV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),V1w_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__RDIV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__DIV2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__neg_2E1(V1w_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__1OVER,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV1,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__ADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V2v_2E0),c_2Ecomplex_2Ecomplex__div_2E2(V1w_2E0,V2v_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0),V2v_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__SUB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V2v_2E0),c_2Ecomplex_2Ecomplex__div_2E2(V1w_2E0,V2v_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0),V2v_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__RMUL__CANCEL,axiom,
! [V0v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0v_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V1z_2E0,V0v_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V2w_2E0,V0v_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(V1z_2E0,V2w_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__LMUL__CANCEL,axiom,
! [V0v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0v_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0v_2E0,V1z_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V0v_2E0,V2w_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(V1z_2E0,V2w_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__DENOM__CANCEL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__div_2E2(V1w_2E0,V0z_2E0),c_2Ecomplex_2Ecomplex__div_2E2(V2v_2E0,V0z_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(V1w_2E0,V2v_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__INNER__CANCEL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__div_2E2(V1w_2E0,V0z_2E0),c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V2v_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(V1w_2E0,V2v_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__OUTER__CANCEL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0),c_2Ecomplex_2Ecomplex__div_2E2(V2v_2E0,V0z_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(V2v_2E0,V1w_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__MUL2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1w_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ! [V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__div_2E2(V2v_2E0,V1w_2E0) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__RAT,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2u_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V1w_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V3v_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0),c_2Ecomplex_2Ecomplex__div_2E2(V2u_2E0,V3v_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V3v_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V2u_2E0)),c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V3v_2E0)) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__RAT,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2u_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V1w_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V3v_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0),c_2Ecomplex_2Ecomplex__div_2E2(V2u_2E0,V3v_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V3v_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V2u_2E0)),c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V3v_2E0)) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__LZERO,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__REFL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__INV2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1w_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__inv_2E1(V1w_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(V1w_2E0,V0z_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__RDIV__EQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V2v_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( ( V0z_2E0 = c_2Ecomplex_2Ecomplex__div_2E2(V1w_2E0,V2v_2E0) )
<=> ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V2v_2E0) = V1w_2E0 ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__LDIV__EQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V2v_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( ( c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V2v_2E0) = V1w_2E0 )
<=> ( V0z_2E0 = c_2Ecomplex_2Ecomplex__mul_2E2(V1w_2E0,V2v_2E0) ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__REAL__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0) = c_2Ecomplex_2Ecomplex__of__real_2E1(V1y_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__REAL__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0),c_2Ecomplex_2Ecomplex__of__real_2E1(V1y_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__REAL__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__REAL__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0),c_2Ecomplex_2Ecomplex__of__real_2E1(V1y_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__REAL__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Erealax_2Einv_2E1(V0x_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__REAL__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0),c_2Ecomplex_2Ecomplex__of__real_2E1(V1y_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__REAL__DIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0),c_2Ecomplex_2Ecomplex__of__real_2E1(V1y_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__NUM__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__of__num_2E1(V0m_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(V1n_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__NUM__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(V0m_2E0),c_2Ecomplex_2Ecomplex__of__num_2E1(V1n_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__OF__NUM__MUL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(V0m_2E0),c_2Ecomplex_2Ecomplex__of__num_2E1(V1n_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2E_2A_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V1l_2E0,V2z_2E0)) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Erealax_2Ereal__mul_2E2(V0k_2E0,V1l_2E0),V2z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__NEG,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0)) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Erealax_2Ereal__neg_2E1(V0k_2E0),V1z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,c_2Ecomplex_2Ecomplex__neg_2E1(V1z_2E0)) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Erealax_2Ereal__neg_2E1(V0k_2E0),V1z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ADD,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Erealax_2Ereal