TPTP Problem File: ITP025^5.p
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%------------------------------------------------------------------------------
% File : ITP025^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Elebesgue_2Epos__fn__integral__cmul.p, 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 : thm_2Elebesgue_2Epos__fn__integral__cmul.p [Gau20]
% : HL412001^5.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 22023 (2762 unt;3326 typ; 0 def)
% Number of atoms : 505697 (19392 equ; 0 cnn)
% Maximal formula atoms : 8851 ( 27 avg)
% Number of connectives : 944572 (2958 ~;1162 |;28763 &;866349 @)
% (6763 <=>;38577 =>; 0 <=; 0 <~>)
% Maximal formula depth : 368 ( 12 avg)
% Number of types : 153 ( 151 usr)
% Number of type conns : 2908 (2908 >; 0 *; 0 +; 0 <<)
% Number of symbols : 3245 (3242 usr;1173 con; 0-6 aty)
% Number of variables : 80934 (5493 ^;61049 !;14392 ?;80934 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
include('Axioms/ITP001/ITP002^5.ax').
include('Axioms/ITP001/ITP003^5.ax').
include('Axioms/ITP001/ITP004^5.ax').
include('Axioms/ITP001/ITP007^5.ax').
include('Axioms/ITP001/ITP006^5.ax').
include('Axioms/ITP001/ITP005^5.ax').
include('Axioms/ITP001/ITP008^5.ax').
include('Axioms/ITP001/ITP009^5.ax').
include('Axioms/ITP001/ITP010^5.ax').
include('Axioms/ITP001/ITP012^5.ax').
include('Axioms/ITP001/ITP011^5.ax').
include('Axioms/ITP001/ITP013^5.ax').
include('Axioms/ITP001/ITP014^5.ax').
include('Axioms/ITP001/ITP015^5.ax').
include('Axioms/ITP001/ITP017^5.ax').
include('Axioms/ITP001/ITP016^5.ax').
include('Axioms/ITP001/ITP019^5.ax').
include('Axioms/ITP001/ITP018^5.ax').
include('Axioms/ITP001/ITP021^5.ax').
include('Axioms/ITP001/ITP022^5.ax').
include('Axioms/ITP001/ITP020^5.ax').
include('Axioms/ITP001/ITP024^5.ax').
include('Axioms/ITP001/ITP023^5.ax').
include('Axioms/ITP001/ITP025^5.ax').
include('Axioms/ITP001/ITP026^5.ax').
include('Axioms/ITP001/ITP027^5.ax').
include('Axioms/ITP001/ITP028^5.ax').
include('Axioms/ITP001/ITP031^5.ax').
include('Axioms/ITP001/ITP029^5.ax').
include('Axioms/ITP001/ITP033^5.ax').
include('Axioms/ITP001/ITP030^5.ax').
include('Axioms/ITP001/ITP032^5.ax').
include('Axioms/ITP001/ITP038^5.ax').
include('Axioms/ITP001/ITP035^5.ax').
include('Axioms/ITP001/ITP034^5.ax').
include('Axioms/ITP001/ITP036^5.ax').
include('Axioms/ITP001/ITP037^5.ax').
include('Axioms/ITP001/ITP039^5.ax').
include('Axioms/ITP001/ITP041^5.ax').
include('Axioms/ITP001/ITP042^5.ax').
include('Axioms/ITP001/ITP040^5.ax').
include('Axioms/ITP001/ITP044^5.ax').
include('Axioms/ITP001/ITP051^5.ax').
include('Axioms/ITP001/ITP045^5.ax').
include('Axioms/ITP001/ITP056^5.ax').
include('Axioms/ITP001/ITP046^5.ax').
include('Axioms/ITP001/ITP043^5.ax').
include('Axioms/ITP001/ITP052^5.ax').
include('Axioms/ITP001/ITP057^5.ax').
include('Axioms/ITP001/ITP048^5.ax').
include('Axioms/ITP001/ITP047^5.ax').
include('Axioms/ITP001/ITP055^5.ax').
include('Axioms/ITP001/ITP053^5.ax').
include('Axioms/ITP001/ITP054^5.ax').
include('Axioms/ITP001/ITP058^5.ax').
include('Axioms/ITP001/ITP049^5.ax').
include('Axioms/ITP001/ITP050^5.ax').
include('Axioms/ITP001/ITP061^5.ax').
include('Axioms/ITP001/ITP069^5.ax').
include('Axioms/ITP001/ITP062^5.ax').
include('Axioms/ITP001/ITP068^5.ax').
include('Axioms/ITP001/ITP078^5.ax').
include('Axioms/ITP001/ITP064^5.ax').
include('Axioms/ITP001/ITP060^5.ax').
