ITP001 Axioms: ITP137_7.ax
%------------------------------------------------------------------------------
% File : ITP137_7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : integral.ax [Gau19]
% : HL4137_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 244 ( 49 unt; 100 typ; 0 def)
% Number of atoms : 465 ( 122 equ)
% Maximal formula atoms : 14 ( 1 avg)
% Number of connectives : 328 ( 7 ~; 5 |; 148 &)
% ( 27 <=>; 141 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 82 ( 47 >; 35 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 96 ( 96 usr; 39 con; 0-4 aty)
% Number of variables : 686 ( 633 !; 27 ?; 686 :)
% ( 26 !>; 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_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_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_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_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_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_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Epred__set_2EFINITE_2E0,type,
c_2Epred__set_2EFINITE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Epred__set_2EFINITE_2E1,type,
c_2Epred__set_2EFINITE_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2EIN_2E0,type,
c_2Ebool_2EIN_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2EIN_2E2,type,
c_2Ebool_2EIN_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
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_2Elim_2Econtl_2E0,type,
c_2Elim_2Econtl_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Elim_2Econtl_2E2,type,
c_2Elim_2Econtl_2E2: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
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_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_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_2Eintegral_2Eintegrable_2E0,type,
c_2Eintegral_2Eintegrable_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_2Ebool)) ).
tff(c_2Eintegral_2Eintegrable_2E2,type,
c_2Eintegral_2Eintegrable_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Emin_2Ebool ).
tff(c_2Eintegral_2Eintegral_2E0,type,
c_2Eintegral_2Eintegral_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_2Erealax_2Ereal)) ).
tff(c_2Eintegral_2Eintegral_2E2,type,
c_2Eintegral_2Eintegral_2E2: ( tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Einv_2E0,type,
c_2Erealax_2Einv_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Einv_2E1,type,
c_2Erealax_2Einv_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Emin_2E0,type,
c_2Ereal_2Emin_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Emin_2E2,type,
c_2Ereal_2Emin_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > 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_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_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_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_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_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(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_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(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_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_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(arityeq2_2Ec_2Elim_2Econtl_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Elim_2Econtl_2E2(X0_2E0,X1_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),c_2Elim_2Econtl_2E0,X0_2E0),X1_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_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(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(arityeq2_2Ec_2Eintegral_2Eintegrable_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Eintegral_2Eintegrable_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,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_2Ebool),c_2Eintegral_2Eintegrable_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eintegral_2Eintegral_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Eintegral_2Eintegral_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_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eintegral_2Eintegral_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(arityeq2_2Ec_2Ereal_2Emin_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_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_2Emin_2E0,X0_2E0),X1_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_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(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(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_2Eintegral_2Eintegrable__def,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2f_2E0))
<=> ? [V3i_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,V3i_2E0)) ) ).
tff(thm_2Eintegral_2Eintegral__def,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3i_2E0: tyop_2Erealax_2Ereal] : ( 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_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,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_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2f_2E0),V3i_2E0) = c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2f_2E0,V3i_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2f_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,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_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2f_2E0)) ) ) ).
tff(thm_2Eintegral_2ESUM__SPLIT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum,V3p_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0m_2E0,V2n_2E0),V1f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2n_2E0),V3p_2E0),V1f_2E0)) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0m_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,V3p_2E0)),V1f_2E0) ) ).