__add_2E2(V0k_2E0,V1l_2E0),V2z_2E0) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2z_2E0),c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V1l_2E0,V2z_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__SUB,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ereal_2Ereal__sub_2E2(V0k_2E0,V1l_2E0),V2z_2E0) = c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2z_2E0),c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V1l_2E0,V2z_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,c_2Ecomplex_2Ecomplex__add_2E2(V1z_2E0,V2w_2E0)) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0),c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,c_2Ecomplex_2Ecomplex__sub_2E2(V1z_2E0,V2w_2E0)) = c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0),c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__SCALAR__LMUL2,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2z_2E0),c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V1l_2E0,V3w_2E0)) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Erealax_2Ereal__mul_2E2(V0k_2E0,V1l_2E0),c_2Ecomplex_2Ecomplex__mul_2E2(V2z_2E0,V3w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__LMUL__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0),V2w_2E0) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,c_2Ecomplex_2Ecomplex__mul_2E2(V1z_2E0,V2w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__RMUL__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V1z_2E0,c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2w_2E0)) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,c_2Ecomplex_2Ecomplex__mul_2E2(V1z_2E0,V2w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ZERO,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ZERO__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ONE,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0z_2E0) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__NEG1,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0z_2E0) = c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DOUBLE,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V0z_2E0) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ENTIRE,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0k_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EQ__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2w_2E0) )
<=> ( ( V0k_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1z_2E0 = V2w_2E0 ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__EQ,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2z_2E0) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V1l_2E0,V2z_2E0) )
<=> ( ( V0k_2E0 = V1l_2E0 )
| ( V2z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__EQ1,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0) = V1z_2E0 )
<=> ( ( V0k_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
| ( V1z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V0k_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0)) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Erealax_2Einv_2E1(V0k_2E0),c_2Ecomplex_2Ecomplex__inv_2E1(V1z_2E0)) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__DIV2,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( V1l_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V3w_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V2z_2E0),c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V1l_2E0,V3w_2E0)) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ereal_2E_2F_2E2(V0k_2E0,V1l_2E0),c_2Ecomplex_2Ecomplex__div_2E2(V2z_2E0,V3w_2E0)) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__MUL__COMM,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0) = c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V1z_2E0,V0k_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2z_2E0,V0k_2E0),V1l_2E0) = c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2z_2E0,c_2Erealax_2Ereal__mul_2E2(V0k_2E0,V1l_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__NEG,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V1z_2E0,V0k_2E0)) = c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V1z_2E0,c_2Erealax_2Ereal__neg_2E1(V0k_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__NEG__SCALAR__RMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(c_2Ecomplex_2Ecomplex__neg_2E1(V1z_2E0),V0k_2E0) = c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V1z_2E0,c_2Erealax_2Ereal__neg_2E1(V0k_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ADD,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2z_2E0,c_2Erealax_2Ereal__add_2E2(V0k_2E0,V1l_2E0)) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2z_2E0,V0k_2E0),c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2z_2E0,V1l_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__RSCALAR__RMUL__SUB,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1l_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2z_2E0,c_2Ereal_2Ereal__sub_2E2(V0k_2E0,V1l_2E0)) = c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2z_2E0,V0k_2E0),c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2z_2E0,V1l_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ADD__RSCALAR__RMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(c_2Ecomplex_2Ecomplex__add_2E2(V1z_2E0,V2w_2E0),V0k_2E0) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V1z_2E0,V0k_2E0),c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2w_2E0,V0k_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SUB__SCALAR__RMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(c_2Ecomplex_2Ecomplex__sub_2E2(V1z_2E0,V2w_2E0),V0k_2E0) = c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V1z_2E0,V0k_2E0),c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V2w_2E0,V0k_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ZERO,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V0z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__ZERO__SCALAR__RMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0),V0k_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ONE,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V0z_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__NEG1,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(V0z_2E0,c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECONJ__REAL__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0) ) ).
tff(thm_2Ecomplex_2ECONJ__NUM__REFL,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(V0n_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(V0n_2E0) ) ).
tff(thm_2Ecomplex_2ECONJ__ADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__add_2E2(c_2Ecomplex_2Econj_2E1(V0z_2E0),c_2Ecomplex_2Econj_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECONJ__NEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__neg_2E1(c_2Ecomplex_2Econj_2E1(V0z_2E0)) ) ).
