ITP001 Axioms: ITP148^5.ax
%------------------------------------------------------------------------------
% File : ITP148^5 : 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 : lebesgue^2.ax [Gau20]
% : HL4148^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 182 ( 0 unt; 21 typ; 0 def)
% Number of atoms : 11490 ( 165 equ; 0 cnn)
% Maximal formula atoms : 171 ( 63 avg)
% Number of connectives : 16494 ( 29 ~; 1 |; 407 &;15341 @)
% ( 8 <=>; 708 =>; 0 <=; 0 <~>)
% Maximal formula depth : 48 ( 21 avg;15341 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 28 ( 28 >; 0 *; 0 +; 0 <<)
% Number of symbols : 120 ( 119 usr; 99 con; 0-3 aty)
% Number of variables : 1023 ( 149 ^ 813 !; 61 ?;1023 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Elebesgue_2ERADON__F,type,
c_2Elebesgue_2ERADON__F: del > $i ).
thf(mem_c_2Elebesgue_2ERADON__F,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2ERADON__F @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ bool ) ) ) ) ).
thf(tp_c_2Elebesgue_2ERADON__F__integrals,type,
c_2Elebesgue_2ERADON__F__integrals: del > $i ).
thf(mem_c_2Elebesgue_2ERADON__F__integrals,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2ERADON__F__integrals @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) ) ) ) ).
thf(tp_c_2Elebesgue_2Efinite__space__integral,type,
c_2Elebesgue_2Efinite__space__integral: del > $i ).
thf(mem_c_2Elebesgue_2Efinite__space__integral,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Efinite__space__integral @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ty_2Eextreal_2Eextreal ) ) ) ).
thf(tp_c_2Elebesgue_2Efn__seq,type,
c_2Elebesgue_2Efn__seq: del > $i ).
thf(mem_c_2Elebesgue_2Efn__seq,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) ) ) ) ).
thf(tp_c_2Elebesgue_2Efn__seq__integral,type,
c_2Elebesgue_2Efn__seq__integral: del > $i ).
thf(mem_c_2Elebesgue_2Efn__seq__integral,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Efn__seq__integral @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal ) ) ) ) ).
thf(tp_c_2Elebesgue_2Eintegrable,type,
c_2Elebesgue_2Eintegrable: del > $i ).
thf(mem_c_2Elebesgue_2Eintegrable,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ bool ) ) ) ).
thf(tp_c_2Elebesgue_2Eintegral,type,
c_2Elebesgue_2Eintegral: del > $i ).
thf(mem_c_2Elebesgue_2Eintegral,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ty_2Eextreal_2Eextreal ) ) ) ).
thf(tp_c_2Elebesgue_2Emax__fn__seq,type,
c_2Elebesgue_2Emax__fn__seq: del > $i ).
thf(mem_c_2Elebesgue_2Emax__fn__seq,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) ) ) ).
thf(tp_c_2Elebesgue_2Emeasure__absolutely__continuous,type,
c_2Elebesgue_2Emeasure__absolutely__continuous: del > $i ).
thf(mem_c_2Elebesgue_2Emeasure__absolutely__continuous,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Emeasure__absolutely__continuous @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ) ).
thf(tp_c_2Elebesgue_2Enegative__set,type,
c_2Elebesgue_2Enegative__set: del > $i ).
thf(mem_c_2Elebesgue_2Enegative__set,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Enegative__set @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Elebesgue_2Epos__fn__integral,type,
c_2Elebesgue_2Epos__fn__integral: del > $i ).
thf(mem_c_2Elebesgue_2Epos__fn__integral,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ty_2Eextreal_2Eextreal ) ) ) ).
thf(tp_c_2Elebesgue_2Epos__simple__fn__integral,type,
c_2Elebesgue_2Epos__simple__fn__integral: del > $i ).
thf(mem_c_2Elebesgue_2Epos__simple__fn__integral,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ty_2Eextreal_2Eextreal ) ) ) ) ) ).
thf(tp_c_2Elebesgue_2Eprod__measure,type,
c_2Elebesgue_2Eprod__measure: del > del > $i ).
thf(mem_c_2Elebesgue_2Eprod__measure,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) ) ).
thf(tp_c_2Elebesgue_2Eprod__measure3,type,
c_2Elebesgue_2Eprod__measure3: del > del > del > $i ).
thf(mem_c_2Elebesgue_2Eprod__measure3,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) ) ) ).
thf(tp_c_2Elebesgue_2Eprod__measure__space,type,
c_2Elebesgue_2Eprod__measure__space: del > del > $i ).
thf(mem_c_2Elebesgue_2Eprod__measure__space,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Elebesgue_2Eprod__measure__space @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ty_2Epair_2Eprod @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(tp_c_2Elebesgue_2Eprod__measure__space3,type,
c_2Elebesgue_2Eprod__measure__space3: del > del > del > $i ).
thf(mem_c_2Elebesgue_2Eprod__measure__space3,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Elebesgue_2Eprod__measure__space3 @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ty_2Epair_2Eprod @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) ) ) ) ) ).
thf(tp_c_2Elebesgue_2Eprod__sets3,type,
c_2Elebesgue_2Eprod__sets3: del > del > del > $i ).
thf(mem_c_2Elebesgue_2Eprod__sets3,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Elebesgue_2Eprod__sets3 @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) ) ) ) ) ).
thf(tp_c_2Elebesgue_2Epsfis,type,
c_2Elebesgue_2Epsfis: del > $i ).
thf(mem_c_2Elebesgue_2Epsfis,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) ) ) ) ).
thf(tp_c_2Elebesgue_2Epsfs,type,
c_2Elebesgue_2Epsfs: del > $i ).
thf(mem_c_2Elebesgue_2Epsfs,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Epsfs @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ) ) ).
thf(tp_c_2Elebesgue_2Eseq__sup,type,
c_2Elebesgue_2Eseq__sup: $i ).
thf(mem_c_2Elebesgue_2Eseq__sup,axiom,
mem @ c_2Elebesgue_2Eseq__sup @ ( arr @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) @ ( arr @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal ) ) ).
thf(tp_c_2Elebesgue_2Esigned__measure__space,type,
c_2Elebesgue_2Esigned__measure__space: del > $i ).
