ITP001 Axioms: ITP148^7.ax
%------------------------------------------------------------------------------
% File : ITP148^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : lebesgue.ax [Gau19]
% : HL4148^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 254 ( 21 unt; 107 typ; 0 def)
% Number of atoms : 1214 ( 166 equ; 30 cnn)
% Maximal formula atoms : 27 ( 4 avg)
% Number of connectives : 6463 ( 30 ~; 2 |; 354 &;5892 @)
% ( 15 <=>; 170 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 13 avg;5892 nst)
% Number of types : 5 ( 4 usr)
% Number of type conns : 2036 (2036 >; 0 *; 0 +; 0 <<)
% Number of symbols : 105 ( 103 usr; 6 con; 0-7 aty)
% Number of variables : 1088 ( 149 ^ 800 !; 62 ?;1088 :)
% ( 77 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Eextreal_2Eextreal,type,
tyop_2Eextreal_2Eextreal: $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Emeasure_2EBorel,type,
c_2Emeasure_2EBorel: tyop_2Epair_2Eprod @ ( tyop_2Eextreal_2Eextreal > $o ) @ ( ( tyop_2Eextreal_2Eextreal > $o ) > $o ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Epred__set_2ECROSS,type,
c_2Epred__set_2ECROSS:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EEXP,type,
c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eextreal_2EEXTREAL__SUM__IMAGE,type,
c_2Eextreal_2EEXTREAL__SUM__IMAGE:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Eextreal_2Eextreal ) > ( A_27a > $o ) > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eextreal_2ENegInf,type,
c_2Eextreal_2ENegInf: tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2ENormal,type,
c_2Eextreal_2ENormal: tyop_2Erealax_2Ereal > tyop_2Eextreal_2Eextreal ).
thf(c_2Epred__set_2EPOW,type,
c_2Epred__set_2EPOW:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EPREIMAGE,type,
c_2Epred__set_2EPREIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27b > $o ) > A_27a > $o ) ).
thf(c_2Eextreal_2EPosInf,type,
c_2Eextreal_2EPosInf: tyop_2Eextreal_2Eextreal ).
thf(c_2Elebesgue_2ERADON__F,type,
c_2Elebesgue_2ERADON__F:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > $o ) ).
thf(c_2Elebesgue_2ERADON__F__integrals,type,
c_2Elebesgue_2ERADON__F__integrals:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > tyop_2Eextreal_2Eextreal > $o ) ).
thf(c_2Ereal__sigma_2EREAL__SUM__IMAGE,type,
c_2Ereal__sigma_2EREAL__SUM__IMAGE:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Erealax_2Ereal ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Epred__set_2Ecount,type,
c_2Epred__set_2Ecount: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emeasure_2Ecountably__additive,type,
c_2Emeasure_2Ecountably__additive:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).
thf(c_2Eextreal_2Eext__mono__increasing,type,
c_2Eextreal_2Eext__mono__increasing: ( tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ) > $o ).
thf(c_2Eextreal_2Eext__suminf,type,
c_2Eextreal_2Eext__suminf: ( tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__abs,type,
c_2Eextreal_2Eextreal__abs: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__add,type,
c_2Eextreal_2Eextreal__add: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__div,type,
c_2Eextreal_2Eextreal__div: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__le,type,
c_2Eextreal_2Eextreal__le: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > $o ).
thf(c_2Eextreal_2Eextreal__lt,type,
c_2Eextreal_2Eextreal__lt: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > $o ).
thf(c_2Eextreal_2Eextreal__max,type,
c_2Eextreal_2Eextreal__max: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__mul,type,
c_2Eextreal_2Eextreal__mul: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__of__num,type,
c_2Eextreal_2Eextreal__of__num: tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__pow,type,
c_2Eextreal_2Eextreal__pow: tyop_2Eextreal_2Eextreal > tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__sub,type,
c_2Eextreal_2Eextreal__sub: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).
thf(c_2Eextreal_2Eextreal__sup,type,
c_2Eextreal_2Eextreal__sup: ( tyop_2Eextreal_2Eextreal > $o ) > tyop_2Eextreal_2Eextreal ).