tff(thm_2Ecomplex_2ECONJ__SUB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__sub_2E2(c_2Ecomplex_2Econj_2E1(V0z_2E0),c_2Ecomplex_2Econj_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECONJ__MUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Econj_2E1(V0z_2E0),c_2Ecomplex_2Econj_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECONJ__INV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Econj_2E1(V0z_2E0)) ) ).
tff(thm_2Ecomplex_2ECONJ__DIV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Econj_2E1(V0z_2E0),c_2Ecomplex_2Econj_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECONJ__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0)) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,c_2Ecomplex_2Econj_2E1(V1z_2E0)) ) ).
tff(thm_2Ecomplex_2ECONJ__CONJ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Econj_2E1(V0z_2E0)) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECONJ__EQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Econj_2E1(V0z_2E0) = V1w_2E0 )
<=> ( V0z_2E0 = c_2Ecomplex_2Econj_2E1(V1w_2E0) ) ) ).
tff(thm_2Ecomplex_2ECONJ__EQ2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Econj_2E1(V0z_2E0) = c_2Ecomplex_2Econj_2E1(V1w_2E0) )
<=> ( V0z_2E0 = V1w_2E0 ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__RCONJ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Econj_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__LCONJ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Econj_2E1(V0z_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ecomplex_2ECONJ__ZERO,axiom,
c_2Ecomplex_2Econj_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ecomplex_2EMODU__POW2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Epow_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).
tff(thm_2Ecomplex_2ERE__IM__LE__MODU,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ecomplex_2ERE_2E1(V0z_2E0)),c_2Ecomplex_2Emodu_2E1(V0z_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ecomplex_2EIM_2E1(V0z_2E0)),c_2Ecomplex_2Emodu_2E1(V0z_2E0))) ) ).
tff(thm_2Ecomplex_2EMODU__POS,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0))) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__RCONJ1,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,c_2Ecomplex_2Econj_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Ereal_2Epow_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__LCONJ1,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Econj_2E1(V0z_2E0),V0z_2E0) = c_2Ecomplex_2Ecomplex__of__real_2E1(c_2Ereal_2Epow_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).
tff(thm_2Ecomplex_2EMODU__NEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0)) = c_2Ecomplex_2Emodu_2E1(V0z_2E0) ) ).
tff(thm_2Ecomplex_2EMODU__SUB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__sub_2E2(V1w_2E0,V0z_2E0)) ) ).
tff(thm_2Ecomplex_2EMODU__CONJ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Econj_2E1(V0z_2E0)) = c_2Ecomplex_2Emodu_2E1(V0z_2E0) ) ).
tff(thm_2Ecomplex_2EMODU__MUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V1w_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2EMODU__0,axiom,
c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ecomplex_2EMODU__1,axiom,
c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Ecomplex_2EMODU__COMPLEX__INV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0)) = c_2Erealax_2Einv_2E1(c_2Ecomplex_2Emodu_2E1(V0z_2E0)) ) ) ).
tff(thm_2Ecomplex_2EMODU__DIV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V1w_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V1w_2E0)) ) ) ).
tff(thm_2Ecomplex_2EMODU__SCALAR__LMUL,axiom,
! [V0k_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V0k_2E0,V1z_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0k_2E0),c_2Ecomplex_2Emodu_2E1(V1z_2E0)) ) ).
tff(thm_2Ecomplex_2EMODU__REAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__of__real_2E1(V0x_2E0)) = c_2Ereal_2Eabs_2E1(V0x_2E0) ) ).
tff(thm_2Ecomplex_2EMODU__NUM,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(V0n_2E0)) = c_2Ereal_2Ereal__of__num_2E1(V0n_2E0) ) ).
tff(thm_2Ecomplex_2EMODU__ZERO,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( c_2Ecomplex_2Emodu_2E1(V0z_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2EMODU__NZ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0))) ) ).
tff(thm_2Ecomplex_2EMODU__CASES,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
| p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0))) ) ).
