ITP001 Axioms: ITP132_7.ax
%------------------------------------------------------------------------------
% File : ITP132_7 : TPTP v9.0.0. 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 : transc.ax [Gau19]
% : HL4132_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 440 ( 125 unt; 131 typ; 0 def)
% Number of atoms : 750 ( 308 equ)
% Maximal formula atoms : 36 ( 1 avg)
% Number of connectives : 466 ( 25 ~; 5 |; 179 &)
% ( 43 <=>; 214 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 100 ( 62 >; 38 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 127 ( 127 usr; 56 con; 0-4 aty)
% Number of variables : 762 ( 711 !; 27 ?; 762 :)
% ( 24 !>; 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_2Eprim__rec_2E_3C_2E0,type,
c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Eprim__rec_2E_3C_2E2,type,
c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2E_3C_3D_2E0,type,
c_2Earithmetic_2E_3C_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3C_3D_2E2,type,
c_2Earithmetic_2E_3C_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
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_2Earithmetic_2E_3E_3D_2E0,type,
c_2Earithmetic_2E_3E_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3E_3D_2E2,type,
c_2Earithmetic_2E_3E_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > 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_2Ebool_2E_3F_21_2E0,type,
c_2Ebool_2E_3F_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_21_2E1,type,
c_2Ebool_2E_3F_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_40_2E0,type,
c_2Emin_2E_40_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).
tff(c_2Emin_2E_40_2E1,type,
c_2Emin_2E_40_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).
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_2Earithmetic_2EDIV_2E0,type,
c_2Earithmetic_2EDIV_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EDIV_2E2,type,
c_2Earithmetic_2EDIV_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Etransc_2EDint_2E0,type,
c_2Etransc_2EDint_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))) ).
tff(c_2Etransc_2EDint_2E3,type,
c_2Etransc_2EDint_2E3: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2EEVEN_2E0,type,
c_2Earithmetic_2EEVEN_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Earithmetic_2EEVEN_2E1,type,
c_2Earithmetic_2EEVEN_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2EFACT_2E0,type,
c_2Earithmetic_2EFACT_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EFACT_2E1,type,
c_2Earithmetic_2EFACT_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_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_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_2Etransc_2Eatn_2E0,type,
c_2Etransc_2Eatn_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Eatn_2E1,type,
c_2Etransc_2Eatn_2E1: 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_2Elim_2Ediffl_2E0,type,
c_2Elim_2Ediffl_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))) ).
tff(c_2Elim_2Ediffl_2E3,type,
c_2Elim_2Ediffl_2E3: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Epowser_2Ediffs_2E0,type,
c_2Epowser_2Ediffs_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) ).
tff(c_2Epowser_2Ediffs_2E2,type,
c_2Epowser_2Ediffs_2E2: ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) * tyop_2Enum_2Enum ) > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Edivision_2E0,type,
c_2Etransc_2Edivision_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)) ).
tff(c_2Etransc_2Edivision_2E2,type,
c_2Etransc_2Edivision_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ) > tyop_2Emin_2Ebool ).
tff(c_2Etransc_2Edsize_2E0,type,
c_2Etransc_2Edsize_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Enum_2Enum) ).
tff(c_2Etransc_2Edsize_2E1,type,
c_2Etransc_2Edsize_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Enum_2Enum ).
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_2Etransc_2Efine_2E0,type,
c_2Etransc_2Efine_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool)) ).
tff(c_2Etransc_2Efine_2E2,type,
c_2Etransc_2Efine_2E2: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) ) > tyop_2Emin_2Ebool ).
tff(c_2Etransc_2Egauge_2E0,type,
c_2Etransc_2Egauge_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)) ).
tff(c_2Etransc_2Egauge_2E2,type,
c_2Etransc_2Egauge_2E2: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Emin_2Ebool ).
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_2Etransc_2Eln_2E0,type,
c_2Etransc_2Eln_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Eln_2E1,type,
c_2Etransc_2Eln_2E1: 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_2Eroot_2E0,type,
c_2Etransc_2Eroot_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Etransc_2Eroot_2E2,type,
c_2Etransc_2Eroot_2E2: ( tyop_2Enum_2Enum * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Erpow_2E0,type,
c_2Etransc_2Erpow_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Etransc_2Erpow_2E2,type,
c_2Etransc_2Erpow_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Ersum_2E0,type,
c_2Etransc_2Ersum_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).
tff(c_2Etransc_2Ersum_2E2,type,
c_2Etransc_2Ersum_2E2: ( tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) * tyop_2Emin_2Efun(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_2Ereal_2Esum_2E0,type,
c_2Ereal_2Esum_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Esum_2E2,type,
c_2Ereal_2Esum_2E2: ( tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ).
tff(c_2Eseq_2Esuminf_2E0,type,
c_2Eseq_2Esuminf_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Eseq_2Esuminf_2E1,type,
c_2Eseq_2Esuminf_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Eseq_2Esums_2E0,type,
c_2Eseq_2Esums_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Eseq_2Esums_2E2,type,
c_2Eseq_2Esums_2E2: ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Etransc_2Etan_2E0,type,
c_2Etransc_2Etan_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Etransc_2Etan_2E1,type,
c_2Etransc_2Etan_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Etransc_2Etdiv_2E0,type,
c_2Etransc_2Etdiv_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool)) ).
tff(c_2Etransc_2Etdiv_2E2,type,
c_2Etransc_2Etdiv_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) ) > tyop_2Emin_2Ebool ).
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_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Erealax_2Ereal_29_20mono_2Etyop_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Erealax_2Ereal_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),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_2Eprim__rec_2E_3C_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3C_3D_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(arityeq2_2Ec_2Earithmetic_2E_3E_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3E_3D_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_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Emin_2E_40_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Emin_2E_40_2E0(tyop_2Erealax_2Ereal),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(arityeq2_2Ec_2Earithmetic_2EDIV_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV_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_2EDIV_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Etransc_2EDint_2E3,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2EDint_2E3(X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Etransc_2EDint_2E0,X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EEVEN_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EEVEN_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,c_2Earithmetic_2EEVEN_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EFACT_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EFACT_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EFACT_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_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_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(arityeq1_2Ec_2Etransc_2Eatn_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eatn_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Eatn_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(arityeq3_2Ec_2Elim_2Ediffl_2E3,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Elim_2Ediffl_2E3(X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Elim_2Ediffl_2E0,X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Epowser_2Ediffs_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Enum_2Enum] : ( c_2Epowser_2Ediffs_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),c_2Epowser_2Ediffs_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Etransc_2Edivision_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Etransc_2Edivision_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),c_2Etransc_2Edivision_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Edsize_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Etransc_2Edsize_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Enum_2Enum,c_2Etransc_2Edsize_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(arityeq2_2Ec_2Etransc_2Efine_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] : ( c_2Etransc_2Efine_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool),c_2Etransc_2Efine_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Etransc_2Egauge_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Etransc_2Egauge_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),c_2Etransc_2Egauge_2E0,X0_2E0),X1_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_2Etransc_2Eln_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eln_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Eln_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(arityeq2_2Ec_2Etransc_2Eroot_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eroot_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Etransc_2Eroot_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Etransc_2Erpow_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Erpow_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_2Etransc_2Erpow_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Etransc_2Ersum_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Etransc_2Ersum_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Etransc_2Ersum_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(arityeq2_2Ec_2Ereal_2Esum_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Ereal_2Esum_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Esuminf_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Esuminf_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Eseq_2Esuminf_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eseq_2Esums_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Eseq_2Esums_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Eseq_2Esums_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Etransc_2Etan_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Etan_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Etransc_2Etan_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Etransc_2Etdiv_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] : ( c_2Etransc_2Etdiv_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool),c_2Etransc_2Etdiv_2E0,X0_2E0),X1_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_2Etransc_2Eatn,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0),V1x_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_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))))),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1x_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))))))
& ( c_2Etransc_2Etan_2E1(V1x_2E0) = V0y_2E0 ) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eatn_2E1(V0y_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0)) ) ) ).