tff(thm_2Eintegral_2EDIVISION__APPEND__EXPLICIT,axiom,
! [F2_2E0: 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))),F1_2E0: 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: 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))),F5_2E0: 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_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F4_2E0: 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_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F3_2E0: 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_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V4d1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6d2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V13n_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)),F2_2E0,V4d1_2E0),V6d2_2E0),V13n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V13n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V4d1_2E0,V13n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V6d2_2E0,c_2Earithmetic_2E_2D_2E2(V13n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)))) )
=> ( ! [V4d1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6d2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V10n_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)),F1_2E0,V4d1_2E0),V6d2_2E0),V10n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V10n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V4d1_2E0,V10n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V6d2_2E0,c_2Earithmetic_2E_2D_2E2(V10n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)))) )
=> ( ! [V4d1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6d2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V8n_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)),F0_2E0,V4d1_2E0),V6d2_2E0),V8n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V8n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V4d1_2E0,V8n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V6d2_2E0,c_2Earithmetic_2E_2D_2E2(V8n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)))) )
=> ( ! [V4d1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5p1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7p2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V14n_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_2Enum_2Enum,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))),F5_2E0,V4d1_2E0),V5p1_2E0),V7p2_2E0),V14n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V14n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V5p1_2E0,V14n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V7p2_2E0,c_2Earithmetic_2E_2D_2E2(V14n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)))) )
=> ( ! [V4d1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5p1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7p2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V11n_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_2Enum_2Enum,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))),F4_2E0,V4d1_2E0),V5p1_2E0),V7p2_2E0),V11n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V11n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V5p1_2E0,V11n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V7p2_2E0,c_2Earithmetic_2E_2D_2E2(V11n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)))) )
=> ( ! [V4d1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5p1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7p2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V9n_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_2Enum_2Enum,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))),F3_2E0,V4d1_2E0),V5p1_2E0),V7p2_2E0),V9n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V9n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V5p1_2E0,V9n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V7p2_2E0,c_2Earithmetic_2E_2D_2E2(V9n_2E0,c_2Etransc_2Edsize_2E1(V4d1_2E0)))) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,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),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,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),V4d1_2E0,V5p1_2E0)))
& p(c_2Etransc_2Efine_2E2(V3g_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_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V2c_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(V3g_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_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2c_2E0),c_2Epair_2E_2C_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_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)),F0_2E0,V4d1_2E0),V6d2_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_2Enum_2Enum,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))),F3_2E0,V4d1_2E0),V5p1_2E0),V7p2_2E0))))
& p(c_2Etransc_2Efine_2E2(V3g_2E0,c_2Epair_2E_2C_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_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)),F1_2E0,V4d1_2E0),V6d2_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_2Enum_2Enum,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))),F4_2E0,V4d1_2E0),V5p1_2E0),V7p2_2E0))))
& ! [V12f_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),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)),F2_2E0,V4d1_2E0),V6d2_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_2Enum_2Enum,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))),F5_2E0,V4d1_2E0),V5p1_2E0),V7p2_2E0)),V12f_2E0) = c_2Erealax_2Ereal__add_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),V4d1_2E0,V5p1_2E0),V12f_2E0),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),V6d2_2E0,V7p2_2E0),V12f_2E0)) ) ) ) ) ) ) ) ) ) ).
tff(thm_2Eintegral_2EDIVISION__APPEND__STRONG,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),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,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)))
& 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)))
& ! [V10f_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),V8D_2E0,V9p_2E0),V10f_2E0) = c_2Erealax_2Ereal__add_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),V4D1_2E0,V5p1_2E0),V10f_2E0),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),V6D2_2E0,V7p2_2E0),V10f_2E0)) ) ) ) ).
tff(thm_2Eintegral_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_2Eintegral_2EINTEGRABLE__DINT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0))
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0,c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0))) ) ).
tff(thm_2Eintegral_2EDIVISION__BOUNDS,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_2Ereal_2Ereal__lte_2E2(V1a_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0d_2E0,V3n_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0d_2E0,V3n_2E0),V2b_2E0)) ) ) ).
tff(thm_2Eintegral_2ETDIV__BOUNDS,axiom,
! [V0d_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),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)))
=> ! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0d_2E0,V4n_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0d_2E0,V4n_2E0),V3b_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1p_2E0,V4n_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1p_2E0,V4n_2E0),V3b_2E0)) ) ) ).
tff(thm_2Eintegral_2ETDIV__LE,axiom,
! [V0d_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),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)))
=> p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3b_2E0)) ) ).
tff(thm_2Eintegral_2EDINT__WRONG,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3i_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V1b_2E0,V0a_2E0))
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2f_2E0,V3i_2E0)) ) ).
tff(thm_2Eintegral_2EDINT__INTEGRAL,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3i_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0,V3i_2E0)) )
=> ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0) = V3i_2E0 ) ) ).
tff(thm_2Eintegral_2EDINT__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0f_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),F0_2E0,V0f_2E0),V4x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3i_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0,V3i_2E0))
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),c_2Erealax_2Ereal__neg_2E1(V3i_2E0))) ) ) ).
tff(thm_2Eintegral_2EDINT__0,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V2x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),F0_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eintegral_2EDINT__ADD,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)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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),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,V0f_2E0),V1g_2E0),V6x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Erealax_2Ereal,V5j_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0,V4i_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0,V5j_2E0)) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_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,V0f_2E0),V1g_2E0),c_2Erealax_2Ereal__add_2E2(V4i_2E0,V5j_2E0))) ) ) ).