tff(thm_2Ecomplex_2ERE__DIV__MODU__BOUNDS,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Ecomplex_2EIM__DIV__MODU__BOUNDS,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2E_2F_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Ecomplex_2ERE__DIV__MODU__ACS__BOUNDS,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eacs_2E1(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Eacs_2E1(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0))),c_2Etransc_2Epi_2E0)) ) ) ).
tff(thm_2Ecomplex_2EIM__DIV__MODU__ASN__BOUNDS,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(c_2Etransc_2Epi_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Etransc_2Easn_2E1(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Easn_2E1(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0))),c_2Ereal_2E_2F_2E2(c_2Etransc_2Epi_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))) ) ) ).
tff(thm_2Ecomplex_2ERE__DIV__MODU__ACS__COS,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Etransc_2Ecos_2E1(c_2Etransc_2Eacs_2E1(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)))) = c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)) ) ) ).
tff(thm_2Ecomplex_2EIM__DIV__MODU__ASN__SIN,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Etransc_2Esin_2E1(c_2Etransc_2Easn_2E1(c_2Ereal_2E_2F_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)))) = c_2Ereal_2E_2F_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)) ) ) ).
tff(thm_2Ecomplex_2EARG__COS,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Etransc_2Ecos_2E1(c_2Ecomplex_2Earg_2E1(V0z_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ecomplex_2ERE_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)) ) ) ).
tff(thm_2Ecomplex_2EARG__SIN,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Etransc_2Esin_2E1(c_2Ecomplex_2Earg_2E1(V0z_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ecomplex_2EIM_2E1(V0z_2E0),c_2Ecomplex_2Emodu_2E1(V0z_2E0)) ) ) ).
tff(thm_2Ecomplex_2ERE__MODU__ARG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2ERE_2E1(V0z_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Etransc_2Ecos_2E1(c_2Ecomplex_2Earg_2E1(V0z_2E0))) ) ).
tff(thm_2Ecomplex_2EIM__MODU__ARG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2EIM_2E1(V0z_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Etransc_2Esin_2E1(c_2Ecomplex_2Earg_2E1(V0z_2E0))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__TRIANGLE,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(c_2Ecomplex_2Earg_2E1(V0z_2E0)),c_2Etransc_2Esin_2E1(c_2Ecomplex_2Earg_2E1(V0z_2E0)))) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MODU__ARG__EQ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 = V1w_2E0 )
<=> ( ( c_2Ecomplex_2Emodu_2E1(V0z_2E0) = c_2Ecomplex_2Emodu_2E1(V1w_2E0) )
& ( c_2Ecomplex_2Earg_2E1(V0z_2E0) = c_2Ecomplex_2Earg_2E1(V1w_2E0) ) ) ) ).
tff(thm_2Ecomplex_2EMODU__UNIT,axiom,
! [A_27a: $tType,V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: A_27a] : ( c_2Ecomplex_2Emodu_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__MUL__ARG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V0x_2E0)),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(V1y_2E0),c_2Etransc_2Esin_2E1(V1y_2E0))) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)),c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__INV__ARG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)),c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__DIV__ARG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__div_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V0x_2E0)),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(V1y_2E0),c_2Etransc_2Esin_2E1(V1y_2E0))) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)),c_2Etransc_2Esin_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ecomplex_2Ecomplex__pow__def__compute,axiom,
( ! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,c_2Enum_2E0_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
& ! [V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(V1z_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V2n_2E0))) = c_2Ecomplex_2Ecomplex__mul_2E2(V1z_2E0,c_2Ecomplex_2Ecomplex__pow_2E2(V1z_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V2n_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
& ! [V3z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(V3z_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V4n_2E0))) = c_2Ecomplex_2Ecomplex__mul_2E2(V3z_2E0,c_2Ecomplex_2Ecomplex__pow_2E2(V3z_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V4n_2E0)))) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0),c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__NZ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,V1n_2E0) != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POWINV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( V0z_2E0 != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,V1n_2E0)) = c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0),V1n_2E0) ) ) ).