tff(thm_2Etransc_2Eacs,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0),V1x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Etransc_2Epi_2E0))
& ( c_2Etransc_2Ecos_2E1(V1x_2E0) = V0y_2E0 ) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eacs_2E1(V0y_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0)) ) ) ).
tff(thm_2Etransc_2Easn,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0),V1x_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))))),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_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))))))
& ( c_2Etransc_2Esin_2E1(V1x_2E0) = V0y_2E0 ) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Easn_2E1(V0y_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0)) ) ) ).
tff(thm_2Etransc_2Etan,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Etan_2E1(V0x_2E0) = c_2Ereal_2E_2F_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0)) ) ).
tff(thm_2Etransc_2Epi,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))
& ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> ( c_2Etransc_2Epi_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_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,F0_2E0)) ) ) ).
tff(thm_2Etransc_2Esqrt,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esqrt_2E1(V0x_2E0) = c_2Etransc_2Eroot_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V0x_2E0) ) ).
tff(thm_2Etransc_2Eroot,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal,V0n_2E0: tyop_2Enum_2Enum,V2u_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1x_2E0),V0n_2E0),V2u_2E0))
<=> ( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2u_2E0)) )
& ( c_2Ereal_2Epow_2E2(V2u_2E0,V0n_2E0) = V1x_2E0 ) ) )
=> ! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eroot_2E2(V0n_2E0,V1x_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1x_2E0),V0n_2E0)) ) ) ).
tff(thm_2Etransc_2Eexp,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(V0x_2E0) = c_2Eseq_2Esuminf_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)) ) ) ).
tff(thm_2Etransc_2Ecos,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V1n_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Eseq_2Esuminf_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)) ) ) ).
tff(thm_2Etransc_2Esin,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0)))),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin_2E1(V0x_2E0) = c_2Eseq_2Esuminf_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)) ) ) ).
tff(thm_2Etransc_2Eln,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1u_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1u_2E0))
<=> ( c_2Etransc_2Eexp_2E1(V1u_2E0) = V0x_2E0 ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eln_2E1(V0x_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0)) ) ) ).
tff(thm_2Etransc_2Edivision,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2D_2E0))
<=> ( ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2D_2E0,c_2Enum_2E0_2E0) = V0a_2E0 )
& ? [V3N_2E0: tyop_2Enum_2Enum] :
( ! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V4n_2E0,V3N_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2D_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2D_2E0,c_2Enum_2ESUC_2E1(V4n_2E0)))) )
& ! [V5n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V5n_2E0,V3N_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2D_2E0,V5n_2E0) = V1b_2E0 ) ) ) ) ) ).
tff(thm_2Etransc_2Edsize,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1N_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0D_2E0),V1N_2E0))
<=> ( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,V1N_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)))) )
& ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3n_2E0,V1N_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V3n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V1N_2E0) ) ) ) )
=> ! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Etransc_2Edsize_2E1(V0D_2E0) = c_2Emin_2E_40_2E1(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0D_2E0)) ) ) ).
tff(thm_2Etransc_2Etdiv,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2D_2E0,V3p_2E0)))
<=> ( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2D_2E0))
& ! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2D_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V3p_2E0,V4n_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V3p_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2D_2E0,c_2Enum_2ESUC_2E1(V4n_2E0)))) ) ) ) ).
tff(thm_2Etransc_2Egauge,axiom,
! [V0E_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Egauge_2E2(V0E_2E0,V1g_2E0))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0E_2E0,V2x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V2x_2E0))) ) ) ).
tff(thm_2Etransc_2Efine,axiom,
! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Efine_2E2(V0g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1D_2E0,V2p_2E0)))
<=> ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3n_2E0,c_2Etransc_2Edsize_2E1(V1D_2E0)))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1D_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1D_2E0,V3n_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0g_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2p_2E0,V3n_2E0)))) ) ) ).
tff(thm_2Etransc_2Ersum,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2f_2E0),V1p_2E0),V0D_2E0),V3n_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1p_2E0,V3n_2E0)),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V3n_2E0))) )
=> ! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Etransc_2Ersum_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0D_2E0,V1p_2E0),V2f_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2f_2E0),V1p_2E0),V0D_2E0)) ) ) ).
tff(thm_2Etransc_2EDint,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1b_2E0),V6x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V1b_2E0)) ) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2f_2E0,V3k_2E0))
<=> ! [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
=> ? [V5g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Egauge_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1b_2E0),V5g_2E0))
& ! [V7D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V8p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7D_2E0,V8p_2E0)))
& p(c_2Etransc_2Efine_2E2(V5g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7D_2E0,V8p_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Etransc_2Ersum_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7D_2E0,V8p_2E0),V2f_2E0),V3k_2E0)),V4e_2E0)) ) ) ) ) ) ).
tff(thm_2Etransc_2Erpow__def,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0) = c_2Etransc_2Eexp_2E1(c_2Erealax_2Ereal__mul_2E2(V1b_2E0,c_2Etransc_2Eln_2E1(V0a_2E0))) ) ).
tff(thm_2Etransc_2ESIN__COS__SQRT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Esin_2E1(V0x_2E0)))
=> ( c_2Etransc_2Esin_2E1(V0x_2E0) = c_2Etransc_2Esqrt_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Etransc_2ECOS__SIN__SQRT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0)))
=> ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Etransc_2Esqrt_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Etransc_2ESIN__ACS__NZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( c_2Etransc_2Esin_2E1(c_2Etransc_2Eacs_2E1(V0x_2E0)) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Etransc_2ECOS__ASN__NZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( c_2Etransc_2Ecos_2E1(c_2Etransc_2Easn_2E1(V0x_2E0)) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Etransc_2ECOS__ATN__NZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(c_2Etransc_2Eatn_2E1(V0x_2E0)) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Etransc_2ECOS__SIN__SQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( 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))))),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_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)))))) )
=> ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Etransc_2Esqrt_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Etransc_2ESIN__COS__SQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Etransc_2Epi_2E0)) )
=> ( c_2Etransc_2Esin_2E1(V0x_2E0) = c_2Etransc_2Esqrt_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Etransc_2ETAN__SEC,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos_2E1(V0x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Etransc_2Etan_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Epow_2E2(c_2Erealax_2Einv_2E1(c_2Etransc_2Ecos_2E1(V0x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Etransc_2ETAN__ATN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_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))))),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_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)))))) )
=> ( c_2Etransc_2Eatn_2E1(c_2Etransc_2Etan_2E1(V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Etransc_2EATN__BOUNDS,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_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_2Eatn_2E1(V0y_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Eatn_2E1(V0y_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_2Etransc_2EATN__TAN,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Etan_2E1(c_2Etransc_2Eatn_2E1(V0y_2E0)) = V0y_2E0 ) ).
tff(thm_2Etransc_2EATN,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_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_2Eatn_2E1(V0y_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Eatn_2E1(V0y_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))))))
& ( c_2Etransc_2Etan_2E1(c_2Etransc_2Eatn_2E1(V0y_2E0)) = V0y_2E0 ) ) ).
tff(thm_2Etransc_2ECOS__ACS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Etransc_2Epi_2E0)) )
=> ( c_2Etransc_2Eacs_2E1(c_2Etransc_2Ecos_2E1(V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Etransc_2EACS__BOUNDS__LT,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eacs_2E1(V0y_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Eacs_2E1(V0y_2E0),c_2Etransc_2Epi_2E0)) ) ) ).
tff(thm_2Etransc_2EACS__BOUNDS,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( 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)))),V0y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eacs_2E1(V0y_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Eacs_2E1(V0y_2E0),c_2Etransc_2Epi_2E0)) ) ) ).