thf(mem_c_2Elebesgue_2Esigned__measure__space,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Esigned__measure__space @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(ax_thm_2Elebesgue_2Epos__simple__fn__integral__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V1s ) @ V2a ) @ V3x ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2ENormal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V3x @ V4i ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ V2a @ V4i ) ) ) ) )
@ V1s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Epsfs__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2Epsfs @ A_27a ) @ V0m ) @ V1f )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ) @ bool ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ bool )
@ ^ [V2s: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ) @ bool ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V3a: $i] :
( lam @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal )
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ) @ V2s ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ V3a ) @ V4x ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Epsfis__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ) @ ty_2Eextreal_2Eextreal )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) @ ty_2Eextreal_2Eextreal )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ bool )
@ ^ [V2s: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ty_2Eextreal_2Eextreal )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V3a: $i] :
( lam @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal )
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfs @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Epos__fn__integral__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V2r: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V2r )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal )
@ ^ [V3g: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ V2r ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V3g ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V3g @ V4x ) ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Eintegral__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Eintegrable__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) ) )
!= fo__c_2Eextreal_2EPosInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Efinite__space__integral__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Efinite__space__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V2r: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ V2r ) @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Eextreal_2Eextreal ) @ V2r ) @ ( c_2Epred__set_2EEMPTY @ ty_2Eextreal_2Eextreal ) ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1f ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Eprod__measure__def,axiom,
! [A_27a: del,A_27b: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b ) @ V0m0 ) @ V1m1 )
= ( lam @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool )
@ ^ [V2a: $i] :
( ap @ c_2Eextreal_2Ereal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m0 )
@ ( lam @ A_27a
@ ^ [V3s0: $i] :
( ap @ c_2Eextreal_2ENormal
@ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27b ) @ V1m1 )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ A_27b
@ ^ [V4s1: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3s0 ) @ V4s1 ) ) )
@ V2a ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Eprod__measure__space__def,axiom,
! [A_27a: del,A_27b: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure__space @ A_27a @ A_27b ) @ V0m0 ) @ V1m1 )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) ) @ ( ap @ ( ap @ ( c_2Eutil__prob_2Eprod__sets @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b ) @ V0m0 ) @ V1m1 ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Eprod__sets3__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__sets3 @ A_27a @ A_27b @ A_27c ) @ V0a ) @ V1b ) @ V2c )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ A_27c @ bool ) ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ A_27c @ bool ) ) @ ( ty_2Epair_2Eprod @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3s: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ ( arr @ A_27b @ bool ) @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V4t: $i] :
( lam @ ( arr @ A_27c @ bool )
@ ^ [V5u: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V3s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ V4t ) @ V5u ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ V0a ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V4t ) @ V1b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27c @ bool ) ) @ V5u ) @ V2c ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Eprod__measure3__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c ) @ V0m0 ) @ V1m1 ) @ V2m2 )
= ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V0m0 ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure__space @ A_27b @ A_27c ) @ V1m1 ) @ V2m2 ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Eprod__measure__space3__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure__space3 @ A_27a @ A_27b @ A_27c ) @ V0m0 ) @ V1m1 ) @ V2m2 )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27c ) @ V2m2 ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27c ) @ V2m2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__sets3 @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27c ) @ V2m2 ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c ) @ V0m0 ) @ V1m1 ) @ V2m2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__present,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V6s_27: $i] :
( ( mem @ V6s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V7b: $i] :
( ( mem @ V7b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V8y: $i] :
( ( mem @ V8y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V5g ) @ V6s_27 ) @ V7b ) @ V8y ) ) )
=> ? [V9z: $i] :
( ( mem @ V9z @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V10z_27: $i] :
( ( mem @ V10z_27 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V11c: $i] :
( ( mem @ V11c @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V12k: $i] :
( ( mem @ V12k @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ! [V13t: $i] :
( ( mem @ V13t @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V13t ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V13t ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V14i: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V9z @ V14i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V11c @ V14i ) ) @ V13t ) ) ) )
@ V12k ) ) ) ) )
& ! [V15t: $i] :
( ( mem @ V15t @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V15t ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V5g @ V15t ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V16i: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V10z_27 @ V16i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V11c @ V16i ) ) @ V15t ) ) ) )
@ V12k ) ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V12k ) @ V11c ) @ V9z ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V6s_27 ) @ V7b ) @ V8y ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V12k ) @ V11c ) @ V10z_27 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V12k ) )
& ! [V17i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V17i ) ) @ V12k ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V9z @ ( inj__ty_2Enum_2Enum @ V17i ) ) ) ) )
& ! [V18i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V18i ) ) @ V12k ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V10z_27 @ ( inj__ty_2Enum_2Enum @ V18i ) ) ) ) )
& ! [V19i: tp__ty_2Enum_2Enum,V20j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V19i ) ) @ V12k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V20j ) ) @ V12k ) )
& ( V19i != V20j ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V11c @ ( inj__ty_2Enum_2Enum @ V19i ) ) ) @ ( ap @ V11c @ ( inj__ty_2Enum_2Enum @ V20j ) ) ) ) )
& ! [V21i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V21i ) ) @ V12k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V11c @ ( inj__ty_2Enum_2Enum @ V21i ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V11c ) @ V12k ) )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__present,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3a: tp__ty_2Eextreal_2Eextreal,V4b: tp__ty_2Eextreal_2Eextreal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V3a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V4b ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V2g ) ) ) )
=> ? [V5z: $i] :
( ( mem @ V5z @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V6z_27: $i] :
( ( mem @ V6z_27 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ? [V7c: $i] :
( ( mem @ V7c @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V8k: $i] :
( ( mem @ V8k @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ! [V9t: $i] :
( ( mem @ V9t @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V9t ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V9t ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10i: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V5z @ V10i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V7c @ V10i ) ) @ V9t ) ) ) )
@ V8k ) ) ) ) )
& ! [V11t: $i] :
( ( mem @ V11t @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V11t ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2g @ V11t ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V12i: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V6z_27 @ V12i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V7c @ V12i ) ) @ V11t ) ) ) )
@ V8k ) ) ) ) )
& ( V3a
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V8k ) @ V7c ) @ V5z ) ) )
& ( V4b
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V8k ) @ V7c ) @ V6z_27 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V8k ) )
& ! [V13i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V13i ) ) @ V8k ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V5z @ ( inj__ty_2Enum_2Enum @ V13i ) ) ) ) )
& ! [V14i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V14i ) ) @ V8k ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V6z_27 @ ( inj__ty_2Enum_2Enum @ V14i ) ) ) ) )
& ! [V15i: tp__ty_2Enum_2Enum,V16j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V15i ) ) @ V8k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V16j ) ) @ V8k ) )
& ( V15i != V16j ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V7c @ ( inj__ty_2Enum_2Enum @ V15i ) ) ) @ ( ap @ V7c @ ( inj__ty_2Enum_2Enum @ V16j ) ) ) ) )