tff(thm_2Eintegral_2EDINT__SUB,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)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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),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,V0f_2E0),V1g_2E0),V6x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Erealax_2Ereal,V5j_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0,V4i_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0,V5j_2E0)) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_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,V0f_2E0),V1g_2E0),c_2Ereal_2Ereal__sub_2E2(V4i_2E0,V5j_2E0))) ) ) ).
tff(thm_2Eintegral_2EDSIZE__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V2D_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),F0_2E0,V2D_2E0),V3n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2D_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2D_2E0,V3n_2E0)) )
=> ! [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))
=> ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,c_2Etransc_2Edsize_2E1(V2D_2E0)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V2D_2E0)),c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V0a_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Eintegral_2EDINT__CONST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V3x_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,V2c_2E0),V3x_2E0) = V2c_2E0 )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] : p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2c_2E0),c_2Erealax_2Ereal__mul_2E2(V2c_2E0,c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V0a_2E0)))) ) ).
tff(thm_2Eintegral_2EDINT__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V3c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5x_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_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,V3c_2E0),V0f_2E0),V5x_2E0) = c_2Erealax_2Ereal__mul_2E2(V3c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0)) )
=> ! [V0f_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,V4i_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0,V4i_2E0))
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(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,V3c_2E0),V0f_2E0),c_2Erealax_2Ereal__mul_2E2(V3c_2E0,V4i_2E0))) ) ) ).
tff(thm_2Eintegral_2EDINT__LINEAR,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))] :
( ! [V1m_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0n_2E0: tyop_2Erealax_2Ereal,V3g_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_2Erealax_2Ereal,tyop_2Emin_2Efun(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_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(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,V1m_2E0),V2f_2E0),V0n_2E0),V3g_2E0),V8x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V1m_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V8x_2E0)),c_2Erealax_2Ereal__mul_2E2(V0n_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,V8x_2E0))) )
=> ! [V0n_2E0: tyop_2Erealax_2Ereal,V1m_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal,V6i_2E0: tyop_2Erealax_2Ereal,V7j_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_2E0),V2f_2E0,V6i_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_2E0),V3g_2E0,V7j_2E0)) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(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_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(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,V1m_2E0),V2f_2E0),V0n_2E0),V3g_2E0),c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V1m_2E0,V6i_2E0),c_2Erealax_2Ereal__mul_2E2(V0n_2E0,V7j_2E0)))) ) ) ).
tff(thm_2Eintegral_2ELT,axiom,
( ! [V0m_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Enum_2E0_2E0) = c_2Ebool_2EF_2E0 )
& ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)))
<=> ( ( V1m_2E0 = V2n_2E0 )
| p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V2n_2E0)) ) ) ) ).
tff(thm_2Eintegral_2ELE__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2E0_2E0,V0n_2E0)) ).
tff(thm_2Eintegral_2ELT__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Enum_2ESUC_2E1(V0n_2E0))) ).
tff(thm_2Eintegral_2EEQ__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC_2E1(V0m_2E0) = c_2Enum_2ESUC_2E1(V1n_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ).
tff(thm_2Eintegral_2ELE__LT,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
| ( V0m_2E0 = V1n_2E0 ) ) ) ).
tff(thm_2Eintegral_2ELT__LE,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
& ( V0m_2E0 != V1n_2E0 ) ) ) ).
tff(thm_2Eintegral_2EREAL__LT__MIN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0)))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V1y_2E0)) ) ) ).
tff(thm_2Eintegral_2EREAL__LE__RMUL1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V2z_2E0))) ) ).
tff(thm_2Eintegral_2EREAL__LE__LMUL1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_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(V1y_2E0,V2z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0))) ) ).
tff(thm_2Eintegral_2EINTEGRAL__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4i_2E0: A_27a,V5j_2E0: A_27b] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3b_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V3b_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6x_2E0))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0),c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0))) ) ).
tff(thm_2Eintegral_2EDINT__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Erealax_2Ereal,V5j_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3b_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0,V4i_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0,V5j_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V3b_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6x_2E0))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V4i_2E0,V5j_2E0)) ) ).
