ITP001 Axioms: ITP137^5.ax
%------------------------------------------------------------------------------
% File : ITP137^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : integral^2.ax [Gau20]
% : HL4137^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 88 ( 0 unt; 2 typ; 0 def)
% Number of atoms : 4476 ( 35 equ; 0 cnn)
% Maximal formula atoms : 200 ( 50 avg)
% Number of connectives : 5635 ( 4 ~; 2 |; 156 &;5253 @)
% ( 10 <=>; 210 =>; 0 <=; 0 <~>)
% Maximal formula depth : 44 ( 18 avg;5253 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 56 usr; 57 con; 0-2 aty)
% Number of variables : 434 ( 35 ^ 373 !; 26 ?; 434 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eintegral_2Eintegrable,type,
c_2Eintegral_2Eintegrable: $i ).
thf(mem_c_2Eintegral_2Eintegrable,axiom,
mem @ c_2Eintegral_2Eintegrable @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ bool ) ) ).
thf(tp_c_2Eintegral_2Eintegral,type,
c_2Eintegral_2Eintegral: $i ).
thf(mem_c_2Eintegral_2Eintegral,axiom,
mem @ c_2Eintegral_2Eintegral @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Eintegral_2ESUM__SPLIT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V1f ) ) @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) @ V1f ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V1f ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__APPEND__EXPLICIT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V4d1: $i] :
( ( mem @ V4d1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5p1: $i] :
( ( mem @ V5p1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V6d2: $i] :
( ( mem @ V6d2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V7p2: $i] :
( ( mem @ V7p2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4d1 ) @ V5p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V3g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4d1 ) @ V5p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6d2 ) @ V7p2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V3g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6d2 ) @ V7p2 ) ) ) )
=> ( ( p
@ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V8n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) @ ( ap @ V4d1 @ V8n ) ) @ ( ap @ V6d2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V8n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V9n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) @ ( ap @ V5p1 @ V9n ) ) @ ( ap @ V7p2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V9n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) ) ) ) ) )
& ( p
@ ( ap @ ( ap @ c_2Etransc_2Efine @ V3g )
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V10n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) @ ( ap @ V4d1 @ V10n ) ) @ ( ap @ V6d2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V10n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V11n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) @ ( ap @ V5p1 @ V11n ) ) @ ( ap @ V7p2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V11n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) ) ) ) ) )
& ! [V12f: $i] :
( ( mem @ V12f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Etransc_2Ersum
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V13n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V13n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) @ ( ap @ V4d1 @ V13n ) ) @ ( ap @ V6d2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V13n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V14n: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V14n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) @ ( ap @ V5p1 @ V14n ) ) @ ( ap @ V7p2 @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V14n ) @ ( ap @ c_2Etransc_2Edsize @ V4d1 ) ) ) ) ) ) )
@ V12f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4d1 ) @ V5p1 ) ) @ V12f ) ) @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6d2 ) @ V7p2 ) ) @ V12f ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__APPEND__STRONG,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal,V4D1: $i] :
( ( mem @ V4D1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5p1: $i] :
( ( mem @ V5p1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V6D2: $i] :
( ( mem @ V6D2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V7p2: $i] :
( ( mem @ V7p2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D1 ) @ V5p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D1 ) @ V5p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6D2 ) @ V7p2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6D2 ) @ V7p2 ) ) ) )
=> ? [V8D: $i] :
( ( mem @ V8D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V9p: $i] :
( ( mem @ V9p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8D ) @ V9p ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8D ) @ V9p ) ) )
& ! [V10f: $i] :
( ( mem @ V10f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8D ) @ V9p ) ) @ V10f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D1 ) @ V5p1 ) ) @ V10f ) ) @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6D2 ) @ V7p2 ) ) @ V10f ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__APPEND,axiom,
! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ? [V4D1: $i] :
( ( mem @ V4D1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V5p1: $i] :
( ( mem @ V5p1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D1 ) @ V5p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V4D1 ) @ V5p1 ) ) ) ) )
& ? [V6D2: $i] :
( ( mem @ V6D2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V7p2: $i] :
( ( mem @ V7p2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6D2 ) @ V7p2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6D2 ) @ V7p2 ) ) ) ) ) )
=> ? [V8D: $i] :
( ( mem @ V8D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V9p: $i] :
( ( mem @ V9p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8D ) @ V9p ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V0g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8D ) @ V9p ) ) ) ) ) ) ) ).
thf(ax_thm_2Eintegral_2Eintegrable__def,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2f ) )
<=> ? [V3i: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V3i ) ) ) ) ) ).