thf(c_2Elebesgue_2Efinite__space__integral,type,
c_2Elebesgue_2Efinite__space__integral:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Emeasure_2Efn__minus,type,
c_2Emeasure_2Efn__minus:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Eextreal_2Eextreal ) > A_27a > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Emeasure_2Efn__plus,type,
c_2Emeasure_2Efn__plus:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Eextreal_2Eextreal ) > A_27a > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Elebesgue_2Efn__seq,type,
c_2Elebesgue_2Efn__seq:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Elebesgue_2Efn__seq__integral,type,
c_2Elebesgue_2Efn__seq__integral:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Emeasure_2Eindicator__fn,type,
c_2Emeasure_2Eindicator__fn:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Elebesgue_2Eintegrable,type,
c_2Elebesgue_2Eintegrable:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > $o ) ).
thf(c_2Elebesgue_2Eintegral,type,
c_2Elebesgue_2Eintegral:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Emeasure_2Em__space,type,
c_2Emeasure_2Em__space:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > A_27a > $o ) ).
thf(c_2Elebesgue_2Emax__fn__seq,type,
c_2Elebesgue_2Emax__fn__seq:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Emeasure_2Emeasurable,type,
c_2Emeasure_2Emeasurable:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2Emeasure_2Emeasurable__sets,type,
c_2Emeasure_2Emeasurable__sets:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > $o ) > $o ) ).
thf(c_2Emeasure_2Emeasure,type,
c_2Emeasure_2Emeasure:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Elebesgue_2Emeasure__absolutely__continuous,type,
c_2Elebesgue_2Emeasure__absolutely__continuous:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).
thf(c_2Emeasure_2Emeasure__space,type,
c_2Emeasure_2Emeasure__space:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).
thf(c_2Elebesgue_2Enegative__set,type,
c_2Elebesgue_2Enegative__set:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > $o ) > $o ) ).
thf(c_2Emeasure_2Enull__set,type,
c_2Emeasure_2Enull__set:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > $o ) > $o ) ).
thf(c_2Elebesgue_2Epos__fn__integral,type,
c_2Elebesgue_2Epos__fn__integral:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Emeasure_2Epos__simple__fn,type,
c_2Emeasure_2Epos__simple__fn:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > ( tyop_2Enum_2Enum > $o ) > ( tyop_2Enum_2Enum > A_27a > $o ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ) ).
thf(c_2Elebesgue_2Epos__simple__fn__integral,type,
c_2Elebesgue_2Epos__simple__fn__integral:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Enum_2Enum > $o ) > ( tyop_2Enum_2Enum > A_27a > $o ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Elebesgue_2Eprod__measure,type,
c_2Elebesgue_2Eprod__measure:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Elebesgue_2Eprod__measure3,type,
c_2Elebesgue_2Eprod__measure3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Elebesgue_2Eprod__measure__space,type,
c_2Elebesgue_2Eprod__measure__space:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) ) ) ) ).
thf(c_2Elebesgue_2Eprod__measure__space3,type,
c_2Elebesgue_2Eprod__measure__space3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) ) ) ) ).
thf(c_2Eutil__prob_2Eprod__sets,type,
c_2Eutil__prob_2Eprod__sets:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( A_27a > $o ) > $o ) > ( ( A_27b > $o ) > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) ).
thf(c_2Elebesgue_2Eprod__sets3,type,
c_2Elebesgue_2Eprod__sets3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( ( A_27a > $o ) > $o ) > ( ( A_27b > $o ) > $o ) > ( ( A_27c > $o ) > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > $o ) ).
thf(c_2Elebesgue_2Epsfis,type,
c_2Elebesgue_2Epsfis:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal > $o ) ).
thf(c_2Elebesgue_2Epsfs,type,
c_2Elebesgue_2Epsfs:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) ) > $o ) ).
thf(c_2Eextreal_2Ereal,type,
c_2Eextreal_2Ereal: tyop_2Eextreal_2Eextreal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Elebesgue_2Eseq__sup,type,
c_2Elebesgue_2Eseq__sup: ( tyop_2Eextreal_2Eextreal > $o ) > tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ).
thf(c_2Emeasure_2Esigma,type,
c_2Emeasure_2Esigma:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( A_27a > $o ) > $o ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) ) ).