tff(thm_2Eintegral_2EDINT__TRIANGLE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5x_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,V0f_2E0),V5x_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3i_2E0: tyop_2Erealax_2Ereal,V4j_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0,V3i_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V4j_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V3i_2E0),V4j_2E0)) ) ) ).
tff(thm_2Eintegral_2EDINT__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Erealax_2Ereal,V5j_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3b_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0,V4i_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0,V5j_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V3b_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6x_2E0) ) ) )
=> ( V4i_2E0 = V5j_2E0 ) ) ).
tff(thm_2Eintegral_2EINTEGRAL__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0,V4i_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V5x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V5x_2E0,V3b_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0) ) ) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0,V4i_2E0)) ) ).
tff(thm_2Eintegral_2EINTEGRATION__BY__PARTS,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_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)))))] :
( ! [V2f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_27_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)),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_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_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,V2f_27_2E0),V1g_2E0),V0f_2E0),V3g_27_2E0),V8x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_27_2E0,V8x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V8x_2E0)),c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_27_2E0,V8x_2E0))) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V4a_2E0,V5b_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V4a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V5b_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(V0f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_27_2E0,V6x_2E0),V6x_2E0)) )
& ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V4a_2E0,V7x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V7x_2E0,V5b_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(V1g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_27_2E0,V7x_2E0),V7x_2E0)) ) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_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)),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_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_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,V2f_27_2E0),V1g_2E0),V0f_2E0),V3g_27_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5b_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5b_2E0)),c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4a_2E0))))) ) ) ).
tff(thm_2Eintegral_2EDIVISION__LE__SUC,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_2Ereal_2Ereal__lte_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)))) ) ).
tff(thm_2Eintegral_2EDIVISION__MONO__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))
=> ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Ereal_2Ereal__lte_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_2Eintegral_2EDIVISION__MONO__LE__SUC,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_2Ereal_2Ereal__lte_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)))) ) ).
tff(thm_2Eintegral_2EDIVISION__DSIZE__LE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] :
( ( 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_2ESUC_2E1(V3n_2E0)) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,V3n_2E0) ) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Etransc_2Edsize_2E1(V2d_2E0),V3n_2E0)) ) ).
tff(thm_2Eintegral_2EDIVISION__DSIZE__GE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)))) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Enum_2ESUC_2E1(V3n_2E0),c_2Etransc_2Edsize_2E1(V2d_2E0))) ) ).
tff(thm_2Eintegral_2EDIVISION__DSIZE__EQ,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Etransc_2Edivision_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),V2d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,c_2Enum_2ESUC_2E1(V3n_2E0))))
& ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(V3n_2E0))) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)) ) )
=> ( c_2Etransc_2Edsize_2E1(V2d_2E0) = c_2Enum_2ESUC_2E1(V3n_2E0) ) ) ).
tff(thm_2Eintegral_2EDIVISION__DSIZE__EQ__ALT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] :
( ( 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_2ESUC_2E1(V3n_2E0)) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,V3n_2E0) )
& ! [V4i_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V4i_2E0,V3n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,V4i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2d_2E0,c_2Enum_2ESUC_2E1(V4i_2E0)))) ) )
=> ( c_2Etransc_2Edsize_2E1(V2d_2E0) = V3n_2E0 ) ) ).
tff(thm_2Eintegral_2Enum__MAX,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( ? [V1x_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1x_2E0))
& ? [V2M_2E0: tyop_2Enum_2Enum] :
! [V3x_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(V3x_2E0,V2M_2E0)) ) )
<=> ? [V4m_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V4m_2E0))
& ! [V5x_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(V5x_2E0,V4m_2E0)) ) ) ) ).
tff(thm_2Eintegral_2EDIVISION__INTERMEDIATE,axiom,
! [V0d_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3c_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,V3c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3c_2E0,V2b_2E0)) )
=> ? [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V4n_2E0,c_2Etransc_2Edsize_2E1(V0d_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0d_2E0,V4n_2E0),V3c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3c_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0d_2E0,c_2Enum_2ESUC_2E1(V4n_2E0)))) ) ) ).
tff(thm_2Eintegral_2EDINT__COMBINE,axiom,
! [V0f_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,V4i_2E0: tyop_2Erealax_2Ereal,V5j_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,V3c_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0,V4i_2E0))
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2b_2E0,V3c_2E0),V0f_2E0,V5j_2E0)) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3c_2E0),V0f_2E0,c_2Erealax_2Ereal__add_2E2(V4i_2E0,V5j_2E0))) ) ).