& ! [V17i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V17i ) ) @ V8k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V7c @ ( inj__ty_2Enum_2Enum @ V17i ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V7c ) @ V8k ) )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__thm1,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5i: tp__ty_2Enum_2Enum,V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5i ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6y ) @ ( ap @ V3a @ ( inj__ty_2Enum_2Enum @ V5i ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V6y ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V4x @ ( inj__ty_2Enum_2Enum @ V5i ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__le,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V6x_27: $i] :
( ( mem @ V6x_27 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V7i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V3s ) @ V4a ) @ V5x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V2g ) @ V3s ) @ V4a ) @ V6x_27 ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V2g @ V8x ) ) @ ( ap @ V1f @ V8x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V7i ) ) @ V3s ) )
& ( ( ap @ V4a @ ( inj__ty_2Enum_2Enum @ V7i ) )
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V6x_27 @ ( inj__ty_2Enum_2Enum @ V7i ) ) ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V5x @ ( inj__ty_2Enum_2Enum @ V7i ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__cmul,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3m: $i] :
( ( mem @ V3m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V3m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V3m ) @ V4f ) @ V0s ) @ V1a ) @ V2x ) )
& ( 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 @ V5z ) ) ) )
=> ? [V6s_27: $i] :
( ( mem @ V6s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V7a_27: $i] :
( ( mem @ V7a_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V8x_27: $i] :
( ( mem @ V8x_27 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V3m )
@ ( lam @ A_27a
@ ^ [V9t: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) @ ( ap @ V4f @ V9t ) ) ) )
@ V6s_27 )
@ V7a_27 )
@ V8x_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__cmul__alt,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( 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 @ V5z ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V6t: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) @ ( ap @ V1f @ V6t ) ) ) )
@ V2s )
@ V3a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( ap @ V4x @ V7i ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__add,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3s_27: $i] :
( ( mem @ V3s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V4a_27: $i] :
( ( mem @ V4a_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V5x_27: $i] :
( ( mem @ V5x_27 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V6m: $i] :
( ( mem @ V6m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V8g: $i] :
( ( mem @ V8g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V6m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V6m ) @ V7f ) @ V0s ) @ V1a ) @ V2x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V6m ) @ V8g ) @ V3s_27 ) @ V4a_27 ) @ V5x_27 ) ) )
=> ? [V9s_27_27: $i] :
( ( mem @ V9s_27_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V10a_27_27: $i] :
( ( mem @ V10a_27_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V11x_27_27: $i] :
( ( mem @ V11x_27_27 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V6m )
@ ( lam @ A_27a
@ ^ [V12t: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V7f @ V12t ) ) @ ( ap @ V8g @ V12t ) ) ) )
@ V9s_27_27 )
@ V10a_27_27 )
@ V11x_27_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__add__alt,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V6y: $i] :
( ( mem @ V6y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V3s ) @ V4a ) @ V5x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V2g ) @ V3s ) @ V4a ) @ V6y ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V7t: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V1f @ V7t ) ) @ ( ap @ V2g @ V7t ) ) ) )
@ V3s )
@ V4a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V5x @ V8i ) ) @ ( ap @ V6y @ V8i ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ? [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1A ) ) @ V2s ) @ V3a ) @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__indicator__alt,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2i ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ V1s ) ) @ V1s ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V3i ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__max,axiom,
! [A_27a: del,A_27b: del,V0s_27: $i] :
( ( mem @ V0s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2m: $i] :
( ( mem @ V2m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V5a: $i] :
( ( mem @ V5a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V8s_27b: $i] :
( ( mem @ V8s_27b @ A_27b )
=> ! [V9y: $i] :
( ( mem @ V9y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V2m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V2m ) @ V3f ) @ V4s ) @ V5a ) @ V6x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V2m ) @ V7g ) @ V0s_27 ) @ V1b ) @ V9y ) ) )
=> ? [V10s_27_27: $i] :
( ( mem @ V10s_27_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V11a_27_27: $i] :
( ( mem @ V11a_27_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V12x_27_27: $i] :
( ( mem @ V12x_27_27 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V2m )
@ ( lam @ A_27a
@ ^ [V13x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__max @ ( ap @ V3f @ V13x ) ) @ ( ap @ V7g @ V13x ) ) ) )
@ V10s_27_27 )
@ V11a_27_27 )
@ V12x_27_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__not__infty,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V5x ) )
!= fo__c_2Eextreal_2ENegInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V5x ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__add,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V6s_27: $i] :
( ( mem @ V6s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V7b: $i] :
( ( mem @ V7b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V8y: $i] :
( ( mem @ V8y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V5g ) @ V6s_27 ) @ V7b ) @ V8y ) ) )
=> ? [V9s_27_27: $i] :
( ( mem @ V9s_27_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V10c: $i] :
( ( mem @ V10c @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V11z: $i] :
( ( mem @ V11z @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V12x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V1f @ V12x ) ) @ ( ap @ V5g @ V12x ) ) ) )
@ V9s_27_27 )
@ V10c )
@ V11z ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V6s_27 ) @ V7b ) @ V8y ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V9s_27_27 ) @ V10c ) @ V11z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__add__alt,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V6y: $i] :
( ( mem @ V6y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V5g ) @ V2s ) @ V3a ) @ V6y ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V6y ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V4x @ V7i ) ) @ ( ap @ V6y @ V7i ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__add,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3a: tp__ty_2Eextreal_2Eextreal,V4b: tp__ty_2Eextreal_2Eextreal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V3a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V4b ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V2g ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( inj__ty_2Eextreal_2Eextreal @ V3a ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V4b ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__mono,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V6s_27: $i] :
( ( mem @ V6s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V7b: $i] :
( ( mem @ V7b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V8y: $i] :
( ( mem @ V8y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V5g ) @ V6s_27 ) @ V7b ) @ V8y ) )
& ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V9x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V1f @ V9x ) ) @ ( ap @ V5g @ V9x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V6s_27 ) @ V7b ) @ V8y ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__mono,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3a: tp__ty_2Eextreal_2Eextreal,V4b: tp__ty_2Eextreal_2Eextreal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V3a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V4b ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V2g ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V3a ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V4b ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__unique,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5s_27: $i] :
( ( mem @ V5s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V6b: $i] :
( ( mem @ V6b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V7y: $i] :
( ( mem @ V7y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V5s_27 ) @ V6b ) @ V7y ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V5s_27 ) @ V6b ) @ V7y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__unique,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2a: tp__ty_2Eextreal_2Eextreal,V3b: tp__ty_2Eextreal_2Eextreal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V2a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V3b ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) ) )
=> ( V2a = V3b ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ? [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1A ) ) @ V2s ) @ V3a ) @ V4x ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1A ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1A ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1A ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__cmul,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
& ( 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 @ V5z ) ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) @ ( ap @ V1f @ V6x ) ) ) )
@ V2s )
@ V3a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( ap @ V4x @ V7i ) ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) @ ( ap @ V4x @ V8i ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__cmul,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2a: tp__ty_2Eextreal_2Eextreal,V3z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V2a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) )
& ( 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 @ V3z ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2a ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__cmul__alt,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( 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 @ V5z ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) ) )