thf(c_2Emeasure_2Esigma__algebra,type,
c_2Emeasure_2Esigma__algebra:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > $o ) ).
thf(c_2Elebesgue_2Esigned__measure__space,type,
c_2Elebesgue_2Esigned__measure__space:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).
thf(c_2Emeasure_2Esubsets,type,
c_2Emeasure_2Esubsets:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: tyop_2Enum_2Enum > $o,V2a: tyop_2Enum_2Enum > A_27a > $o,V3x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V1s @ V2a @ V3x )
= ( c_2Eextreal_2ENormal
@ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V3x @ V4i ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( V2a @ V4i ) ) )
@ V1s ) ) ) ).
thf(thm_2Elebesgue_2Epsfs__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Epsfs @ A_27a @ V0m @ V1f )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) )
@ ( c_2Epair_2EUNCURRY @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) ) @ $o )
@ ^ [V2s: tyop_2Enum_2Enum > $o] :
( c_2Epair_2EUNCURRY @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) ) @ $o )
@ ^ [V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) ) @ $o @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) @ V2s @ ( c_2Epair_2E_2C @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ V3a @ V4x ) ) @ ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f )
= ( c_2Epred__set_2EIMAGE @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) ) @ tyop_2Eextreal_2Eextreal
@ ( c_2Epair_2EUNCURRY @ ( tyop_2Enum_2Enum > $o ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) ) @ tyop_2Eextreal_2Eextreal
@ ^ [V2s: tyop_2Enum_2Enum > $o] :
( c_2Epair_2EUNCURRY @ ( tyop_2Enum_2Enum > A_27a > $o ) @ ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) @ tyop_2Eextreal_2Eextreal
@ ^ [V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] : ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x ) ) )
@ ( c_2Elebesgue_2Epsfs @ A_27a @ V0m @ V1f ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f )
= ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
@ ^ [V2r: tyop_2Eextreal_2Eextreal] :
( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V2r
@ ( c_2Ebool_2E_3F @ ( A_27a > tyop_2Eextreal_2Eextreal )
@ ^ [V3g: A_27a > tyop_2Eextreal_2Eextreal] :
( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V2r @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V3g ) )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( V3g @ V4x ) @ ( V1f @ V4x ) ) ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Eextreal_2Eextreal__sub @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f ) ) @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ( (~)
@ ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f ) )
= c_2Eextreal_2EPosInf ) )
& ( (~)
@ ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f ) )
= c_2Eextreal_2EPosInf ) ) ) ) ).
thf(thm_2Elebesgue_2Efinite__space__integral__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Efinite__space__integral @ A_27a @ V0m @ V1f )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Eextreal_2Eextreal
@ ^ [V2r: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul @ V2r @ ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1f @ ( c_2Epred__set_2EINSERT @ tyop_2Eextreal_2Eextreal @ V2r @ ( c_2Epred__set_2EEMPTY @ tyop_2Eextreal_2Eextreal ) ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1f @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) ).
thf(thm_2Elebesgue_2Eprod__measure__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b @ V0m0 @ V1m1 )
= ( ^ [V2a: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( c_2Eextreal_2Ereal
@ ( c_2Elebesgue_2Eintegral @ A_27a @ V0m0
@ ^ [V3s0: A_27a] :
( c_2Eextreal_2ENormal
@ ( c_2Emeasure_2Emeasure @ A_27b @ V1m1
@ ( c_2Epred__set_2EPREIMAGE @ A_27b @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V4s1: A_27b] : ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3s0 @ V4s1 )
@ V2a ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eprod__measure__space__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Elebesgue_2Eprod__measure__space @ A_27a @ A_27b @ V0m0 @ V1m1 )
= ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) ) @ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Emeasure_2Esigma @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) ) @ ( c_2Eutil__prob_2Eprod__sets @ A_27a @ A_27b @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) ) ) ) @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b @ V0m0 @ V1m1 ) ) ) ) ).