tff(thm_2Etransc_2EACS__COS,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( 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)))),V0y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( c_2Etransc_2Ecos_2E1(c_2Etransc_2Eacs_2E1(V0y_2E0)) = V0y_2E0 ) ) ).
tff(thm_2Etransc_2EACS,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( 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)))),V0y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eacs_2E1(V0y_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Eacs_2E1(V0y_2E0),c_2Etransc_2Epi_2E0))
& ( c_2Etransc_2Ecos_2E1(c_2Etransc_2Eacs_2E1(V0y_2E0)) = V0y_2E0 ) ) ) ).
tff(thm_2Etransc_2ESIN__ASN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( 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))))),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_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)))))) )
=> ( c_2Etransc_2Easn_2E1(c_2Etransc_2Esin_2E1(V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Etransc_2EASN__BOUNDS__LT,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( p(c_2Erealax_2Ereal__lt_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(V0y_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Easn_2E1(V0y_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_2Etransc_2EASN__BOUNDS,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( 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)))),V0y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_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(V0y_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Easn_2E1(V0y_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_2Etransc_2EASN__SIN,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( 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)))),V0y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( c_2Etransc_2Esin_2E1(c_2Etransc_2Easn_2E1(V0y_2E0)) = V0y_2E0 ) ) ).
tff(thm_2Etransc_2EASN,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( 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)))),V0y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_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(V0y_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Easn_2E1(V0y_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))))))
& ( c_2Etransc_2Esin_2E1(c_2Etransc_2Easn_2E1(V0y_2E0)) = V0y_2E0 ) ) ) ).
tff(thm_2Etransc_2ETAN__TOTAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0),V1x_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_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))))),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1x_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))))))
& ( c_2Etransc_2Etan_2E1(V1x_2E0) = V0y_2E0 ) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0))) ) ).
tff(thm_2Etransc_2ETAN__TOTAL__POS,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0))
=> ? [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1x_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))))))
& ( c_2Etransc_2Etan_2E1(V1x_2E0) = V0y_2E0 ) ) ) ).
tff(thm_2Etransc_2ETAN__TOTAL__LEMMA,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0))
=> ? [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V1x_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))))))
& p(c_2Erealax_2Ereal__lt_2E2(V0y_2E0,c_2Etransc_2Etan_2E1(V1x_2E0))) ) ) ).
tff(thm_2Etransc_2EDIFF__TAN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos_2E1(V0x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Etan_2E0,c_2Erealax_2Einv_2E1(c_2Ereal_2Epow_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0)) ) ).
tff(thm_2Etransc_2ETAN__POS__PI2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_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)))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Etan_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ETAN__DOUBLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Ecos_2E1(V0x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0)) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Etransc_2Etan_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0)) = c_2Ereal_2E_2F_2E2(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_2Etan_2E1(V0x_2E0)),c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Etransc_2Etan_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Etransc_2ETAN__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( ( c_2Etransc_2Ecos_2E1(V0x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Etransc_2Ecos_2E1(V1y_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Etransc_2Etan_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2E_2F_2E2(c_2Erealax_2Ereal__add_2E2(c_2Etransc_2Etan_2E1(V0x_2E0),c_2Etransc_2Etan_2E1(V1y_2E0)),c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Etan_2E1(V0x_2E0),c_2Etransc_2Etan_2E1(V1y_2E0)))) ) ) ).
tff(thm_2Etransc_2ETAN__PERIODIC,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Etan_2E1(c_2Erealax_2Ereal__add_2E2(V0x_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))) = c_2Etransc_2Etan_2E1(V0x_2E0) ) ).
tff(thm_2Etransc_2ETAN__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Etan_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Etan_2E1(V0x_2E0)) ) ).
tff(thm_2Etransc_2ETAN__NPI,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Etransc_2Etan_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Etransc_2Epi_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Etransc_2ETAN__PI,axiom,
c_2Etransc_2Etan_2E1(c_2Etransc_2Epi_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Etransc_2ETAN__0,axiom,
c_2Etransc_2Etan_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Etransc_2ESIN__ZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Esin_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ? [V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(V1n_2E0))
& ( V0x_2E0 = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_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))))) ) )
| ? [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(V2n_2E0))
& ( V0x_2E0 = c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2n_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_2Etransc_2ECOS__ZERO,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ? [V1n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Earithmetic_2EEVEN_2E1(V1n_2E0))
& ( V0x_2E0 = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_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))))) ) )
| ? [V2n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Earithmetic_2EEVEN_2E1(V2n_2E0))
& ( V0x_2E0 = c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2n_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_2Etransc_2ESIN__ZERO__LEMMA,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& ( c_2Etransc_2Esin_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ? [V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(V1n_2E0))
& ( V0x_2E0 = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_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_2Etransc_2ECOS__ZERO__LEMMA,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ? [V1n_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Earithmetic_2EEVEN_2E1(V1n_2E0))
& ( V0x_2E0 = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_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_2Etransc_2ESIN__TOTAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0),V1x_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))))),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_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))))))
& ( c_2Etransc_2Esin_2E1(V1x_2E0) = V0y_2E0 ) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( 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)))),V0y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0))) ) ) ).
tff(thm_2Etransc_2ECOS__TOTAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0),V1x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Etransc_2Epi_2E0))
& ( c_2Etransc_2Ecos_2E1(V1x_2E0) = V0y_2E0 ) ) )
=> ! [V0y_2E0: tyop_2Erealax_2Ereal] :
( ( 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)))),V0y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0y_2E0))) ) ) ).
tff(thm_2Etransc_2ESIN__POS__PI__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Etransc_2Epi_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESIN__POS__PI2__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_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)))))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ECOS__POS__PI__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( 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))))),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_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)))))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ECOS__POS__PI2__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_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)))))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESIN__POS__PI,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Etransc_2Epi_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ECOS__POS__PI,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_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))))),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_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)))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ECOS__POS__PI2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_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)))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESIN__POS__PI2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_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)))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESIN__NPI,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Etransc_2Epi_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Etransc_2ECOS__NPI,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Etransc_2Epi_2E0)) = c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0n_2E0) ) ).
tff(thm_2Etransc_2ECOS__PERIODIC,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__add_2E2(V0x_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))) = c_2Etransc_2Ecos_2E1(V0x_2E0) ) ).
tff(thm_2Etransc_2ESIN__PERIODIC,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__add_2E2(V0x_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))) = c_2Etransc_2Esin_2E1(V0x_2E0) ) ).
tff(thm_2Etransc_2ECOS__PERIODIC__PI,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Etransc_2Epi_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Ecos_2E1(V0x_2E0)) ) ).
tff(thm_2Etransc_2ESIN__PERIODIC__PI,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,c_2Etransc_2Epi_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Esin_2E1(V0x_2E0)) ) ).
tff(thm_2Etransc_2ECOS__SIN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Etransc_2Esin_2E1(c_2Ereal_2Ereal__sub_2E2(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)))),V0x_2E0)) ) ).
tff(thm_2Etransc_2ESIN__COS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin_2E1(V0x_2E0) = c_2Etransc_2Ecos_2E1(c_2Ereal_2Ereal__sub_2E2(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)))),V0x_2E0)) ) ).
tff(thm_2Etransc_2ESIN__PI,axiom,
c_2Etransc_2Esin_2E1(c_2Etransc_2Epi_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Etransc_2ECOS__PI,axiom,
c_2Etransc_2Ecos_2E1(c_2Etransc_2Epi_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ).
tff(thm_2Etransc_2ESIN__PI2,axiom,
c_2Etransc_2Esin_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_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Etransc_2EPI__POS,axiom,
p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Epi_2E0)) ).