tff(thm_2Eintegral_2ESUM__EQ__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V3r_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V3r_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V3r_2E0,c_2Earithmetic_2E_2B_2E2(V1m_2E0,V0n_2E0))) )
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2f_2E0,V3r_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1m_2E0,V0n_2E0),V2f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Eintegral_2EDINT__DELTA__LEFT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0a_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),F1_2E0,V2x_2E0),V0a_2E0))
<=> ( V2x_2E0 = V0a_2E0 ) )
=> ( ! [V0a_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,V0a_2E0),V2x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V2x_2E0),V0a_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_2Enum_2E0_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ).
tff(thm_2Eintegral_2EDINT__DELTA__RIGHT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V1b_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),F1_2E0,V2x_2E0),V1b_2E0))
<=> ( V2x_2E0 = V1b_2E0 ) )
=> ( ! [V1b_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,V1b_2E0),V2x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V2x_2E0),V1b_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_2Enum_2E0_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1b_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ).
tff(thm_2Eintegral_2EDINT__DELTA,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V2c_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),F1_2E0,V3x_2E0),V2c_2E0))
<=> ( V3x_2E0 = V2c_2E0 ) )
=> ( ! [V2c_2E0: tyop_2Erealax_2Ereal,V3x_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,V2c_2E0),V3x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V3x_2E0),V2c_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_2Enum_2E0_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] : p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2c_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ).
tff(thm_2Eintegral_2EDINT__POINT__SPIKE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4c_2E0: tyop_2Erealax_2Ereal,V5i_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V3b_2E0))
& ( V6x_2E0 != V4c_2E0 ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6x_2E0) ) )
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0,V5i_2E0)) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0,V5i_2E0)) ) ).
tff(thm_2Eintegral_2EDINT__FINITE__SPIKE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5i_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V4s_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V3b_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V4s_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6x_2E0) ) )
& p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0,V5i_2E0)) )
=> p(c_2Etransc_2EDint_2E3(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0,V5i_2E0)) ) ).
tff(thm_2Eintegral_2EREAL__POW__LBOUND,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( 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))),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),V0x_2E0)),c_2Ereal_2Epow_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),V1n_2E0))) ) ).
tff(thm_2Eintegral_2EREAL__ARCH__POW,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_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))),V0x_2E0))
=> ? [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Ereal_2Epow_2E2(V0x_2E0,V2n_2E0))) ) ).
tff(thm_2Eintegral_2EREAL__ARCH__POW2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V1n_2E0))) ).
tff(thm_2Eintegral_2EREAL__POW__LE__1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_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))),V1x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0))) ) ).
tff(thm_2Eintegral_2EREAL__POW__MONO,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2x_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))),V2x_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Epow_2E2(V2x_2E0,V0m_2E0),c_2Ereal_2Epow_2E2(V2x_2E0,V1n_2E0))) ) ).
tff(thm_2Eintegral_2EREAL__LE__INV2,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_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Einv_2E1(V1y_2E0),c_2Erealax_2Einv_2E1(V0x_2E0))) ) ).
tff(thm_2Eintegral_2EGAUGE__MIN__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1gs_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2n_2E0: tyop_2Enum_2Enum] :
( ! [V3m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V2n_2E0))
=> p(c_2Etransc_2Egauge_2E2(V0s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1gs_2E0,V3m_2E0))) )
=> ? [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Egauge_2E2(V0s_2E0,V4g_2E0))
& ! [V5d_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Etransc_2Efine_2E2(V4g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5d_2E0,V6p_2E0)))
=> ! [V7m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V7m_2E0,V2n_2E0))
=> p(c_2Etransc_2Efine_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1gs_2E0,V7m_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5d_2E0,V6p_2E0))) ) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__CAUCHY,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V5x_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,V1a_2E0),V2b_2E0),V5x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V5x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V5x_2E0,V2b_2E0)) ) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0))
<=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4g_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,V1a_2E0),V2b_2E0),V4g_2E0))
& ! [V6d1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7p1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V8d2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V9p2_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),V6d1_2E0,V7p1_2E0)))
& p(c_2Etransc_2Efine_2E2(V4g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6d1_2E0,V7p1_2E0)))
& 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),V8d2_2E0,V9p2_2E0)))
& p(c_2Etransc_2Efine_2E2(V4g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V8d2_2E0,V9p2_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),V6d1_2E0,V7p1_2E0),V0f_2E0),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),V8d2_2E0,V9p2_2E0),V0f_2E0))),V3e_2E0)) ) ) ) ) ) ).