thf(thm_2Elebesgue_2Eprod__sets3__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0a: ( A_27a > $o ) > $o,V1b: ( A_27b > $o ) > $o,V2c: ( A_27c > $o ) > $o] :
( ( c_2Elebesgue_2Eprod__sets3 @ A_27a @ A_27b @ A_27c @ V0a @ V1b @ V2c )
= ( c_2Epred__set_2EGSPEC @ ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) @ ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( A_27c > $o ) ) )
@ ( c_2Epair_2EUNCURRY @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( A_27c > $o ) ) @ ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) @ $o )
@ ^ [V3s: A_27a > $o] :
( c_2Epair_2EUNCURRY @ ( A_27b > $o ) @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) @ $o )
@ ^ [V4t: A_27b > $o,V5u: A_27c > $o] : ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) @ $o @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V3s @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V4t @ V5u ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ V0a ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V4t @ V1b ) @ ( c_2Ebool_2EIN @ ( A_27c > $o ) @ V5u @ V2c ) ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eprod__measure3__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2m2: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c @ V0m0 @ V1m1 @ V2m2 )
= ( c_2Elebesgue_2Eprod__measure @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V0m0 @ ( c_2Elebesgue_2Eprod__measure__space @ A_27b @ A_27c @ V1m1 @ V2m2 ) ) ) ).
thf(thm_2Elebesgue_2Eprod__measure__space3__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2m2: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Elebesgue_2Eprod__measure__space3 @ A_27a @ A_27b @ A_27c @ V0m0 @ V1m1 @ V2m2 )
= ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) @ ( c_2Emeasure_2Em__space @ A_27c @ V2m2 ) ) ) @ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Emeasure_2Esigma @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) @ ( c_2Emeasure_2Em__space @ A_27c @ V2m2 ) ) ) @ ( c_2Elebesgue_2Eprod__sets3 @ A_27a @ A_27b @ A_27c @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27c @ V2m2 ) ) ) ) @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c @ V0m0 @ V1m1 @ V2m2 ) ) ) ) ).
thf(thm_2Elebesgue_2Efn__seq__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f )
= ( ^ [V2n: tyop_2Enum_2Enum,V3x: A_27a] :
( c_2Eextreal_2Eextreal__add
@ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V4k: tyop_2Enum_2Enum] :
( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__of__num @ V4k ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) )
@ ( c_2Emeasure_2Eindicator__fn @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__of__num @ V4k ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) @ ( V1f @ V5x ) ) @ ( c_2Eextreal_2Eextreal__lt @ ( V1f @ V5x ) @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__add @ ( c_2Eextreal_2Eextreal__of__num @ V4k ) @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) ) ) ) ) )
@ V3x ) )
@ ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) )
@ ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n )
@ ( c_2Emeasure_2Eindicator__fn @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V6x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ ( V1f @ V6x ) ) ) ) )
@ V3x ) ) ) ) ) ).
thf(thm_2Elebesgue_2Efn__seq__integral__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Efn__seq__integral @ A_27a @ V0m @ V1f )
= ( ^ [V2n: tyop_2Enum_2Enum] :
( c_2Eextreal_2ENormal
@ ( c_2Erealax_2Ereal__add
@ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V3k: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ V3k ) @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) )
@ ( c_2Emeasure_2Emeasure @ A_27a @ V0m
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__of__num @ V3k ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) @ ( V1f @ V4x ) ) @ ( c_2Eextreal_2Eextreal__lt @ ( V1f @ V4x ) @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__add @ ( c_2Eextreal_2Eextreal__of__num @ V3k ) @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) ) ) ) ) ) ) )
@ ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n )
@ ( c_2Emeasure_2Emeasure @ A_27a @ V0m
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ ( V1f @ V5x ) ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eseq__sup__def,axiom,
( ! [V0P: tyop_2Eextreal_2Eextreal > $o] :
( ( c_2Elebesgue_2Eseq__sup @ V0P @ c_2Enum_2E0 )
= ( c_2Emin_2E_40 @ tyop_2Eextreal_2Eextreal
@ ^ [V1r: tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V1r @ V0P ) @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Eextreal_2Eextreal__sup @ V0P ) @ ( c_2Eextreal_2Eextreal__add @ V1r @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ! [V2P: tyop_2Eextreal_2Eextreal > $o,V3n: tyop_2Enum_2Enum] :
( ( c_2Elebesgue_2Eseq__sup @ V2P @ ( c_2Enum_2ESUC @ V3n ) )
= ( c_2Emin_2E_40 @ tyop_2Eextreal_2Eextreal
@ ^ [V4r: tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V4r @ V2P ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Eextreal_2Eextreal__sup @ V2P ) @ ( c_2Eextreal_2Eextreal__add @ V4r @ ( c_2Eextreal_2ENormal @ ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Enum_2ESUC @ V3n ) ) ) ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Elebesgue_2Eseq__sup @ V2P @ V3n ) @ V4r ) @ ( c_2Eextreal_2Eextreal__lt @ V4r @ ( c_2Eextreal_2Eextreal__sup @ V2P ) ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Emax__fn__seq__def,axiom,
! [A_27a: $tType] :
( ! [V0g: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V1x: A_27a] :
( ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V0g @ c_2Enum_2E0 @ V1x )
= ( V0g @ c_2Enum_2E0 @ V1x ) )
& ! [V2g: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V3n: tyop_2Enum_2Enum,V4x: A_27a] :
( ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V2g @ ( c_2Enum_2ESUC @ V3n ) @ V4x )
= ( c_2Eextreal_2Eextreal__max @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V2g @ V3n @ V4x ) @ ( V2g @ ( c_2Enum_2ESUC @ V3n ) @ V4x ) ) ) ) ).