=> ? [V6s_27: $i] :
( ( mem @ V6s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V7a_27: $i] :
( ( mem @ V7a_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V8x_27: $i] :
( ( mem @ V8x_27 @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V9t: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) @ ( ap @ V1f @ V9t ) ) ) )
@ V6s_27 )
@ V7a_27 )
@ V8x_27 ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V6s_27 ) @ V7a_27 ) @ V8x_27 ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V5z ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2EIN__psfis,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1r: tp__ty_2Eextreal_2Eextreal,V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V1r ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V2f ) ) )
=> ? [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V4a: $i] :
( ( mem @ V4a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V5x: $i] :
( ( mem @ V5x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V2f ) @ V3s ) @ V4a ) @ V5x ) )
& ( V1r
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V3s ) @ V4a ) @ V5x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2EIN__psfis__eq,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1r: tp__ty_2Eextreal_2Eextreal,V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V1r ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V2f ) ) )
<=> ? [V3s: $i] :
( ( mem @ V3s @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V4a: $i] :
( ( mem @ V4a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V5x: $i] :
( ( mem @ V5x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V2f ) @ V3s ) @ V4a ) @ V5x ) )
& ( V1r
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V3s ) @ V4a ) @ V5x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__pos,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2a: tp__ty_2Eextreal_2Eextreal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V2a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__zero,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4t: $i] : ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
@ V1s )
@ V2a )
@ V3x ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V1s ) @ V2a ) @ V3x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__zero__alt,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V1m ) @ V0g ) @ V2s ) @ V3a ) @ V4x ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V0g @ V5x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V1m ) @ V2s ) @ V3a ) @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__zero,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1a: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V1a ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
<=> ( V1a
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__not__infty,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V1s ) @ V2a ) @ V3x ) )
!= fo__c_2Eextreal_2ENegInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V1s ) @ V2a ) @ V3x ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__not__infty,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2a: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V2a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) )
=> ( ( V2a != fo__c_2Eextreal_2ENegInf )
& ( V2a != fo__c_2Eextreal_2EPosInf ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__sum,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ A_27b @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) )
=> ! [V5P: $i] :
( ( mem @ V5P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V6i: $i] :
( ( mem @ V6i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6i ) @ V5P ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ ( ap @ V1f @ V6i ) ) @ V2s ) @ V3a ) @ ( ap @ V4x @ V6i ) ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V5P ) )
& ( V5P
!= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
=> ( ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V7t: $i] :
( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V8i: $i] : ( ap @ ( ap @ V1f @ V8i ) @ V7t ) ) )
@ V5P ) ) )
@ V2s )
@ V3a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9i: $i] :
( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V10j: $i] : ( ap @ ( ap @ V4x @ V10j ) @ V9i ) ) )
@ V5P ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11j: $i] :
( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V12i: $i] : ( ap @ ( ap @ V4x @ V12i ) @ V11j ) ) )
@ V5P ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V13i: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ ( ap @ V4x @ V13i ) ) ) )
@ V5P ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__sum__alt,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ A_27b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ A_27b @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) )
=> ! [V5P: $i] :
( ( mem @ V5P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V6i: $i] :
( ( mem @ V6i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6i ) @ V5P ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ ( ap @ V1f @ V6i ) ) @ ( ap @ V2s @ V6i ) ) @ ( ap @ V3a @ V6i ) ) @ ( ap @ V4x @ V6i ) ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V5P ) )
& ( V5P
!= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
=> ? [V7c: $i] :
( ( mem @ V7c @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V8k: $i] :
( ( mem @ V8k @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V9z: $i] :
( ( mem @ V9z @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V10t: $i] :
( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V11i: $i] : ( ap @ ( ap @ V1f @ V11i ) @ V10t ) ) )
@ V5P ) ) )
@ V8k )
@ V7c )
@ V9z ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V8k ) @ V7c ) @ V9z ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V12i: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ ( ap @ V2s @ V12i ) ) @ ( ap @ V3a @ V12i ) ) @ ( ap @ V4x @ V12i ) ) ) )
@ V5P ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__sum,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ A_27b @ ty_2Eextreal_2Eextreal ) )
=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V4i: $i] :
( ( mem @ V4i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4i ) @ V3P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ V2a @ V4i ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ ( ap @ V1f @ V4i ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3P ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b ) @ V2a ) @ V3P ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V5t: $i] :
( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V6i: $i] : ( ap @ ( ap @ V1f @ V6i ) @ V5t ) ) )
@ V3P ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__intro,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V4i: $i] :
( ( mem @ V4i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4i ) @ V3P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V1a @ V4i ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) )
& ! [V5i: $i] :
( ( mem @ V5i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5i ) @ V3P ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2x @ V5i ) ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3P ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal )
@ ( ap @ c_2Eextreal_2ENormal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V6i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2x @ V6i ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ V1a @ V6i ) ) ) ) )
@ V3P ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V7t: $i] :
( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V8i: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V2x @ V8i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V1a @ V8i ) ) @ V7t ) ) ) )
@ V3P ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__simple__fn__integral__sub,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V6s_27: $i] :
( ( mem @ V6s_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V7b: $i] :
( ( mem @ V7b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V8y: $i] :
( ( mem @ V8y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V5g @ V9x ) ) @ ( ap @ V1f @ V9x ) ) ) )
& ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V5g @ V10x ) )
!= fo__c_2Eextreal_2EPosInf ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V5g ) @ V6s_27 ) @ V7b ) @ V8y ) ) )
=> ? [V11s_27_27: $i] :
( ( mem @ V11s_27_27 @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V12c: $i] :
( ( mem @ V12c @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V13z: $i] :
( ( mem @ V13z @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V14x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( ap @ V1f @ V14x ) ) @ ( ap @ V5g @ V14x ) ) ) )
@ V11s_27_27 )
@ V12c )
@ V13z ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V6s_27 ) @ V7b ) @ V8y ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V11s_27_27 ) @ V12c ) @ V13z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__sub,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3a: tp__ty_2Eextreal_2Eextreal,V4b: tp__ty_2Eextreal_2Eextreal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V2g @ V5x ) ) @ ( ap @ V1f @ V5x ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2g @ V6x ) )
!= fo__c_2Eextreal_2EPosInf ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V3a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V4b ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V2g ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( inj__ty_2Eextreal_2Eextreal @ V3a ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V4b ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V7x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( ap @ V1f @ V7x ) ) @ ( ap @ V2g @ V7x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__pos__simple__fn,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a ) @ V0m ) @ V2s ) @ V3a ) @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__mspace,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__zero,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__mono,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V2g ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__mono__mspace,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V2g @ V3x ) ) @ ( ap @ V1f @ V3x ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V4x ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2g @ V5x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V2g ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__pos,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__cmul,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) )