tff(thm_2Etransc_2EPI2__BOUNDS,axiom,
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_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))))))
& p(c_2Erealax_2Ereal__lt_2E2(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_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Etransc_2ECOS__PI2,axiom,
c_2Etransc_2Ecos_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_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Etransc_2EPI2,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))
& ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_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)))) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,F0_2E0) ) ) ).
tff(thm_2Etransc_2ECOS__ISZERO,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F0_2E0,V0x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))
& ( c_2Etransc_2Ecos_2E1(V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> p(c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,F0_2E0)) ) ).
tff(thm_2Etransc_2ECOS__2,axiom,
p(c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Ecos_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ).
tff(thm_2Etransc_2ECOS__PAIRED,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1n_2E0)))),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1n_2E0))) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESIN__POS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESIN__PAIRED,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1n_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1n_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ECOS__DOUBLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Epow_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).
tff(thm_2Etransc_2ESIN__DOUBLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_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_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ECOS__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Etransc_2Ecos_2E1(V0x_2E0) ) ).
tff(thm_2Etransc_2ESIN__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Esin_2E1(V0x_2E0)) ) ).
tff(thm_2Etransc_2ECOS__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Ecos_2E1(V1y_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V1y_2E0))) ) ).
tff(thm_2Etransc_2ESIN__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Etransc_2Ecos_2E1(V1y_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V1y_2E0))) ) ).
tff(thm_2Etransc_2ESIN__COS__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__add_2E2(c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)),c_2Etransc_2Esin_2E1(V0x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)),c_2Etransc_2Ecos_2E1(V0x_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_2Etransc_2ESIN__COS__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Etransc_2Esin_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Etransc_2Ecos_2E1(V1y_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V1y_2E0)))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Etransc_2Ecos_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)),c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Etransc_2Ecos_2E1(V1y_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Etransc_2Esin_2E1(V1y_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_2Etransc_2ECOS__BOUNDS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( 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_2Etransc_2Ecos_2E1(V0x_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Etransc_2ECOS__BOUND,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Etransc_2Ecos_2E1(V0x_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Etransc_2ESIN__BOUNDS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( 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_2Etransc_2Esin_2E1(V0x_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(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_2Etransc_2ESIN__BOUND,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(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_2Etransc_2ESIN__CIRCLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Epow_2E2(c_2Etransc_2Esin_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(c_2Etransc_2Ecos_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_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_2Etransc_2ECOS__0,axiom,
c_2Etransc_2Ecos_2E1(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_2Etransc_2ESIN__0,axiom,
c_2Etransc_2Esin_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Etransc_2ESQRT__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = V1y_2E0 )
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) )
=> ( V0x_2E0 = c_2Etransc_2Esqrt_2E1(V1y_2E0) ) ) ).
tff(thm_2Etransc_2EREAL__DIV__SQRT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Ereal_2E_2F_2E2(V0x_2E0,c_2Etransc_2Esqrt_2E1(V0x_2E0)) = c_2Etransc_2Esqrt_2E1(V0x_2E0) ) ) ).
tff(thm_2Etransc_2ESQRT__EVEN__POW2,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2EEVEN_2E1(V0n_2E0))
=> ( c_2Etransc_2Esqrt_2E1(c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0n_2E0)) = c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2EDIV_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ).
tff(thm_2Etransc_2ESQRT__MONO__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Esqrt_2E1(V0x_2E0),c_2Etransc_2Esqrt_2E1(V1y_2E0))) ) ).
tff(thm_2Etransc_2ESQRT__DIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ( c_2Etransc_2Esqrt_2E1(c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2E_2F_2E2(c_2Etransc_2Esqrt_2E1(V0x_2E0),c_2Etransc_2Esqrt_2E1(V1y_2E0)) ) ) ).
tff(thm_2Etransc_2ESQRT__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Etransc_2Esqrt_2E1(c_2Erealax_2Einv_2E1(V0x_2E0)) = c_2Erealax_2Einv_2E1(c_2Etransc_2Esqrt_2E1(V0x_2E0)) ) ) ).
tff(thm_2Etransc_2ESQRT__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ( c_2Etransc_2Esqrt_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Esqrt_2E1(V0x_2E0),c_2Etransc_2Esqrt_2E1(V1y_2E0)) ) ) ).
tff(thm_2Etransc_2ESQRT__POS__UNIQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0))
& ( c_2Ereal_2Epow_2E2(V1y_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = V0x_2E0 ) )
=> ( c_2Etransc_2Esqrt_2E1(V0x_2E0) = V1y_2E0 ) ) ).
tff(thm_2Etransc_2EPOW__2__SQRT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Etransc_2Esqrt_2E1(c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0x_2E0 ) ) ).
tff(thm_2Etransc_2ESQRT__POW__2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Ereal_2Epow_2E2(c_2Etransc_2Esqrt_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = V0x_2E0 ) ) ).
tff(thm_2Etransc_2ESQRT__POW2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow_2E2(c_2Etransc_2Esqrt_2E1(V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = V0x_2E0 )
<=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) ) ).
tff(thm_2Etransc_2ESQRT__POS__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Esqrt_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESQRT__POS__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Esqrt_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESQRT__1,axiom,
c_2Etransc_2Esqrt_2E1(c_2Ereal_2Ereal__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_2Etransc_2ESQRT__0,axiom,
c_2Etransc_2Esqrt_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Etransc_2EROOT__MONO__LE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V2y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0),c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V2y_2E0))) ) ).
tff(thm_2Etransc_2EROOT__DIV,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2y_2E0)) )
=> ( c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Ereal_2E_2F_2E2(V1x_2E0,V2y_2E0)) = c_2Ereal_2E_2F_2E2(c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0),c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V2y_2E0)) ) ) ).
tff(thm_2Etransc_2EROOT__INV,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> ( c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Erealax_2Einv_2E1(V1x_2E0)) = c_2Erealax_2Einv_2E1(c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0)) ) ) ).
tff(thm_2Etransc_2EROOT__MUL,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2y_2E0)) )
=> ( c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Erealax_2Ereal__mul_2E2(V1x_2E0,V2y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0),c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V2y_2E0)) ) ) ).
tff(thm_2Etransc_2EROOT__POS__UNIQ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2y_2E0))
& ( c_2Ereal_2Epow_2E2(V2y_2E0,c_2Enum_2ESUC_2E1(V0n_2E0)) = V1x_2E0 ) )
=> ( c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0) = V2y_2E0 ) ) ).
tff(thm_2Etransc_2EROOT__POS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0))) ) ).
tff(thm_2Etransc_2EPOW__ROOT__POS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> ( c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Ereal_2Epow_2E2(V1x_2E0,c_2Enum_2ESUC_2E1(V0n_2E0))) = V1x_2E0 ) ) ).
tff(thm_2Etransc_2EROOT__POW__POS,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> ( c_2Ereal_2Epow_2E2(c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0),c_2Enum_2ESUC_2E1(V0n_2E0)) = V1x_2E0 ) ) ).
tff(thm_2Etransc_2EROOT__POS__LT,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0))) ) ).
tff(thm_2Etransc_2EROOT__1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Ereal_2Ereal__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_2Etransc_2EROOT__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Etransc_2EROOT__LN,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> ( c_2Etransc_2Eroot_2E2(c_2Enum_2ESUC_2E1(V0n_2E0),V1x_2E0) = c_2Etransc_2Eexp_2E1(c_2Ereal_2E_2F_2E2(c_2Etransc_2Eln_2E1(V1x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)))) ) ) ).
tff(thm_2Etransc_2EROOT__LT__LEMMA,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> ( c_2Ereal_2Epow_2E2(c_2Etransc_2Eexp_2E1(c_2Ereal_2E_2F_2E2(c_2Etransc_2Eln_2E1(V1x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2ESUC_2E1(V0n_2E0)))),c_2Enum_2ESUC_2E1(V0n_2E0)) = V1x_2E0 ) ) ).