thf(thm_2Elebesgue_2Esigned__measure__space__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Elebesgue_2Esigned__measure__space @ A_27a @ V0m )
<=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
& ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0m ) ) ) ).
thf(thm_2Elebesgue_2Enegative__set__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1A: A_27a > $o] :
( ( c_2Elebesgue_2Enegative__set @ A_27a @ V0m @ V1A )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ! [V2s: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2s @ V1A ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2s ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Elebesgue_2ERADON__F__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1v: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Elebesgue_2ERADON__F @ A_27a @ V0m @ V1v )
= ( c_2Epred__set_2EGSPEC @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( A_27a > tyop_2Eextreal_2Eextreal )
@ ^ [V2f: A_27a > tyop_2Eextreal_2Eextreal] :
( c_2Epair_2E_2C @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ $o @ V2f
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
@ ( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2f @ V3x ) ) )
@ ( c_2Ebool_2E_21 @ ( A_27a > $o )
@ ^ [V4A: A_27a > $o] :
( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
@ ( c_2Eextreal_2Eextreal__le
@ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V2f @ V5x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V4A @ V5x ) ) )
@ ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ V4A ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2ERADON__F__integrals__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1v: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Elebesgue_2ERADON__F__integrals @ A_27a @ V0m @ V1v )
= ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
@ ^ [V2r: tyop_2Eextreal_2Eextreal] :
( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V2r
@ ( c_2Ebool_2E_3F @ ( A_27a > tyop_2Eextreal_2Eextreal )
@ ^ [V3f: A_27a > tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ V2r @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V3f ) ) @ ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V3f @ ( c_2Elebesgue_2ERADON__F @ A_27a @ V0m @ V1v ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Emeasure__absolutely__continuous__def,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1v: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Elebesgue_2Emeasure__absolutely__continuous @ A_27a @ V0m @ V1v )
<=> ! [V2A: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ V2A )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2A )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__present,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5g: A_27a > tyop_2Eextreal_2Eextreal,V6s_27: tyop_2Enum_2Enum > $o,V7b: tyop_2Enum_2Enum > A_27a > $o,V8y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V5g @ V6s_27 @ V7b @ V8y ) )
=> ? [V9z: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V10z_27: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V11c: tyop_2Enum_2Enum > A_27a > $o,V12k: tyop_2Enum_2Enum > $o] :
( ! [V13t: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V13t @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( V1f @ V13t )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V14i: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ ( V9z @ V14i ) ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V11c @ V14i ) @ V13t ) )
@ V12k ) ) )
& ! [V15t: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V15t @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( V5g @ V15t )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V16i: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ ( V10z_27 @ V16i ) ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V11c @ V16i ) @ V15t ) )
@ V12k ) ) )
& ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x )
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V12k @ V11c @ V9z ) )
& ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V6s_27 @ V7b @ V8y )
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V12k @ V11c @ V10z_27 ) )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V12k )
& ! [V17i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V17i @ V12k )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V9z @ V17i ) ) )
& ! [V18i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V18i @ V12k )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V10z_27 @ V18i ) ) )
& ! [V19i: tyop_2Enum_2Enum,V20j: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V19i @ V12k )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V20j @ V12k )
& ( (~) @ ( V19i = V20j ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V11c @ V19i ) @ ( V11c @ V20j ) ) )
& ! [V21i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V21i @ V12k )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V11c @ V21i ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
& ( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V11c @ V12k ) )
= ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__present,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3a: tyop_2Eextreal_2Eextreal,V4b: tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V3a @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V4b @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V2g ) ) )
=> ? [V5z: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V6z_27: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V7c: tyop_2Enum_2Enum > A_27a > $o,V8k: tyop_2Enum_2Enum > $o] :