include('Axioms/ITP001/ITP067^5.ax').
include('Axioms/ITP001/ITP075^5.ax').
include('Axioms/ITP001/ITP074^5.ax').
include('Axioms/ITP001/ITP063^5.ax').
include('Axioms/ITP001/ITP059^5.ax').
include('Axioms/ITP001/ITP065^5.ax').
include('Axioms/ITP001/ITP076^5.ax').
include('Axioms/ITP001/ITP066^5.ax').
include('Axioms/ITP001/ITP077^5.ax').
include('Axioms/ITP001/ITP070^5.ax').
include('Axioms/ITP001/ITP071^5.ax').
include('Axioms/ITP001/ITP072^5.ax').
include('Axioms/ITP001/ITP073^5.ax').
include('Axioms/ITP001/ITP080^5.ax').
include('Axioms/ITP001/ITP083^5.ax').
include('Axioms/ITP001/ITP088^5.ax').
include('Axioms/ITP001/ITP082^5.ax').
include('Axioms/ITP001/ITP081^5.ax').
include('Axioms/ITP001/ITP079^5.ax').
include('Axioms/ITP001/ITP087^5.ax').
include('Axioms/ITP001/ITP085^5.ax').
include('Axioms/ITP001/ITP086^5.ax').
include('Axioms/ITP001/ITP090^5.ax').
include('Axioms/ITP001/ITP089^5.ax').
include('Axioms/ITP001/ITP084^5.ax').
include('Axioms/ITP001/ITP104^5.ax').
include('Axioms/ITP001/ITP097^5.ax').
include('Axioms/ITP001/ITP094^5.ax').
include('Axioms/ITP001/ITP095^5.ax').
include('Axioms/ITP001/ITP093^5.ax').
include('Axioms/ITP001/ITP102^5.ax').
include('Axioms/ITP001/ITP091^5.ax').
include('Axioms/ITP001/ITP098^5.ax').
include('Axioms/ITP001/ITP096^5.ax').
include('Axioms/ITP001/ITP100^5.ax').
include('Axioms/ITP001/ITP092^5.ax').
include('Axioms/ITP001/ITP105^5.ax').
include('Axioms/ITP001/ITP101^5.ax').
include('Axioms/ITP001/ITP103^5.ax').
include('Axioms/ITP001/ITP099^5.ax').
include('Axioms/ITP001/ITP111^5.ax').
include('Axioms/ITP001/ITP108^5.ax').
include('Axioms/ITP001/ITP106^5.ax').
include('Axioms/ITP001/ITP107^5.ax').
include('Axioms/ITP001/ITP114^5.ax').
include('Axioms/ITP001/ITP110^5.ax').
include('Axioms/ITP001/ITP109^5.ax').
include('Axioms/ITP001/ITP112^5.ax').
include('Axioms/ITP001/ITP113^5.ax').
include('Axioms/ITP001/ITP117^5.ax').
include('Axioms/ITP001/ITP116^5.ax').
include('Axioms/ITP001/ITP118^5.ax').
include('Axioms/ITP001/ITP120^5.ax').
include('Axioms/ITP001/ITP119^5.ax').
include('Axioms/ITP001/ITP121^5.ax').
include('Axioms/ITP001/ITP123^5.ax').
include('Axioms/ITP001/ITP124^5.ax').
include('Axioms/ITP001/ITP126^5.ax').
include('Axioms/ITP001/ITP127^5.ax').
include('Axioms/ITP001/ITP129^5.ax').
include('Axioms/ITP001/ITP128^5.ax').
include('Axioms/ITP001/ITP130^5.ax').
include('Axioms/ITP001/ITP131^5.ax').
include('Axioms/ITP001/ITP132^5.ax').
include('Axioms/ITP001/ITP134^5.ax').
include('Axioms/ITP001/ITP133^5.ax').
include('Axioms/ITP001/ITP135^5.ax').
include('Axioms/ITP001/ITP136^5.ax').
include('Axioms/ITP001/ITP138^5.ax').
include('Axioms/ITP001/ITP137^5.ax').
include('Axioms/ITP001/ITP140^5.ax').
include('Axioms/ITP001/ITP141^5.ax').
include('Axioms/ITP001/ITP139^5.ax').
include('Axioms/ITP001/ITP142^5.ax').
include('Axioms/ITP001/ITP145^5.ax').
include('Axioms/ITP001/ITP144^5.ax').
include('Axioms/ITP001/ITP143^5.ax').
include('Axioms/ITP001/ITP147^5.ax').
include('Axioms/ITP001/ITP146^5.ax').
include('Axioms/ITP001/ITP149^5.ax').
%------------------------------------------------------------------------------
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,conjecture,
! [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 ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------