tff(thm_2Etransc_2EEXP__CONVERGES,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Etransc_2Eexp_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ESIN__CONVERGES,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0)))),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Etransc_2Esin_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2ECOS__CONVERGES,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V1n_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Etransc_2Ecos_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2EEXP__FDIFF,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V1n_2E0) = c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0))) )
=> ! [V0x_2E0: tyop_2Enum_2Enum] : ( c_2Epowser_2Ediffs_2E2(F0_2E0,V0x_2E0) = c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V0x_2E0))) ) ) ).
tff(thm_2Etransc_2ESIN__FDIFF,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V1n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0)))) )
=> ! [V0x_2E0: tyop_2Enum_2Enum] : ( c_2Epowser_2Ediffs_2E2(F0_2E0,V0x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V0x_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V0x_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ).
tff(thm_2Etransc_2ECOS__FDIFF,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V1n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V1n_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0x_2E0: tyop_2Enum_2Enum] : ( c_2Epowser_2Ediffs_2E2(F0_2E0,V0x_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2D_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V0x_2E0))))) ) ) ).
tff(thm_2Etransc_2ESIN__NEGLEMMA,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2EEVEN_2E1(V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_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_2Earithmetic_2EDIV_2E2(c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0)))),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0))) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Esin_2E1(V0x_2E0)) = c_2Eseq_2Esuminf_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)) ) ) ).
tff(thm_2Etransc_2EDIFF__EXP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Eexp_2E0,c_2Etransc_2Eexp_2E1(V0x_2E0),V0x_2E0)) ).
tff(thm_2Etransc_2EDIFF__SIN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Esin_2E0,c_2Etransc_2Ecos_2E1(V0x_2E0),V0x_2E0)) ).
tff(thm_2Etransc_2EDIFF__COS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Ecos_2E0,c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Esin_2E1(V0x_2E0)),V0x_2E0)) ).
tff(thm_2Etransc_2EDIFF__COMPOSITE,axiom,
! [F8_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F5_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F7_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F9_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F6_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V14x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F8_2E0,V4g_2E0),V14x_2E0) = c_2Etransc_2Esin_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V14x_2E0)) )
=> ( ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V11x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F5_2E0,V5f_2E0),V11x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V11x_2E0)) )
=> ( ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V5f_2E0),V6x_2E0) = c_2Erealax_2Einv_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V6x_2E0)) )
=> ( ! [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V13x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F7_2E0,V4g_2E0),V13x_2E0) = c_2Etransc_2Eexp_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V13x_2E0)) )
=> ( ! [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V15x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F9_2E0,V4g_2E0),V15x_2E0) = c_2Etransc_2Ecos_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V15x_2E0)) )
=> ( ! [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum,V12x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F6_2E0,V4g_2E0),V1n_2E0),V12x_2E0) = c_2Ereal_2Epow_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V12x_2E0),V1n_2E0) )
=> ( ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V10x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F4_2E0,V5f_2E0),V4g_2E0),V10x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V10x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V10x_2E0)) )
=> ( ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V9x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F3_2E0,V5f_2E0),V4g_2E0),V9x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V9x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V9x_2E0)) )
=> ( ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F2_2E0,V5f_2E0),V4g_2E0),V8x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V8x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V8x_2E0)) )
=> ( ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V5f_2E0),V4g_2E0),V7x_2E0) = c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V7x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V7x_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Erealax_2Ereal,V3l_2E0: tyop_2Erealax_2Ereal,V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( ( p(c_2Elim_2Ediffl_2E3(V5f_2E0,V3l_2E0,V0x_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V0x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V5f_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(V3l_2E0,c_2Ereal_2Epow_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))),V0x_2E0)) )
& ( ( p(c_2Elim_2Ediffl_2E3(V5f_2E0,V3l_2E0,V0x_2E0))
& p(c_2Elim_2Ediffl_2E3(V4g_2E0,V2m_2E0,V0x_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V0x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V5f_2E0),V4g_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(V3l_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V0x_2E0)),c_2Erealax_2Ereal__mul_2E2(V2m_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V0x_2E0))),c_2Ereal_2Epow_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V0x_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0)) )
& ( ( p(c_2Elim_2Ediffl_2E3(V5f_2E0,V3l_2E0,V0x_2E0))
& p(c_2Elim_2Ediffl_2E3(V4g_2E0,V2m_2E0,V0x_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F2_2E0,V5f_2E0),V4g_2E0),c_2Erealax_2Ereal__add_2E2(V3l_2E0,V2m_2E0),V0x_2E0)) )
& ( ( p(c_2Elim_2Ediffl_2E3(V5f_2E0,V3l_2E0,V0x_2E0))
& p(c_2Elim_2Ediffl_2E3(V4g_2E0,V2m_2E0,V0x_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F3_2E0,V5f_2E0),V4g_2E0),c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V3l_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V0x_2E0)),c_2Erealax_2Ereal__mul_2E2(V2m_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V0x_2E0))),V0x_2E0)) )
& ( ( p(c_2Elim_2Ediffl_2E3(V5f_2E0,V3l_2E0,V0x_2E0))
& p(c_2Elim_2Ediffl_2E3(V4g_2E0,V2m_2E0,V0x_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F4_2E0,V5f_2E0),V4g_2E0),c_2Ereal_2Ereal__sub_2E2(V3l_2E0,V2m_2E0),V0x_2E0)) )
& ( p(c_2Elim_2Ediffl_2E3(V5f_2E0,V3l_2E0,V0x_2E0))
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F5_2E0,V5f_2E0),c_2Erealax_2Ereal__neg_2E1(V3l_2E0),V0x_2E0)) )
& ( p(c_2Elim_2Ediffl_2E3(V4g_2E0,V2m_2E0,V0x_2E0))
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F6_2E0,V4g_2E0),V1n_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),c_2Ereal_2Epow_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V0x_2E0),c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),V2m_2E0),V0x_2E0)) )
& ( p(c_2Elim_2Ediffl_2E3(V4g_2E0,V2m_2E0,V0x_2E0))
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F7_2E0,V4g_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Eexp_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V0x_2E0)),V2m_2E0),V0x_2E0)) )
& ( p(c_2Elim_2Ediffl_2E3(V4g_2E0,V2m_2E0,V0x_2E0))
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F8_2E0,V4g_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Ecos_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V0x_2E0)),V2m_2E0),V0x_2E0)) )
& ( p(c_2Elim_2Ediffl_2E3(V4g_2E0,V2m_2E0,V0x_2E0))
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F9_2E0,V4g_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Esin_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V0x_2E0))),V2m_2E0),V0x_2E0)) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(thm_2Etransc_2EEXP__0,axiom,
c_2Etransc_2Eexp_2E1(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_2Etransc_2EEXP__LE__X,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0),c_2Etransc_2Eexp_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2EEXP__LT__1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Etransc_2Eexp_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2EEXP__ADD__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Eexp_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)),c_2Etransc_2Eexp_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0))) = c_2Etransc_2Eexp_2E1(V1y_2E0) ) ).
tff(thm_2Etransc_2EEXP__NEG__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Eexp_2E1(V0x_2E0),c_2Etransc_2Eexp_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Etransc_2EEXP__NEG__MUL2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Eexp_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)),c_2Etransc_2Eexp_2E1(V0x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).
tff(thm_2Etransc_2EEXP__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(c_2Erealax_2Ereal__neg_2E1(V0x_2E0)) = c_2Erealax_2Einv_2E1(c_2Etransc_2Eexp_2E1(V0x_2E0)) ) ).
tff(thm_2Etransc_2EEXP__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Eexp_2E1(V0x_2E0),c_2Etransc_2Eexp_2E1(V1y_2E0)) ) ).