& ( 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 @ V2c ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ V1f @ V4x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1s ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__cmul__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( 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 @ V2c ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1s ) @ V3x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__sum__cmul__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) )
& ! [V4i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V3x @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) )
& ! [V5i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5i ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V2a @ ( inj__ty_2Enum_2Enum @ V5i ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V6t: $i] :
( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V3x @ V7i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V2a @ V7i ) ) @ V6t ) ) ) )
@ V1s ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2ENormal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V3x @ V8i ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ V2a @ V8i ) ) ) ) )
@ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elebesgue__monotone__convergence__lemma,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2fi: $i] :
( ( mem @ V2fi @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V4r_27: tp__ty_2Eextreal_2Eextreal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V5i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V5i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( p
@ ( ap @ c_2Eextreal_2Eext__mono__increasing
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ V2fi @ V7i ) @ V6x ) ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9i: $i] : ( ap @ ( ap @ V2fi @ V9i ) @ V8x ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V8x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V4r_27 ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V3g ) ) )
& ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V3g @ V10x ) ) @ ( ap @ V1f @ V10x ) ) ) )
& ! [V11i: tp__ty_2Enum_2Enum,V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V11i ) ) @ V12x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V4r_27 ) )
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V13i: $i] : ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ V2fi @ V13i ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elebesgue__monotone__convergence,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2fi: $i] :
( ( mem @ V2fi @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V3i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V3i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V4i: tp__ty_2Enum_2Enum,V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V5x ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V6x ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( p
@ ( ap @ c_2Eextreal_2Eext__mono__increasing
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8i: $i] : ( ap @ ( ap @ V2fi @ V8i ) @ V7x ) ) ) ) )
& ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V9x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10i: $i] : ( ap @ ( ap @ V2fi @ V10i ) @ V9x ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V9x ) ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11i: $i] : ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ V2fi @ V11i ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elebesgue__monotone__convergence__subset,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2fi: $i] :
( ( mem @ V2fi @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V3A: $i] :
( ( mem @ V3A @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V4i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V5i: tp__ty_2Enum_2Enum,V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V5i ) ) @ V6x ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V7x ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9i: $i] : ( ap @ ( ap @ V2fi @ V9i ) @ V8x ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V8x ) ) ) ) )
& ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ( p
@ ( ap @ c_2Eextreal_2Eext__mono__increasing
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11i: $i] : ( ap @ ( ap @ V2fi @ V11i ) @ V10x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V12x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V12x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V3A ) @ V12x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V13i: $i] :
( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V14x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ ( ap @ V2fi @ V13i ) @ V14x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V3A ) @ V14x ) ) ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Efn__seq__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f )
= ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] :
( lam @ A_27a
@ ^ [V3x: $i] :
( ap
@ ( ap @ c_2Eextreal_2Eextreal__add
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4k: $i] :
( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ c_2Eextreal_2Eextreal__of__num @ V4k ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) )
@ ( ap
@ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ c_2Eextreal_2Eextreal__of__num @ V4k ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) ) @ ( ap @ V1f @ V5x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V1f @ V5x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ c_2Eextreal_2Eextreal__of__num @ V4k ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) ) ) ) ) ) ) )
@ V3x ) ) ) )
@ ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) ) )
@ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) )
@ ( ap
@ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) @ ( ap @ V1f @ V6x ) ) ) ) ) ) )
@ V3x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Efn__seq__integral__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq__integral @ A_27a ) @ V0m ) @ V1f )
= ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] :
( ap @ c_2Eextreal_2ENormal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3k: $i] :
( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ V3k ) ) @ ( 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 ) ) ) ) ) @ V2n ) ) )
@ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ c_2Eextreal_2Eextreal__of__num @ V3k ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) ) @ ( ap @ V1f @ V4x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V1f @ V4x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ c_2Eextreal_2Eextreal__of__num @ V3k ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( 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 ) ) ) ) ) @ V2n ) )
@ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2n ) ) @ ( ap @ V1f @ V5x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__1,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4k ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ V4k ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V1f @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ V4k ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ V4k ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__2,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V1f @ V3x ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__3,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V1f @ V3x ) ) )
| ? [V4k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4k ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ V4k ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V1f @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ V4k ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__4,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__5,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__mono__increasing,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) )
=> ( p
@ ( ap @ c_2Eextreal_2Eext__mono__increasing
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ V3n ) @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__upper__bounded,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3x ) ) @ ( ap @ V1f @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__sup,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ V3n ) @ V2x ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__fn__in__psfis,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq__integral @ A_27a ) @ V0m ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__sequence,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Efn__seq @ A_27a ) @ V0m ) @ V1f ) @ V3i ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Emeasurable__sequence,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( ? [V2fi: $i] :
( ( mem @ V2fi @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
& ? [V3ri: $i] :
( ( mem @ V3ri @ ( arr @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p
@ ( ap @ c_2Eextreal_2Eext__mono__increasing
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5i: $i] : ( ap @ ( ap @ V2fi @ V5i ) @ V4x ) ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ V2fi @ V7i ) @ V6x ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) @ V6x ) ) ) ) )
& ! [V8i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ V3ri @ ( inj__ty_2Enum_2Enum @ V8i ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V8i ) ) ) ) )
& ! [V9i: tp__ty_2Enum_2Enum,V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ V10x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) @ V10x ) ) ) )
& ! [V11i: tp__ty_2Enum_2Enum,V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V2fi @ ( inj__ty_2Enum_2Enum @ V11i ) ) @ V12x ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V13i: $i] : ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ V2fi @ V13i ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) )
& ? [V14gi: $i] :
( ( mem @ V14gi @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
& ? [V15vi: $i] :
( ( mem @ V15vi @ ( arr @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal ) )
& ! [V16x: $i] :
( ( mem @ V16x @ A_27a )
=> ( p
@ ( ap @ c_2Eextreal_2Eext__mono__increasing
@ ( lam @ ty_2Enum_2Enum
@ ^ [V17i: $i] : ( ap @ ( ap @ V14gi @ V17i ) @ V16x ) ) ) ) )
& ! [V18x: $i] :
( ( mem @ V18x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V18x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V19i: $i] : ( ap @ ( ap @ V14gi @ V19i ) @ V18x ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) @ V18x ) ) ) ) )