thf(ax_thm_2Eintegral_2Eintegral__def,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3i: $i] : ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2f ) @ V3i ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__DINT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__BOUNDS,axiom,
! [V0d: $i] :
( ( mem @ V0d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0d ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ETDIV__BOUNDS,axiom,
! [V0d: $i] :
( ( mem @ V0d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V0d ) @ V1p ) ) )
=> ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ V1p @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1p @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ETDIV__LE,axiom,
! [V0d: $i] :
( ( mem @ V0d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V0d ) @ V1p ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__WRONG,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3i: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V3i ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__INTEGRAL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3i ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) )
= V3i ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__NEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3i ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V4x ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V3i ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__0,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V6x ) ) @ ( ap @ V1g @ V6x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V6x ) ) @ ( ap @ V1g @ V6x ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDSIZE__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2D: $i] :
( ( mem @ V2D @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2D ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Ereal_2Ereal__sub
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Etransc_2Edsize @ V2D ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V2D @ ( ap @ c_2Enum_2ESUC @ V3n ) ) ) @ ( ap @ V2D @ V3n ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__CONST,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( inj__ty_2Erealax_2Ereal @ V2c ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__CMUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( ap @ V0f @ V5x ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__LINEAR,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal,V6i: tp__ty_2Erealax_2Ereal,V7j: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V6i ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) @ V3g ) @ ( inj__ty_2Erealax_2Ereal @ V7j ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( ap @ V2f @ V8x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1n ) ) @ ( ap @ V3g @ V8x ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0m ) ) @ ( inj__ty_2Erealax_2Ereal @ V6i ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1n ) ) @ ( inj__ty_2Erealax_2Ereal @ V7j ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ELT,axiom,
( ! [V0m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> $false )
& ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
<=> ( ( V1m = V2n )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ELE__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ).
thf(conj_thm_2Eintegral_2ELT__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Eintegral_2EEQ__SUC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( V0m = V1n ) ) ).
thf(conj_thm_2Eintegral_2ELE__LT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
| ( V0m = V1n ) ) ) ).
thf(conj_thm_2Eintegral_2ELT__LE,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( V0m != V1n ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__LT__MIN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__LE__RMUL1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__LE__LMUL1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__LE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: $i] :
( ( mem @ V4i @ A_27a )
=> ! [V5j: $i] :
( ( mem @ V5j @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__TRIANGLE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Erealax_2Ereal,V4j: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3i ) ) )
& ( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V5x ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V4j ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3i ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4j ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) ) )
=> ( V4i = V5j ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRATION__BY__PARTS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3g_27: $i] :
( ( mem @ V3g_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( ap @ V2f_27 @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( ap @ V3g_27 @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2f_27 @ V8x ) ) @ ( ap @ V1g @ V8x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V8x ) ) @ ( ap @ V3g_27 @ V8x ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__LE__SUC,axiom,
! [V0d: $i] :
( ( mem @ V0d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0d ) )
=> ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__MONO__LE,axiom,
! [V0d: $i] :
( ( mem @ V0d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0d ) )
=> ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__MONO__LE__SUC,axiom,
! [V0d: $i] :
( ( mem @ V0d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0d ) )
=> ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__DSIZE__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( ( mem @ V2d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2d ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2d @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Etransc_2Edsize @ V2d ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__DSIZE__GE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( ( mem @ V2d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2d ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V2d @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V2d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ c_2Etransc_2Edsize @ V2d ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__DSIZE__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( ( mem @ V2d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2d ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V2d @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V2d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2d @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Etransc_2Edsize @ V2d ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__DSIZE__EQ__ALT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( ( mem @ V2d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ V2d ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2d @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ! [V4i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V2d @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) @ ( ap @ V2d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Etransc_2Edsize @ V2d ) )
= V3n ) ) ) ).