tff(thm_2Etransc_2EEXP__POS__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eexp_2E1(V0x_2E0))) ).
tff(thm_2Etransc_2EEXP__NZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(V0x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Etransc_2EEXP__POS__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eexp_2E1(V0x_2E0))) ).
tff(thm_2Etransc_2EEXP__N,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),V1x_2E0)) = c_2Ereal_2Epow_2E2(c_2Etransc_2Eexp_2E1(V1x_2E0),V0n_2E0) ) ).
tff(thm_2Etransc_2EEXP__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2E_2F_2E2(c_2Etransc_2Eexp_2E1(V0x_2E0),c_2Etransc_2Eexp_2E1(V1y_2E0)) ) ).
tff(thm_2Etransc_2EEXP__MONO__IMP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Eexp_2E1(V0x_2E0),c_2Etransc_2Eexp_2E1(V1y_2E0))) ) ).
tff(thm_2Etransc_2EEXP__MONO__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Eexp_2E1(V0x_2E0),c_2Etransc_2Eexp_2E1(V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Etransc_2EEXP__MONO__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Eexp_2E1(V0x_2E0),c_2Etransc_2Eexp_2E1(V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Etransc_2EEXP__INJ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eexp_2E1(V0x_2E0) = c_2Etransc_2Eexp_2E1(V1y_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Etransc_2EEXP__TOTAL__LEMMA,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0y_2E0))
=> ? [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Ereal_2Ereal__sub_2E2(V0y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))))
& ( c_2Etransc_2Eexp_2E1(V1x_2E0) = V0y_2E0 ) ) ) ).
tff(thm_2Etransc_2EEXP__TOTAL,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0y_2E0))
=> ? [V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eexp_2E1(V1x_2E0) = V0y_2E0 ) ) ).
tff(thm_2Etransc_2ELN__EXP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Eln_2E1(c_2Etransc_2Eexp_2E1(V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Etransc_2EEXP__LN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Etransc_2Eexp_2E1(c_2Etransc_2Eln_2E1(V0x_2E0)) = V0x_2E0 )
<=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) ) ).
tff(thm_2Etransc_2ELN__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ( c_2Etransc_2Eln_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Etransc_2Eln_2E1(V0x_2E0),c_2Etransc_2Eln_2E1(V1y_2E0)) ) ) ).
tff(thm_2Etransc_2ELN__INJ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ( ( c_2Etransc_2Eln_2E1(V0x_2E0) = c_2Etransc_2Eln_2E1(V1y_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ) ).
tff(thm_2Etransc_2ELN__1,axiom,
c_2Etransc_2Eln_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Etransc_2ELN__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Etransc_2Eln_2E1(c_2Erealax_2Einv_2E1(V0x_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Eln_2E1(V0x_2E0)) ) ) ).
tff(thm_2Etransc_2ELN__DIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ( c_2Etransc_2Eln_2E1(c_2Ereal_2E_2F_2E2(V0x_2E0,V1y_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Etransc_2Eln_2E1(V0x_2E0),c_2Etransc_2Eln_2E1(V1y_2E0)) ) ) ).
tff(thm_2Etransc_2ELN__MONO__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Eln_2E1(V0x_2E0),c_2Etransc_2Eln_2E1(V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Etransc_2ELN__MONO__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0)) )
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Eln_2E1(V0x_2E0),c_2Etransc_2Eln_2E1(V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0) ) ) ).
tff(thm_2Etransc_2ELN__POW,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
=> ( c_2Etransc_2Eln_2E1(c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),c_2Etransc_2Eln_2E1(V1x_2E0)) ) ) ).
tff(thm_2Etransc_2ELN__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Eln_2E1(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0)),V0x_2E0)) ) ).
tff(thm_2Etransc_2ELN__LT__X,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Eln_2E1(V0x_2E0),V0x_2E0)) ) ).
tff(thm_2Etransc_2ELN__POS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Eln_2E1(V0x_2E0))) ) ).
tff(thm_2Etransc_2EDIFF__LN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Eln_2E0,c_2Erealax_2Einv_2E1(V0x_2E0),V0x_2E0)) ) ).
tff(thm_2Etransc_2EDIFF__ASN__LEMMA,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Easn_2E0,c_2Erealax_2Einv_2E1(c_2Etransc_2Ecos_2E1(c_2Etransc_2Easn_2E1(V0x_2E0))),V0x_2E0)) ) ).
tff(thm_2Etransc_2EDIFF__ASN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Easn_2E0,c_2Erealax_2Einv_2E1(c_2Etransc_2Esqrt_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))),V0x_2E0)) ) ).
tff(thm_2Etransc_2EDIFF__ACS__LEMMA,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Eacs_2E0,c_2Erealax_2Einv_2E1(c_2Erealax_2Ereal__neg_2E1(c_2Etransc_2Esin_2E1(c_2Etransc_2Eacs_2E1(V0x_2E0)))),V0x_2E0)) ) ).
tff(thm_2Etransc_2EDIFF__ACS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Eacs_2E0,c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Einv_2E1(c_2Etransc_2Esqrt_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))),V0x_2E0)) ) ).
tff(thm_2Etransc_2EDIFF__ATN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Elim_2Ediffl_2E3(c_2Etransc_2Eatn_2E0,c_2Erealax_2Einv_2E1(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))),V0x_2E0)) ).
tff(thm_2Etransc_2EDIVISION__0,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1b_2E0),V2n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V2n_2E0),V0a_2E0,V1b_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 = V1b_2E0 )
=> ( c_2Etransc_2Edsize_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1b_2E0)) = c_2Enum_2E0_2E0 ) ) ) ) ).
tff(thm_2Etransc_2EDIVISION__1,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1b_2E0),V2n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V2n_2E0),V0a_2E0,V1b_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V1b_2E0))
=> ( c_2Etransc_2Edsize_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1b_2E0)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ) ) ) ) ).
tff(thm_2Etransc_2EDIVISION__SINGLE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V2n_2E0))
<=> ( V2n_2E0 = c_2Enum_2E0_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1b_2E0),V2n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V2n_2E0),V0a_2E0,V1b_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
=> p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1b_2E0))) ) ) ) ).
tff(thm_2Etransc_2EDIVISION__LHS,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Enum_2E0_2E0) = V1a_2E0 ) ) ).
tff(thm_2Etransc_2EDIVISION__THM,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
<=> ( ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Enum_2E0_2E0) = V1a_2E0 )
& ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3n_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0)))
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)))) )
& ! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V4n_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0)))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V4n_2E0) = V2b_2E0 ) ) ) ) ).
tff(thm_2Etransc_2EDIVISION__RHS,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0)) = V2b_2E0 ) ) ).
tff(thm_2Etransc_2EDIVISION__LT__GEN,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V3m_2E0,V4n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V4n_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V4n_2E0))) ) ).
tff(thm_2Etransc_2EDIVISION__LT,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
=> ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3n_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0)))
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)))) ) ) ).
tff(thm_2Etransc_2EDIVISION__LE,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0)) ) ).
tff(thm_2Etransc_2EDIVISION__GT,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
=> ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3n_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0)))
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0)))) ) ) ).
tff(thm_2Etransc_2EDIVISION__EQ,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
=> ( ( V1a_2E0 = V2b_2E0 )
<=> ( c_2Etransc_2Edsize_2E1(V0D_2E0) = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Etransc_2EDIVISION__LBOUND,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
=> ! [V3r_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V3r_2E0))) ) ).
tff(thm_2Etransc_2EDIVISION__LBOUND__LT,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
& ( c_2Etransc_2Edsize_2E1(V0D_2E0) != c_2Enum_2E0_2E0 ) )
=> ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)))) ) ).
tff(thm_2Etransc_2EDIVISION__UBOUND,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
=> ! [V3r_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V3r_2E0),V2b_2E0)) ) ).