tff(thm_2Ecomplex_2EMODU__COMPLEX__POW,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Emodu_2E1(c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,V1n_2E0)) = c_2Ereal_2Epow_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),V1n_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__ADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V2n_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,V1m_2E0),c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,V2n_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__1,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Ecomplex_2Ecomplex__mul_2E2(V0z_2E0,V0z_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__ONE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0n_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__MUL,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__mul_2E2(V1z_2E0,V2w_2E0),V0n_2E0) = c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__pow_2E2(V1z_2E0,V0n_2E0),c_2Ecomplex_2Ecomplex__pow_2E2(V2w_2E0,V0n_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__INV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__inv_2E1(V0z_2E0),V1n_2E0) = c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,V1n_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__DIV,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__div_2E2(V0z_2E0,V1w_2E0),V2n_2E0) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,V2n_2E0),c_2Ecomplex_2Ecomplex__pow_2E2(V1w_2E0,V2n_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__L,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1k_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(V1k_2E0,V2z_2E0),V0n_2E0) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ereal_2Epow_2E2(V1k_2E0,V0n_2E0),c_2Ecomplex_2Ecomplex__pow_2E2(V2z_2E0,V0n_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__ZERO,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__pow_2E2(V1z_2E0,V0n_2E0) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( V1z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__ZERO__EQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( c_2Ecomplex_2Ecomplex__pow_2E2(V1z_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V1z_2E0 = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__POW__POW,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,V1m_2E0),V2n_2E0) = c_2Ecomplex_2Ecomplex__pow_2E2(V0z_2E0,c_2Earithmetic_2E_2A_2E2(V1m_2E0,V2n_2E0)) ) ).
tff(thm_2Ecomplex_2EDE__MOIVRE__LEMMA,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V0x_2E0)),V1n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),V0x_2E0)),c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),V0x_2E0))) ) ).
tff(thm_2Ecomplex_2EDE__MOIVRE__THM,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(c_2Ecomplex_2Earg_2E1(V0z_2E0)),c_2Etransc_2Esin_2E1(c_2Ecomplex_2Earg_2E1(V0z_2E0)))),V1n_2E0) = c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ereal_2Epow_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),V1n_2E0),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),c_2Ecomplex_2Earg_2E1(V0z_2E0))),c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),c_2Ecomplex_2Earg_2E1(V0z_2E0))))) ) ).
tff(thm_2Ecomplex_2EEXP__IMAGINARY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(c_2Ecomplex_2Ei_2E0,V0x_2E0)) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V0x_2E0)) ) ).
tff(thm_2Ecomplex_2EEULER__FORMULE,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ecomplex_2Emodu_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__scalar__rmul_2E2(c_2Ecomplex_2Ei_2E0,c_2Ecomplex_2Earg_2E1(V0z_2E0)))) = V0z_2E0 ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__ADD,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__exp_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__NEG,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__inv_2E1(c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__SUB,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__sub_2E2(V0z_2E0,V1w_2E0)) = c_2Ecomplex_2Ecomplex__div_2E2(c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__exp_2E1(V1w_2E0)) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__N,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__scalar__lmul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),V0z_2E0)) = c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0),V1n_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__N2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__of__num_2E1(V1n_2E0),V0z_2E0)) = c_2Ecomplex_2Ecomplex__pow_2E2(c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0),V1n_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__0,axiom,
c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__NZ,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0) != c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__ADD__MUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1w_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__add_2E2(V0z_2E0,V1w_2E0)),c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0))) = c_2Ecomplex_2Ecomplex__exp_2E1(V1w_2E0) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__NEG__MUL,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0),c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0))) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Ecomplex_2ECOMPLEX__EXP__NEG__MUL2,axiom,
! [V0z_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecomplex_2Ecomplex__mul_2E2(c_2Ecomplex_2Ecomplex__exp_2E1(c_2Ecomplex_2Ecomplex__neg_2E1(V0z_2E0)),c_2Ecomplex_2Ecomplex__exp_2E1(V0z_2E0)) = c_2Ecomplex_2Ecomplex__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
%------------------------------------------------------------------------------