( ! [V9t: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V9t @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( V1f @ V9t )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V10i: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ ( V5z @ V10i ) ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V7c @ V10i ) @ V9t ) )
@ V8k ) ) )
& ! [V11t: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V11t @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( V2g @ V11t )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V12i: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ ( V6z_27 @ V12i ) ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V7c @ V12i ) @ V11t ) )
@ V8k ) ) )
& ( V3a
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V8k @ V7c @ V5z ) )
& ( V4b
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V8k @ V7c @ V6z_27 ) )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V8k )
& ! [V13i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V13i @ V8k )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V5z @ V13i ) ) )
& ! [V14i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V14i @ V8k )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V6z_27 @ V14i ) ) )
& ! [V15i: tyop_2Enum_2Enum,V16j: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V15i @ V8k )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V16j @ V8k )
& ( (~) @ ( V15i = V16j ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V7c @ V15i ) @ ( V7c @ V16j ) ) )
& ! [V17i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V17i @ V8k )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V7c @ V17i ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
& ( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V7c @ V8k ) )
= ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__thm1,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5i: tyop_2Enum_2Enum,V6y: A_27a] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5i @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V6y @ ( V3a @ V5i ) ) )
=> ( ( V1f @ V6y )
= ( c_2Eextreal_2ENormal @ ( V4x @ V5i ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__le,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3s: tyop_2Enum_2Enum > $o,V4a: tyop_2Enum_2Enum > A_27a > $o,V5x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V6x_27: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V7i: tyop_2Enum_2Enum] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V3s @ V4a @ V5x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V2g @ V3s @ V4a @ V6x_27 )
& ! [V8x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V8x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( c_2Eextreal_2Eextreal__le @ ( V2g @ V8x ) @ ( V1f @ V8x ) ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V7i @ V3s )
& ( (~)
@ ( ( V4a @ V7i )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2ENormal @ ( V6x_27 @ V7i ) ) @ ( c_2Eextreal_2ENormal @ ( V5x @ V7i ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__cmul,axiom,
! [A_27a: $tType,V0x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1s: tyop_2Enum_2Enum > $o,V2a: tyop_2Enum_2Enum > A_27a > $o,V3m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V4f: A_27a > tyop_2Eextreal_2Eextreal,V5z: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V3m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V3m @ V4f @ V1s @ V2a @ V0x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5z ) )
=> ? [V6s_27: tyop_2Enum_2Enum > $o,V7a_27: tyop_2Enum_2Enum > A_27a > $o,V8x_27: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( c_2Emeasure_2Epos__simple__fn @ A_27a @ V3m
@ ^ [V9t: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V5z ) @ ( V4f @ V9t ) )
@ V6s_27
@ V7a_27
@ V8x_27 ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__cmul__alt,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5z: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5z )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x ) )
=> ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V6t: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V5z ) @ ( V1f @ V6t ) )
@ V2s
@ V3a
@ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ V5z @ ( V4x @ V7i ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__add,axiom,
! [A_27a: $tType,V0x_27: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2s_27: tyop_2Enum_2Enum > $o,V3s: tyop_2Enum_2Enum > $o,V4a_27: tyop_2Enum_2Enum > A_27a > $o,V5a: tyop_2Enum_2Enum > A_27a > $o,V6m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V7f: A_27a > tyop_2Eextreal_2Eextreal,V8g: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V6m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V6m @ V7f @ V3s @ V5a @ V1x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V6m @ V8g @ V2s_27 @ V4a_27 @ V0x_27 ) )
=> ? [V9s_27_27: tyop_2Enum_2Enum > $o,V10a_27_27: tyop_2Enum_2Enum > A_27a > $o,V11x_27_27: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( c_2Emeasure_2Epos__simple__fn @ A_27a @ V6m
@ ^ [V12t: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V7f @ V12t ) @ ( V8g @ V12t ) )
@ V9s_27_27
@ V10a_27_27
@ V11x_27_27 ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__add__alt,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3s: tyop_2Enum_2Enum > $o,V4a: tyop_2Enum_2Enum > A_27a > $o,V5x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V6y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V3s @ V4a @ V5x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V2g @ V3s @ V4a @ V6y ) )
=> ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V7t: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V1f @ V7t ) @ ( V2g @ V7t ) )
@ V3s
@ V4a