tff(thm_2Etransc_2EDIVISION__UBOUND__LT,axiom,
! [V0D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0D_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V3n_2E0,c_2Etransc_2Edsize_2E1(V0D_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0D_2E0,V3n_2E0),V2b_2E0)) ) ).
tff(thm_2Etransc_2EDIVISION__APPEND,axiom,
! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3c_2E0: tyop_2Erealax_2Ereal] :
( ( ? [V4D1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5p1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4D1_2E0,V5p1_2E0)))
& p(c_2Etransc_2Efine_2E2(V0g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4D1_2E0,V5p1_2E0))) )
& ? [V6D2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7p2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2b_2E0,V3c_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6D2_2E0,V7p2_2E0)))
& p(c_2Etransc_2Efine_2E2(V0g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6D2_2E0,V7p2_2E0))) ) )
=> ? [V8D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V9p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3c_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V8D_2E0,V9p_2E0)))
& p(c_2Etransc_2Efine_2E2(V0g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V8D_2E0,V9p_2E0))) ) ) ).
tff(thm_2Etransc_2EDIVISION__EXISTS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1b_2E0),V3x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V1b_2E0)) ) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
& p(c_2Etransc_2Egauge_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1b_2E0),V2g_2E0)) )
=> ? [V4D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4D_2E0,V5p_2E0)))
& p(c_2Etransc_2Efine_2E2(V2g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4D_2E0,V5p_2E0))) ) ) ) ).
tff(thm_2Etransc_2EGAUGE__MIN,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1g1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1g1_2E0),V2g2_2E0),V3x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g1_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g2_2E0,V3x_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g1_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g2_2E0,V3x_2E0)) )
=> ! [V0E_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Etransc_2Egauge_2E2(V0E_2E0,V1g1_2E0))
& p(c_2Etransc_2Egauge_2E2(V0E_2E0,V2g2_2E0)) )
=> p(c_2Etransc_2Egauge_2E2(V0E_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1g1_2E0),V2g2_2E0))) ) ) ).
tff(thm_2Etransc_2EFINE__MIN,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0g1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0g1_2E0),V1g2_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0g1_2E0,V4x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g2_2E0,V4x_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0g1_2E0,V4x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g2_2E0,V4x_2E0)) )
=> ! [V0g1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Efine_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0g1_2E0),V1g2_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2D_2E0,V3p_2E0)))
=> ( p(c_2Etransc_2Efine_2E2(V0g1_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2D_2E0,V3p_2E0)))
& p(c_2Etransc_2Efine_2E2(V1g2_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2D_2E0,V3p_2E0))) ) ) ) ).
tff(thm_2Etransc_2EDINT__UNIQ,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3k1_2E0: tyop_2Erealax_2Ereal,V4k2_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2f_2E0,V3k1_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2f_2E0,V4k2_2E0)) )
=> ( V3k1_2E0 = V4k2_2E0 ) ) ).
tff(thm_2Etransc_2EINTEGRAL__NULL,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal] : p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V1a_2E0),V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ).
tff(thm_2Etransc_2EFTC1,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3b_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V4x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V3b_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(V0f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_27_2E0,V4x_2E0),V4x_2E0)) ) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1f_27_2E0,c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3b_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2a_2E0)))) ) ).
tff(thm_2Etransc_2EMCLAURIN,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2h_2E0: tyop_2Erealax_2Ereal,V7m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1diff_2E0),V2h_2E0),V7m_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V7m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V7m_2E0))),c_2Ereal_2Epow_2E2(V2h_2E0,V7m_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2h_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2h_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V3n_2E0))
& ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,c_2Enum_2E0_2E0) = V0f_2E0 )
& ! [V4m_2E0: tyop_2Enum_2Enum,V5t_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V4m_2E0,V3n_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5t_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V5t_2E0,V2h_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V4m_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,c_2Enum_2ESUC_2E1(V4m_2E0)),V5t_2E0),V5t_2E0)) ) )
=> ? [V6t_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V6t_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V6t_2E0,V2h_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2h_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V3n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1diff_2E0),V2h_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V3n_2E0),V6t_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V3n_2E0))),c_2Ereal_2Epow_2E2(V2h_2E0,V3n_2E0))) ) ) ) ) ).
tff(thm_2Etransc_2EMCLAURIN__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2h_2E0: tyop_2Erealax_2Ereal,V7m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1diff_2E0),V2h_2E0),V7m_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V7m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V7m_2E0))),c_2Ereal_2Epow_2E2(V2h_2E0,V7m_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2h_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V2h_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V3n_2E0))
& ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,c_2Enum_2E0_2E0) = V0f_2E0 )
& ! [V4m_2E0: tyop_2Enum_2Enum,V5t_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V4m_2E0,V3n_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2h_2E0,V5t_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V5t_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V4m_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,c_2Enum_2ESUC_2E1(V4m_2E0)),V5t_2E0),V5t_2E0)) ) )
=> ? [V6t_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V2h_2E0,V6t_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V6t_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2h_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V3n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1diff_2E0),V2h_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V3n_2E0),V6t_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V3n_2E0))),c_2Ereal_2Epow_2E2(V2h_2E0,V3n_2E0))) ) ) ) ) ).
tff(thm_2Etransc_2EMCLAURIN__ALL__LT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V4x_2E0: tyop_2Erealax_2Ereal,V7m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1diff_2E0),V4x_2E0),V7m_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V7m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V7m_2E0))),c_2Ereal_2Epow_2E2(V4x_2E0,V7m_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ( ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,c_2Enum_2E0_2E0) = V0f_2E0 )
& ! [V2m_2E0: tyop_2Enum_2Enum,V3x_2E0: tyop_2Erealax_2Ereal] : p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V2m_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,c_2Enum_2ESUC_2E1(V2m_2E0)),V3x_2E0),V3x_2E0)) )
=> ! [V4x_2E0: tyop_2Erealax_2Ereal,V5n_2E0: tyop_2Enum_2Enum] :
( ( ( V4x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V5n_2E0)) )
=> ? [V6t_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Eabs_2E1(V6t_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V6t_2E0),c_2Ereal_2Eabs_2E1(V4x_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V5n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1diff_2E0),V4x_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V5n_2E0),V6t_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V5n_2E0))),c_2Ereal_2Epow_2E2(V4x_2E0,V5n_2E0))) ) ) ) ) ) ).
tff(thm_2Etransc_2EMCLAURIN__ZERO,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2x_2E0: tyop_2Erealax_2Ereal,V3m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0diff_2E0),V2x_2E0),V3m_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0diff_2E0,V3m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V3m_2E0))),c_2Ereal_2Epow_2E2(V2x_2E0,V3m_2E0)) )
=> ! [V0diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1n_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( ( V2x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0)) )
=> ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0diff_2E0),V2x_2E0)) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0diff_2E0,c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ) ).
tff(thm_2Etransc_2EMCLAURIN__ALL__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V4x_2E0: tyop_2Erealax_2Ereal,V7m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1diff_2E0),V4x_2E0),V7m_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V7m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V7m_2E0))),c_2Ereal_2Epow_2E2(V4x_2E0,V7m_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ( ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,c_2Enum_2E0_2E0) = V0f_2E0 )
& ! [V2m_2E0: tyop_2Enum_2Enum,V3x_2E0: tyop_2Erealax_2Ereal] : p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V2m_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,c_2Enum_2ESUC_2E1(V2m_2E0)),V3x_2E0),V3x_2E0)) )
=> ! [V4x_2E0: tyop_2Erealax_2Ereal,V5n_2E0: tyop_2Enum_2Enum] :
? [V6t_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V6t_2E0),c_2Ereal_2Eabs_2E1(V4x_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V5n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1diff_2E0),V4x_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1diff_2E0,V5n_2E0),V6t_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V5n_2E0))),c_2Ereal_2Epow_2E2(V4x_2E0,V5n_2E0))) ) ) ) ) ).