tff(thm_2Eintegral_2ESUM__DIFFS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_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),F0_2E0,V0f_2E0),V3n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum,V2d_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1n_2E0,V2d_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2d_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)) ) ) ).
tff(thm_2Eintegral_2ERSUM__BOUND,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),V4e_2E0: tyop_2Erealax_2Ereal,V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,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)))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V1b_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V6x_2E0)),V4e_2E0)) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(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),V2d_2E0,V3p_2E0),V5f_2E0)),c_2Erealax_2Ereal__mul_2E2(V4e_2E0,c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V0a_2E0)))) ) ).
tff(thm_2Eintegral_2ERSUM__DIFF__BOUND,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),V4e_2E0: tyop_2Erealax_2Ereal,V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,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)))
& ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V7x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V7x_2E0,V1b_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V7x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6g_2E0,V7x_2E0))),V4e_2E0)) ) )
=> p(c_2Ereal_2Ereal__lte_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),V2d_2E0,V3p_2E0),V5f_2E0),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),V2d_2E0,V3p_2E0),V6g_2E0))),c_2Erealax_2Ereal__mul_2E2(V4e_2E0,c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V0a_2E0)))) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__LIMIT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V5x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V5x_2E0,V2b_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V5x_2E0))),V3e_2E0)) )
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V4g_2E0)) ) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0)) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__CONST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V3x_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,V2c_2E0),V3x_2E0) = V2c_2E0 )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] : p(c_2Eintegral_2Eintegrable_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_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2c_2E0))) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__ADD,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)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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,V0f_2E0),V1g_2E0),V4x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0)) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_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,V0f_2E0),V1g_2E0))) ) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V3c_2E0: tyop_2Erealax_2Ereal,V0f_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_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,V3c_2E0),V0f_2E0),V4x_2E0) = c_2Erealax_2Ereal__mul_2E2(V3c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)) )
=> ! [V0f_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] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0)) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(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,V3c_2E0),V0f_2E0))) ) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__COMBINE,axiom,
! [V0f_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] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,V3c_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2b_2E0,V3c_2E0),V0f_2E0)) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3c_2E0),V0f_2E0)) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__POINT__SPIKE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4c_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V5x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V5x_2E0,V3b_2E0))
& ( V5x_2E0 != V4c_2E0 ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0) ) )
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0)) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0)) ) ).
tff(thm_2Eintegral_2ESUP__INTERVAL,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ? [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0)) )
=> ? [V4s_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V4s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4s_2E0,V2b_2E0))
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V4s_2E0))
<=> ? [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V2b_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V5y_2E0,V6x_2E0)) ) ) ) ) ).
tff(thm_2Eintegral_2EBOLZANO__LEMMA__ALT,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,V3c_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V1a_2E0),V2b_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V2b_2E0),V3c_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V1a_2E0),V3c_2E0)) )
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
? [V5d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5d_2E0))
& ! [V6a_2E0: tyop_2Erealax_2Ereal,V7b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V6a_2E0,V4x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V7b_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__sub_2E2(V7b_2E0,V6a_2E0),V5d_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V6a_2E0),V7b_2E0)) ) ) )
=> ! [V8a_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V8a_2E0,V9b_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V8a_2E0),V9b_2E0)) ) ) ).