& ! [V20i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ V15vi @ ( inj__ty_2Enum_2Enum @ V20i ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ ( ap @ V14gi @ ( inj__ty_2Enum_2Enum @ V20i ) ) ) ) )
& ! [V21i: tp__ty_2Enum_2Enum,V22x: $i] :
( ( mem @ V22x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ V14gi @ ( inj__ty_2Enum_2Enum @ V21i ) ) @ V22x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) @ V22x ) ) ) )
& ! [V23i: tp__ty_2Enum_2Enum,V24x: $i] :
( ( mem @ V24x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V14gi @ ( inj__ty_2Enum_2Enum @ V23i ) ) @ V24x ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V25i: $i] : ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ V14gi @ V25i ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__add,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2g @ V3x ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V2g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__sum,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V3i: $i] :
( ( mem @ V3i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3i ) @ V2s ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V1f @ V3i ) @ V4x ) ) ) ) ) )
& ! [V5i: $i] :
( ( mem @ V5i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5i ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V1f @ V5i ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V6x: $i] :
( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V7i: $i] : ( ap @ ( ap @ V1f @ V7i ) @ V6x ) ) )
@ V2s ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V8i: $i] : ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ V1f @ V8i ) ) ) )
@ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__disjoint__sets,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V2s ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V4x ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V5x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2s ) @ V3t ) ) @ V5x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ c_2Eextreal_2Eextreal__add
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V6x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V2s ) @ V6x ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V7x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V7x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V3t ) @ V7x ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__disjoint__sets__sum,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V4i: $i] :
( ( mem @ V4i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4i ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V3a @ V4i ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V5x ) ) ) )
& ! [V6i: $i] :
( ( mem @ V6i @ A_27b )
=> ! [V7j: $i] :
( ( mem @ V7j @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6i ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7j ) @ V2s ) )
& ( V6i != V7j ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V3a @ V6i ) ) @ ( ap @ V3a @ V7j ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V8x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V8x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( arr @ A_27a @ bool ) ) @ V3a ) @ V2s ) ) ) @ V8x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V9i: $i] :
( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V10x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V10x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V3a @ V9i ) ) @ V10x ) ) ) ) ) )
@ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__split,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ c_2Eextreal_2Eextreal__add
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V4x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V2s ) @ V4x ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V5x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ V2s ) ) @ V5x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__cmul__infty,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1s ) @ V2x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__suminf,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V2i: tp__ty_2Enum_2Enum,V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V3x ) ) ) )
& ! [V4i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V5x: $i] :
( ap @ c_2Eextreal_2Eext__suminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6i: $i] : ( ap @ ( ap @ V1f @ V6i ) @ V5x ) ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eext__suminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ ( ap @ V1f @ V7i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__pos__fn,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__pos,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__infty,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V3x ) )
= fo__c_2Eextreal_2EPosInf ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__infty__null,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Emeasure_2Enull__set @ A_27a ) @ V0m )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V2x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__bounded,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__abs @ ( ap @ V2g @ V3x ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V2g ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__fn__plus,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__fn__minus,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__const,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__zero,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ A_27b )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__plus__minus,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__add__pos,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V2g ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2g @ V4x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__add__lemma,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V2g ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) @ V3x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V2g ) @ V3x ) ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) @ V4x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V2g ) @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__add,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V2f2 ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V1f1 @ V3x ) ) @ ( ap @ V2f2 @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__cmul,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__sub,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V2f2 ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( ap @ V1f1 @ V3x ) ) @ ( ap @ V2f2 @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__not__infty,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V3g ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V3g @ V4x ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V3g @ V5x ) )
!= fo__c_2Eextreal_2EPosInf ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V3g ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__not__infty__alt,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V4x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__not__infty__alt2,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V4x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegrable__not__infty__alt3,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V2x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V2x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1s ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__add__lemma,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V2f1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V3f2 ) )
& ( V1f
= ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( ap @ V2f1 @ V4x ) ) @ ( ap @ V3f2 @ V4x ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2f1 @ V5x ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V3f2 @ V6x ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V2f1 ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V3f2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__add,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V2g ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V2g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__cmul,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegrable @ A_27a ) @ V0m ) @ V1f ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__cmul__indicator,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1s ) @ V3x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__zero,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__mspace,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V2x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eintegral__mono,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V3t: $i] :
( ( mem @ V3t @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3t ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V1f1 @ V3t ) ) @ ( ap @ V2f2 @ V3t ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f1 ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V2f2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Efinite__space__integral__reduce,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V2x ) )
!= fo__c_2Eextreal_2ENegInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V2x ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Efinite__space__integral @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Efinite__support__integral__reduce,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V2x ) )
!= fo__c_2Eextreal_2ENegInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V2x ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1f ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Efinite__space__integral @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Efinite__space__POW__integral__reduce,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V2x ) )
!= fo__c_2Eextreal_2ENegInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V2x ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Eintegral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V3x ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Efinite__POW__prod__measure__reduce,axiom,
! [A_27a: del,A_27b: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) ) )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ ( arr @ A_27a @ bool ) )
=> ! [V3a1: $i] :
( ( mem @ V3a1 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2a0 ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3a1 ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b ) @ V0m0 ) @ V1m1 ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V2a0 ) @ V3a1 ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m0 ) @ V2a0 ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27b ) @ V1m1 ) @ V3a1 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW1,axiom,