tff(thm_2Etransc_2EMCLAURIN__EXP__LT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V3m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V3m_2E0) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V3m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V3m_2E0))) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( ( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1n_2E0)) )
=> ? [V2t_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Eabs_2E1(V2t_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V2t_2E0),c_2Ereal_2Eabs_2E1(V0x_2E0)))
& ( c_2Etransc_2Eexp_2E1(V0x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(c_2Etransc_2Eexp_2E1(V2t_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0))) ) ) ) ) ).
tff(thm_2Etransc_2EMCLAURIN__EXP__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V3m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V3m_2E0) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V3m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V3m_2E0))) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
? [V2t_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V2t_2E0),c_2Ereal_2Eabs_2E1(V0x_2E0)))
& ( c_2Etransc_2Eexp_2E1(V0x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(c_2Etransc_2Eexp_2E1(V2t_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2EFACT_2E1(V1n_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0))) ) ) ) ).
tff(thm_2Etransc_2EDIFF__LN__COMPOSITE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0g_2E0),V3x_2E0) = c_2Etransc_2Eln_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0g_2E0,V3x_2E0)) )
=> ! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Elim_2Ediffl_2E3(V0g_2E0,V1m_2E0,V2x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0g_2E0,V2x_2E0))) )
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0g_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0g_2E0,V2x_2E0)),V1m_2E0),V2x_2E0)) ) ) ).
tff(thm_2Etransc_2EGEN__RPOW,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> ( c_2Ereal_2Epow_2E2(V0a_2E0,V1n_2E0) = c_2Etransc_2Erpow_2E2(V0a_2E0,c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) ) ) ).
tff(thm_2Etransc_2ERPOW__SUC__N,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> ( c_2Etransc_2Erpow_2E2(V0a_2E0,c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Ereal_2Epow_2E2(V0a_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)) ) ) ).
tff(thm_2Etransc_2ERPOW__0,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> ( c_2Etransc_2Erpow_2E2(V0a_2E0,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_2Etransc_2ERPOW__1,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> ( c_2Etransc_2Erpow_2E2(V0a_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = V0a_2E0 ) ) ).
tff(thm_2Etransc_2EONE__RPOW,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> ( c_2Etransc_2Erpow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0a_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Etransc_2ERPOW__POS__LT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0))) ) ).
tff(thm_2Etransc_2ERPOW__NZ,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) != V0a_2E0 )
=> ( c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Etransc_2ELN__RPOW,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> ( c_2Etransc_2Eln_2E1(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0)) = c_2Erealax_2Ereal__mul_2E2(V1b_2E0,c_2Etransc_2Eln_2E1(V0a_2E0)) ) ) ).
tff(thm_2Etransc_2ERPOW__ADD,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Erpow_2E2(V0a_2E0,c_2Erealax_2Ereal__add_2E2(V1b_2E0,V2c_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0),c_2Etransc_2Erpow_2E2(V0a_2E0,V2c_2E0)) ) ).
tff(thm_2Etransc_2ERPOW__ADD__MUL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,c_2Erealax_2Ereal__add_2E2(V1b_2E0,V2c_2E0)),c_2Etransc_2Erpow_2E2(V0a_2E0,c_2Erealax_2Ereal__neg_2E1(V1b_2E0))) = c_2Etransc_2Erpow_2E2(V0a_2E0,V2c_2E0) ) ).
tff(thm_2Etransc_2ERPOW__SUB,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] : ( c_2Etransc_2Erpow_2E2(V0a_2E0,c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V2c_2E0)) = c_2Ereal_2E_2F_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0),c_2Etransc_2Erpow_2E2(V0a_2E0,V2c_2E0)) ) ).
tff(thm_2Etransc_2ERPOW__DIV,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1b_2E0)) )
=> ( c_2Etransc_2Erpow_2E2(c_2Ereal_2E_2F_2E2(V0a_2E0,V1b_2E0),V2c_2E0) = c_2Ereal_2E_2F_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V2c_2E0),c_2Etransc_2Erpow_2E2(V1b_2E0,V2c_2E0)) ) ) ).
tff(thm_2Etransc_2ERPOW__INV,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> ( c_2Etransc_2Erpow_2E2(c_2Erealax_2Einv_2E1(V0a_2E0),V1b_2E0) = c_2Erealax_2Einv_2E1(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0)) ) ) ).
tff(thm_2Etransc_2ERPOW__MUL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1b_2E0)) )
=> ( c_2Etransc_2Erpow_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V1b_2E0),V2c_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V2c_2E0),c_2Etransc_2Erpow_2E2(V1b_2E0,V2c_2E0)) ) ) ).
tff(thm_2Etransc_2ERPOW__RPOW,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
=> ( c_2Etransc_2Erpow_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0),V2c_2E0) = c_2Etransc_2Erpow_2E2(V0a_2E0,c_2Erealax_2Ereal__mul_2E2(V1b_2E0,V2c_2E0)) ) ) ).
tff(thm_2Etransc_2ERPOW__LT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0a_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0),c_2Etransc_2Erpow_2E2(V0a_2E0,V2c_2E0)) = c_2Erealax_2Ereal__lt_2E2(V1b_2E0,V2c_2E0) ) ) ).
tff(thm_2Etransc_2ERPOW__LE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0a_2E0))
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0),c_2Etransc_2Erpow_2E2(V0a_2E0,V2c_2E0)) = c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2c_2E0) ) ) ).
tff(thm_2Etransc_2EBASE__RPOW__LE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2c_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1b_2E0)) )
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0),c_2Etransc_2Erpow_2E2(V2c_2E0,V1b_2E0)) = c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V2c_2E0) ) ) ).
tff(thm_2Etransc_2EBASE__RPOW__LT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2c_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1b_2E0)) )
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0),c_2Etransc_2Erpow_2E2(V2c_2E0,V1b_2E0)) = c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V2c_2E0) ) ) ).
tff(thm_2Etransc_2ERPOW__UNIQ__BASE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0a_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2c_2E0))
& ( c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) != V1b_2E0 )
& ( c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0) = c_2Etransc_2Erpow_2E2(V2c_2E0,V1b_2E0) ) )
=> ( V0a_2E0 = V2c_2E0 ) ) ).
tff(thm_2Etransc_2ERPOW__UNIQ__EXP,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0a_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2c_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1b_2E0))
& ( c_2Etransc_2Erpow_2E2(V0a_2E0,V1b_2E0) = c_2Etransc_2Erpow_2E2(V0a_2E0,V2c_2E0) ) )
=> ( V1b_2E0 = V2c_2E0 ) ) ).
tff(thm_2Etransc_2ERPOW__DIV__BASE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> ( c_2Ereal_2E_2F_2E2(c_2Etransc_2Erpow_2E2(V0x_2E0,V1t_2E0),V0x_2E0) = c_2Etransc_2Erpow_2E2(V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V1t_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ).
tff(thm_2Etransc_2EDIFF__COMPOSITE__EXP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0g_2E0),V3x_2E0) = c_2Etransc_2Eexp_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0g_2E0,V3x_2E0)) )
=> ! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Elim_2Ediffl_2E3(V0g_2E0,V1m_2E0,V2x_2E0))
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0g_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Etransc_2Eexp_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0g_2E0,V2x_2E0)),V1m_2E0),V2x_2E0)) ) ) ).
tff(thm_2Etransc_2EDIFF__RPOW,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1y_2E0),V2x_2E0) = c_2Etransc_2Erpow_2E2(V2x_2E0,V1y_2E0) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Elim_2Ediffl_2E3(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,c_2Etransc_2Erpow_2E2(V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V1y_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))),V0x_2E0)) ) ) ).
%------------------------------------------------------------------------------