@ ^ [V8i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V5x @ V8i ) @ ( V6y @ V8i ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1A: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ? [V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] : ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1A ) @ V2s @ V3a @ V4x ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__indicator__alt,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1s ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2i @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ V1s ) @ V1s )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3i @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__max,axiom,
! [A_27a: $tType,A_27b: $tType,V0s_27: tyop_2Enum_2Enum > $o,V1b: tyop_2Enum_2Enum > A_27a > $o,V2m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V3f: A_27a > tyop_2Eextreal_2Eextreal,V4s: tyop_2Enum_2Enum > $o,V5a: tyop_2Enum_2Enum > A_27a > $o,V6x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V7g: A_27a > tyop_2Eextreal_2Eextreal,V8s_27b: A_27b,V9y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V2m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V2m @ V3f @ V4s @ V5a @ V6x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V2m @ V7g @ V0s_27 @ V1b @ V9y ) )
=> ? [V10s_27_27: tyop_2Enum_2Enum > $o,V11a_27_27: tyop_2Enum_2Enum > A_27a > $o,V12x_27_27: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( c_2Emeasure_2Epos__simple__fn @ A_27a @ V2m
@ ^ [V13x: A_27a] : ( c_2Eextreal_2Eextreal__max @ ( V3f @ V13x ) @ ( V7g @ V13x ) )
@ V10s_27_27
@ V11a_27_27
@ V12x_27_27 ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__not__infty,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
=> ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( (~)
@ ( ( V1f @ V5x )
= c_2Eextreal_2ENegInf ) )
& ( (~)
@ ( ( V1f @ V5x )
= c_2Eextreal_2EPosInf ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__add,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5g: A_27a > tyop_2Eextreal_2Eextreal,V6s_27: tyop_2Enum_2Enum > $o,V7b: tyop_2Enum_2Enum > A_27a > $o,V8y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V5g @ V6s_27 @ V7b @ V8y ) )
=> ? [V9s_27_27: tyop_2Enum_2Enum > $o,V10c: tyop_2Enum_2Enum > A_27a > $o,V11z: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V12x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V1f @ V12x ) @ ( V5g @ V12x ) )
@ V9s_27_27
@ V10c
@ V11z )
& ( ( c_2Eextreal_2Eextreal__add @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x ) @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V6s_27 @ V7b @ V8y ) )
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V9s_27_27 @ V10c @ V11z ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__add__alt,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5g: A_27a > tyop_2Eextreal_2Eextreal,V6y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V5g @ V2s @ V3a @ V6y ) )
=> ( ( c_2Eextreal_2Eextreal__add @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x ) @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V6y ) )
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a
@ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V4x @ V7i ) @ ( V6y @ V7i ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__add,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3a: tyop_2Eextreal_2Eextreal,V4b: tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V3a @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V4b @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V2g ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( c_2Eextreal_2Eextreal__add @ V3a @ V4b )
@ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m
@ ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V1f @ V5x ) @ ( V2g @ V5x ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__mono,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5g: A_27a > tyop_2Eextreal_2Eextreal,V6s_27: tyop_2Enum_2Enum > $o,V7b: tyop_2Enum_2Enum > A_27a > $o,V8y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V5g @ V6s_27 @ V7b @ V8y )
& ! [V9x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V9x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( c_2Eextreal_2Eextreal__le @ ( V1f @ V9x ) @ ( V5g @ V9x ) ) ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x ) @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V6s_27 @ V7b @ V8y ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__mono,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3a: tyop_2Eextreal_2Eextreal,V4b: tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V3a @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V4b @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V2g ) )
& ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( c_2Eextreal_2Eextreal__le @ ( V1f @ V5x ) @ ( V2g @ V5x ) ) ) )
=> ( c_2Eextreal_2Eextreal__le @ V3a @ V4b ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__unique,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5s_27: tyop_2Enum_2Enum > $o,V6b: tyop_2Enum_2Enum > A_27a > $o,V7y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V5s_27 @ V6b @ V7y ) )
=> ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x )
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V5s_27 @ V6b @ V7y ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__unique,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2a: tyop_2Eextreal_2Eextreal,V3b: tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V2a @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V3b @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) ) )