tff(thm_2Eintegral_2ECONT__UNIFORM,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) )
=> p(c_2Elim_2Econtl_2E2(V0f_2E0,V3x_2E0)) ) )
=> ! [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
=> ? [V5d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5d_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal,V7y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V2b_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V7y_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V7y_2E0,V2b_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V6x_2E0,V7y_2E0)),V5d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7y_2E0))),V4e_2E0)) ) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__CONTINUOUS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) )
=> p(c_2Elim_2Econtl_2E2(V0f_2E0,V3x_2E0)) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0)) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__SPLIT__SIDES,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V7x_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,V1a_2E0),V2b_2E0),V7x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V7x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V7x_2E0,V2b_2E0)) ) )
=> ! [V0f_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] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3c_2E0,V2b_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0)) )
=> ? [V4i_2E0: tyop_2Erealax_2Ereal] :
! [V5e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5e_2E0))
=> ? [V6g_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,V1a_2E0),V2b_2E0),V6g_2E0))
& ! [V8d1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V9p1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V10d2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V11p2_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),V8d1_2E0,V9p1_2E0)))
& p(c_2Etransc_2Efine_2E2(V6g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V8d1_2E0,V9p1_2E0)))
& p(c_2Etransc_2Etdiv_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3c_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),V10d2_2E0,V11p2_2E0)))
& p(c_2Etransc_2Efine_2E2(V6g_2E0,c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V10d2_2E0,V11p2_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__add_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),V8d1_2E0,V9p1_2E0),V0f_2E0),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),V10d2_2E0,V11p2_2E0),V0f_2E0)),V4i_2E0)),V5e_2E0)) ) ) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL__LEFT,axiom,
! [V0f_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] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3c_2E0,V2b_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0)) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3c_2E0),V0f_2E0)) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL__RIGHT,axiom,
! [V0f_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] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3c_2E0,V2b_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0)) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3c_2E0,V2b_2E0),V0f_2E0)) ) ).
tff(thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL,axiom,
! [V0f_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,V4d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3c_2E0,V4d_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4d_2E0,V2b_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0)) )
=> p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3c_2E0,V4d_2E0),V0f_2E0)) ) ).
tff(thm_2Eintegral_2EINTEGRAL__0,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V2x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
=> ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),F0_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRAL__CONST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V3x_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,V2c_2E0),V3x_2E0) = V2c_2E0 )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
=> ( c_2Eintegral_2Eintegral_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_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2c_2E0)) = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V0a_2E0)) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRAL__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_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_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,V1c_2E0),V0f_2E0),V4x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3b_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0)) )
=> ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(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,V1c_2E0),V0f_2E0)) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0)) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRAL__ADD,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)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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,V0f_2E0),V1g_2E0),V4x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0)) )
=> ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_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,V0f_2E0),V1g_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0),c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0)) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRAL__SUB,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)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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,V0f_2E0),V1g_2E0),V4x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0)) )
=> ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_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,V0f_2E0),V1g_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V0f_2E0),c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),V1g_2E0)) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRAL__BY__PARTS,axiom,
! [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))),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)))] :
( ! [V2f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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),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,V2f_27_2E0),V1g_2E0),V11x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_27_2E0,V11x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V11x_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_27_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)),F2_2E0,V0f_2E0),V3g_27_2E0),V10x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V10x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_27_2E0,V10x_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_27_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)),F1_2E0,V0f_2E0),V3g_27_2E0),V9x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V9x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_27_2E0,V9x_2E0)) )
=> ( ! [V2f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_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)),F0_2E0,V2f_27_2E0),V1g_2E0),V8x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_27_2E0,V8x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V8x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V4a_2E0,V5b_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V4a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V5b_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(V0f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_27_2E0,V6x_2E0),V6x_2E0)) )
& ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V4a_2E0,V7x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V7x_2E0,V5b_2E0)) )
=> p(c_2Elim_2Ediffl_2E3(V1g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_27_2E0,V7x_2E0),V7x_2E0)) )
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_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,V2f_27_2E0),V1g_2E0)))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_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)),F1_2E0,V0f_2E0),V3g_27_2E0))) )
=> ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_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)),F2_2E0,V0f_2E0),V3g_27_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5b_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5b_2E0)),c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4a_2E0))),c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_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)),F3_2E0,V2f_27_2E0),V1g_2E0))) ) ) ) ) ) ) ).
tff(thm_2Eintegral_2EINTEGRAL__COMBINE,axiom,
! [V0f_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] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,V3c_2E0))
& p(c_2Eintegral_2Eintegrable_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3c_2E0),V0f_2E0)) )
=> ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3c_2E0),V0f_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0),c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2b_2E0,V3c_2E0),V0f_2E0)) ) ) ).
tff(thm_2Eintegral_2EINTEGRAL__MVT1,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) )
=> p(c_2Elim_2Econtl_2E2(V0f_2E0,V3x_2E0)) ) )
=> ? [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V4x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V2b_2E0))
& ( c_2Eintegral_2Eintegral_2E2(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),V0f_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),c_2Ereal_2Ereal__sub_2E2(V2b_2E0,V1a_2E0)) ) ) ) ).
%------------------------------------------------------------------------------