thf(conj_thm_2Eintegral_2Enum__MAX,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ? [V1x: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1x ) ) )
& ? [V2M: tp__ty_2Enum_2Enum] :
! [V3x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2M ) ) ) ) )
<=> ? [V4m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4m ) ) )
& ! [V5x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V5x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5x ) ) @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDIVISION__INTERMEDIATE,axiom,
! [V0d: $i] :
( ( mem @ V0d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Edivision @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0d ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ? [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( ap @ c_2Etransc_2Edsize @ V0d ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( ap @ V0d @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__COMBINE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ V0f ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) @ ( inj__ty_2Erealax_2Ereal @ V5j ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ESUM__EQ__0,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V3r: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V3r ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3r ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3r ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V2f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__DELTA__LEFT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__DELTA__RIGHT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V2x ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__DELTA,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ V3x ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__POINT__SPIKE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4c: tp__ty_2Erealax_2Ereal,V5i: tp__ty_2Erealax_2Ereal] :
( ( ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( V6x != V4c ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5i ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V5i ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EDINT__FINITE__SPIKE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V5i: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V4s ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ V4s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5i ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Etransc_2EDint @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V5i ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__POW__LBOUND,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__ARCH__POW,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ? [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__ARCH__POW2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
? [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__POW__LE__1,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__POW__MONO,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EREAL__LE__INV2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EGAUGE__MIN__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1gs: $i] :
( ( mem @ V1gs @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ! [V3m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Etransc_2Egauge @ V0s ) @ ( ap @ V1gs @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
=> ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Egauge @ V0s ) @ V4g ) )
& ! [V5d: $i] :
( ( mem @ V5d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V6p: $i] :
( ( mem @ V6p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V4g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V5d ) @ V6p ) ) )
=> ! [V7m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V7m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ ( ap @ V1gs @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V5d ) @ V6p ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__CAUCHY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) )
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap @ c_2Etransc_2Egauge
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V5x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V5x ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
@ V4g ) )
& ! [V6d1: $i] :
( ( mem @ V6d1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V7p1: $i] :
( ( mem @ V7p1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V8d2: $i] :
( ( mem @ V8d2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V9p2: $i] :
( ( mem @ V9p2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6d1 ) @ V7p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V4g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6d1 ) @ V7p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8d2 ) @ V9p2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V4g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8d2 ) @ V9p2 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V6d1 ) @ V7p1 ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8d2 ) @ V9p2 ) ) @ V0f ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ESUM__DIFFS,axiom,
! [V0d: $i] :
( ( mem @ V0d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0d @ ( ap @ c_2Enum_2ESUC @ V3i ) ) ) @ ( ap @ V0d @ V3i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0d @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V0d @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ERSUM__BOUND,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( ( mem @ V2d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V4e: tp__ty_2Erealax_2Ereal,V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2d ) @ V3p ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2d ) @ V3p ) ) @ V5f ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ERSUM__DIFF__BOUND,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( ( mem @ V2d @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V4e: tp__ty_2Erealax_2Ereal,V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V6g: $i] :
( ( mem @ V6g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2d ) @ V3p ) ) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V5f @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) @ ( ap @ V6g @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2d ) @ V3p ) ) @ V5f ) ) @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V2d ) @ V3p ) ) @ V6g ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__LIMIT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ! [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) @ ( ap @ V4g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V4g ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__CONST,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__CMUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( ap @ V0f @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__COMBINE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__POINT__SPIKE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4c: tp__ty_2Erealax_2Ereal] :
( ( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( V5x != V4c ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ESUP__INTERVAL,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ? [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
=> ? [V4s: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4s ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4s ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ ( inj__ty_2Erealax_2Ereal @ V4s ) ) )
<=> ? [V6x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EBOLZANO__LEMMA__ALT,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
? [V5d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) )
& ! [V6a: tp__ty_2Erealax_2Ereal,V7b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) ) ) ) )
=> ! [V8a: tp__ty_2Erealax_2Ereal,V9b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2ECONT__UNIFORM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) )
=> ? [V5d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal,V7y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__CONTINUOUS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__SPLIT__SIDES,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) )
=> ? [V4i: tp__ty_2Erealax_2Ereal] :
! [V5e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) )
=> ? [V6g: $i] :
( ( mem @ V6g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap @ c_2Etransc_2Egauge
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ V7x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V7x ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
@ V6g ) )
& ! [V8d1: $i] :
( ( mem @ V8d1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V9p1: $i] :
( ( mem @ V9p1 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V10d2: $i] :
( ( mem @ V10d2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V11p2: $i] :
( ( mem @ V11p2 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8d1 ) @ V9p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V6g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8d1 ) @ V9p1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Etdiv @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V10d2 ) @ V11p2 ) ) )
& ( p @ ( ap @ ( ap @ c_2Etransc_2Efine @ V6g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V10d2 ) @ V11p2 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V8d1 ) @ V9p1 ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Etransc_2Ersum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V10d2 ) @ V11p2 ) ) @ V0f ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4i ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5e ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL__LEFT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL__RIGHT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal,V4d: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) @ ( inj__ty_2Erealax_2Ereal @ V4d ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__0,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__CONST,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__CMUL,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0f @ V4x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__SUB,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b ) ) ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__BY__PARTS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V3g_27: $i] :
( ( mem @ V3g_27 @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V0f ) @ ( ap @ V2f_27 @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) ) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Ediffl @ V1g ) @ ( ap @ V3g_27 @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) ) )
& ( p
@ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V8x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2f_27 @ V8x ) ) @ ( ap @ V1g @ V8x ) ) ) ) )
& ( p
@ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V9x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V9x ) ) @ ( ap @ V3g_27 @ V9x ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V10x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V10x ) ) @ ( ap @ V3g_27 @ V10x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) ) ) )
@ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4a ) ) @ ( inj__ty_2Erealax_2Ereal @ V5b ) ) )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V11x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2f_27 @ V11x ) ) @ ( ap @ V1g @ V11x ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__COMBINE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ ( ap @ c_2Eintegral_2Eintegrable @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ V0f ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eintegral_2EINTEGRAL__MVT1,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Elim_2Econtl @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ? [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Eintegral_2Eintegral @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------