! [A_27a: del,A_27b: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure__space @ A_27a @ A_27b ) @ V0m0 ) @ V1m1 ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW2,axiom,
! [A_27a: del,A_27b: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b ) @ V0m0 ) @ V1m1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Efinite__prod__measure__space__POW,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s2 ) ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure__space @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V0s1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s1 ) ) @ V2u ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V1s2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ V1s2 ) ) @ V3v ) ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V0s1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s1 ) ) @ V2u ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V1s2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ V1s2 ) ) @ V3v ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Efinite__POW__prod__measure__reduce3,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m1 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27c ) @ V2m2 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27c ) @ V2m2 ) ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27c ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27c ) @ V2m2 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27c ) @ V2m2 ) ) )
=> ! [V3a0: $i] :
( ( mem @ V3a0 @ ( arr @ A_27a @ bool ) )
=> ! [V4a1: $i] :
( ( mem @ V4a1 @ ( arr @ A_27b @ bool ) )
=> ! [V5a2: $i] :
( ( mem @ V5a2 @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3a0 ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V4a1 ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27c @ bool ) ) @ V5a2 ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27c ) @ V2m2 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c ) @ V0m0 ) @ V1m1 ) @ V2m2 ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V3a0 ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ V4a1 ) @ V5a2 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m0 ) @ V3a0 ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27b ) @ V1m1 ) @ V4a1 ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27c ) @ V2m2 ) @ V5a2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW3,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m1 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27c ) @ V2m2 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27c ) @ V2m2 ) ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m1 ) )
& ( ( ap @ ( c_2Epred__set_2EPOW @ A_27c ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27c ) @ V2m2 ) )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27c ) @ V2m2 ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27c ) @ V2m2 ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m1 ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27c ) @ V2m2 ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c ) @ V0m0 ) @ V1m1 ) @ V2m2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Efinite__prod__measure__space__POW3,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ! [V2s3: $i] :
( ( mem @ V2s3 @ ( arr @ A_27c @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) )
=> ! [V5w: $i] :
( ( mem @ V5w @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s2 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c ) @ V2s3 ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure__space3 @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V0s1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s1 ) ) @ V3u ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V1s2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ V1s2 ) ) @ V4v ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V2s3 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27c ) @ V2s3 ) ) @ V5w ) ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V0s1 ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ V1s2 ) @ V2s3 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V0s1 ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ V1s2 ) @ V2s3 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V0s1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s1 ) ) @ V3u ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V1s2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ V1s2 ) ) @ V4v ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27c @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V2s3 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27c ) @ V2s3 ) ) @ V5w ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Eseq__sup__def,axiom,
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Elebesgue_2Eseq__sup @ V0P ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ V1r ) @ V0P ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ c_2Eextreal_2Eextreal__sup @ V0P ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ V1r ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) )
& ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Elebesgue_2Eseq__sup @ V2P ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V4r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ V4r ) @ V2P ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ c_2Eextreal_2Eextreal__sup @ V2P ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ V4r ) @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( 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 @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ ( ap @ c_2Elebesgue_2Eseq__sup @ V2P ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V4r ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V4r ) @ ( ap @ c_2Eextreal_2Eextreal__sup @ V2P ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Eseq__sup__def__compute,axiom,
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Elebesgue_2Eseq__sup @ V0P ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ V1r ) @ V0P ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ c_2Eextreal_2Eextreal__sup @ V0P ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ V1r ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) )
& ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Elebesgue_2Eseq__sup @ V2P ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V4r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ V4r ) @ V2P ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ c_2Eextreal_2Eextreal__sup @ V2P ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ V4r ) @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( 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 @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ ( ap @ c_2Elebesgue_2Eseq__sup @ V2P ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V4r ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V4r ) @ ( ap @ c_2Eextreal_2Eextreal__sup @ V2P ) ) ) ) ) ) ) ) ) )
& ! [V5P: $i] :
( ( mem @ V5P @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ! [V6n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Elebesgue_2Eseq__sup @ V5P ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V7r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ V7r ) @ V5P ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ c_2Eextreal_2Eextreal__sup @ V5P ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ V7r ) @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( 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 @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ ( ap @ c_2Elebesgue_2Eseq__sup @ V5P ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) @ V7r ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V7r ) @ ( ap @ c_2Eextreal_2Eextreal__sup @ V5P ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2EEXTREAL__SUP__SEQ,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ( ( ? [V1x: tp__ty_2Eextreal_2Eextreal] : ( p @ ( ap @ V0P @ ( inj__ty_2Eextreal_2Eextreal @ V1x ) ) )
& ? [V2z: tp__ty_2Eextreal_2Eextreal] :
( ( V2z != fo__c_2Eextreal_2EPosInf )
& ! [V3x: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Eextreal_2Eextreal @ V3x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2z ) ) ) ) ) )
=> ? [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal ) )
& ! [V5n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ V4x @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ V0P ) )
& ! [V6n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V4x @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ V4x @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__sup @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal ) @ V4x ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__sup @ V0P ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2EEXTREAL__SUP__FUN__SEQ__IMAGE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ty_2Eextreal_2Eextreal ) )
=> ( ( ? [V3x: tp__ty_2Eextreal_2Eextreal] : ( p @ ( ap @ V0P @ ( inj__ty_2Eextreal_2Eextreal @ V3x ) ) )
& ? [V4z: tp__ty_2Eextreal_2Eextreal] :
( ( V4z != fo__c_2Eextreal_2EPosInf )
& ! [V5x: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Eextreal_2Eextreal @ V5x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V5x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V4z ) ) ) ) )
& ( V0P
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ty_2Eextreal_2Eextreal ) @ V2f ) @ V1P_27 ) ) )
=> ? [V6g: $i] :
( ( mem @ V6g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
& ! [V7n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V6g @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) @ V1P_27 ) )
& ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8n: $i] : ( ap @ V2f @ ( ap @ V6g @ V8n ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__sup @ V0P ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Emax__fn__seq__def,axiom,
! [A_27a: del] :
( ! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V0g ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ V0g @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1x ) ) ) ) )
& ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V3n: tp__ty_2Enum_2Enum,V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V2g ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__max @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V2g ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V4x ) ) @ ( ap @ ( ap @ V2g @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Emax__fn__seq__def__compute,axiom,
! [A_27a: del] :
( ! [V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V0g ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ V0g @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1x ) ) ) ) )
& ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V3n: tp__ty_2Enum_2Enum,V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V2g ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__max @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V2g ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V4x ) ) @ ( ap @ ( ap @ V2g @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V4x ) ) ) ) ) )
& ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V6n: tp__ty_2Enum_2Enum,V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V5g ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ V7x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__max @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V5g ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ V7x ) ) @ ( ap @ ( ap @ V5g @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ V7x ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Emax__fn__seq__mono,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V0g ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a ) @ V0g ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V2x ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2EEXTREAL__SUP__FUN__SEQ__MONO__IMAGE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ty_2Eextreal_2Eextreal ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ! [V2P_27: $i] :
( ( mem @ V2P_27 @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ bool ) )
=> ( ( ? [V3x: tp__ty_2Eextreal_2Eextreal] : ( p @ ( ap @ V1P @ ( inj__ty_2Eextreal_2Eextreal @ V3x ) ) )
& ? [V4z: tp__ty_2Eextreal_2Eextreal] :
( ( V4z != fo__c_2Eextreal_2EPosInf )
& ! [V5x: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ V1P @ ( inj__ty_2Eextreal_2Eextreal @ V5x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V5x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V4z ) ) ) ) )
& ( V1P
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ty_2Eextreal_2Eextreal ) @ V0f ) @ V2P_27 ) )
& ! [V6g1: $i] :
( ( mem @ V6g1 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V7g2: $i] :
( ( mem @ V7g2 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V6g1 ) @ V2P_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V7g2 ) @ V2P_27 ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V6g1 @ V8x ) ) @ ( ap @ V7g2 @ V8x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V6g1 ) ) @ ( ap @ V0f @ V7g2 ) ) ) ) ) )
& ! [V9g1: $i] :
( ( mem @ V9g1 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V10g2: $i] :
( ( mem @ V10g2 @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V9g1 ) @ V2P_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V10g2 ) @ V2P_27 ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ A_27a
@ ^ [V11x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__max @ ( ap @ V9g1 @ V11x ) ) @ ( ap @ V10g2 @ V11x ) ) ) )
@ V2P_27 ) ) ) ) ) )
=> ? [V12g: $i] :
( ( mem @ V12g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
& ! [V13n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V12g @ ( inj__ty_2Enum_2Enum @ V13n ) ) ) @ V2P_27 ) )
& ! [V14x: $i] :
( ( mem @ V14x @ A_27a )
=> ! [V15n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ V12g @ ( inj__ty_2Enum_2Enum @ V15n ) ) @ V14x ) ) @ ( ap @ ( ap @ V12g @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V15n ) ) ) @ V14x ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V16n: $i] : ( ap @ V0f @ ( ap @ V12g @ V16n ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__sup @ V1P ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Esigned__measure__space__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Elebesgue_2Esigned__measure__space @ A_27a ) @ V0m ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Ecountably__additive @ A_27a ) @ V0m ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Enegative__set__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Enegative__set @ A_27a ) @ V0m ) @ V1A ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s ) @ V1A ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2s ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2ERADON__F__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2ERADON__F @ A_27a ) @ V0m ) @ V1v )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal )
@ ^ [V2f: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ bool ) @ V2f )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2f @ V3x ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V4A: $i] :
( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
@ ( ap
@ ( ap @ c_2Eextreal_2Eextreal__le
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V2f @ V5x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V4A ) @ V5x ) ) ) ) )
@ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ V4A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2ERADON__F__integrals__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elebesgue_2ERADON__F__integrals @ A_27a ) @ V0m ) @ V1v )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V2r: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V2r )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal )
@ ^ [V3f: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ V2r ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V3f ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Elebesgue_2ERADON__F @ A_27a ) @ V0m ) @ V1v ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Emeasure__absolutely__continuous__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Emeasure__absolutely__continuous @ A_27a ) @ V0m ) @ V1v ) )
<=> ! [V2A: $i] :
( ( mem @ V2A @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ V2A ) )
= ( 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_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2A ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__radon__max__in__F,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2m: $i] :
( ( mem @ V2m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V2m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V3v ) )
& ( ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V3v )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V2m ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V3v )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V2m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elebesgue_2ERADON__F @ A_27a ) @ V2m ) @ V3v ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1g ) @ ( ap @ ( ap @ ( c_2Elebesgue_2ERADON__F @ A_27a ) @ V2m ) @ V3v ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__max @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2ERADON__F @ A_27a ) @ V2m ) @ V3v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Elemma__radon__seq__conv__sup,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ A_27a )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V2v ) )
& ( ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V2v )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V2v )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m ) ) )
=> ? [V3f__n: $i] :
( ( mem @ V3f__n @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ty_2Eextreal_2Eextreal ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V3f__n @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2ERADON__F @ A_27b ) @ V1m ) @ V2v ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27b )
=> ! [V6n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ V3f__n @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V5x ) ) @ ( ap @ ( ap @ V3f__n @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ V5x ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27b ) @ V1m ) @ ( ap @ V3f__n @ V7n ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__sup @ ( ap @ ( ap @ ( c_2Elebesgue_2ERADON__F__integrals @ A_27b ) @ V1m ) @ V2v ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2ERN__lemma1,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1v ) )
& ( 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 @ V2e ) ) )
& ( ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1v )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1v )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
=> ? [V3A: $i] :
( ( mem @ V3A @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V3A ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ V3A ) ) ) )
& ! [V4B: $i] :
( ( mem @ V4B @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4B ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V4B ) @ V3A ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V4B ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ V4B ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2ERN__lemma2,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1v ) )
& ( ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1v )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1v )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
=> ? [V2A: $i] :
( ( mem @ V2A @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2A ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ V2A ) ) ) )
& ! [V3B: $i] :
( ( mem @ V3B @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3B ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3B ) @ V2A ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V3B ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ V3B ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2ERadon__Nikodym,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1v ) )
& ( ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1v )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1v )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Emeasure__absolutely__continuous @ A_27a ) @ V1v ) @ V0m ) ) )
=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2f @ V3x ) ) ) )
& ! [V4A: $i] :
( ( mem @ V4A @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V2f @ V5x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V4A ) @ V5x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ V4A ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2ERadon__Nikodym2,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1v ) )
& ( ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1v )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1v )
= ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Elebesgue_2Emeasure__absolutely__continuous @ A_27a ) @ V1v ) @ V0m ) ) )
=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2f @ V3x ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2f @ V4x ) )
!= fo__c_2Eextreal_2EPosInf ) )
& ! [V5A: $i] :
( ( mem @ V5A @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V5A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V2f @ V6x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V5A ) @ V6x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1v ) @ V5A ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------