=> ( V2a = V3b ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1A: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ? [V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1A ) @ V2s @ V3a @ V4x )
& ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x )
= ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1A ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1A: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1A ) ) @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1A ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__cmul,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5z: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5z ) )
=> ( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V6x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V5z ) @ ( V1f @ V6x ) )
@ V2s
@ V3a
@ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ V5z @ ( V4x @ V7i ) ) )
& ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a
@ ^ [V8i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ V5z @ ( V4x @ V8i ) ) )
= ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V5z ) @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__cmul,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2a: tyop_2Eextreal_2Eextreal,V3z: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V2a @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3z ) )
=> ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V3z ) @ V2a )
@ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m
@ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V3z ) @ ( V1f @ V4x ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__cmul__alt,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5z: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5z )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x ) )
=> ? [V6s_27: tyop_2Enum_2Enum > $o,V7a_27: tyop_2Enum_2Enum > A_27a > $o,V8x_27: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V9t: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V5z ) @ ( V1f @ V9t ) )
@ V6s_27
@ V7a_27
@ V8x_27 )
& ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V6s_27 @ V7a_27 @ V8x_27 )
= ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V5z ) @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x ) ) ) ) ) ).
thf(thm_2Elebesgue_2EIN__psfis,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1r: tyop_2Eextreal_2Eextreal,V2f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V1r @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V2f ) )
=> ? [V3s: tyop_2Enum_2Enum > $o,V4a: tyop_2Enum_2Enum > A_27a > $o,V5x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V2f @ V3s @ V4a @ V5x )
& ( V1r
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V3s @ V4a @ V5x ) ) ) ) ).
thf(thm_2Elebesgue_2EIN__psfis__eq,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1r: tyop_2Eextreal_2Eextreal,V2f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V1r @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V2f ) )
<=> ? [V3s: tyop_2Enum_2Enum > $o,V4a: tyop_2Enum_2Enum > A_27a > $o,V5x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V2f @ V3s @ V4a @ V5x )
& ( V1r
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V3s @ V4a @ V5x ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__pos,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2a: tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V2a @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) ) )
=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__zero,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: tyop_2Enum_2Enum > $o,V2a: tyop_2Enum_2Enum > A_27a > $o,V3x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V4t: A_27a] : ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 )
@ V1s
@ V2a
@ V3x ) )
=> ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V1s @ V2a @ V3x )
= ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__zero__alt,axiom,
! [A_27a: $tType,V0g: A_27a > tyop_2Eextreal_2Eextreal,V1m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V1m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V1m @ V0g @ V2s @ V3a @ V4x )
& ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
=> ( ( V0g @ V5x )
= ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V1m @ V2s @ V3a @ V4x )
= ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__zero,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1a: tyop_2Eextreal_2Eextreal] :
( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
=> ( ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V1a
@ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m
@ ^ [V2x: A_27a] : ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) )
<=> ( V1a
= ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__not__infty,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: tyop_2Enum_2Enum > $o,V2a: tyop_2Enum_2Enum > A_27a > $o,V3x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( (~)
@ ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V1s @ V2a @ V3x )
= c_2Eextreal_2ENegInf ) )
& ( (~)
@ ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V1s @ V2a @ V3x )
= c_2Eextreal_2EPosInf ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__not__infty,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2a: tyop_2Eextreal_2Eextreal] :
( ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V2a @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) )
=> ( ( (~) @ ( V2a = c_2Eextreal_2ENegInf ) )
& ( (~) @ ( V2a = c_2Eextreal_2EPosInf ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__sum,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27b > A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: A_27b > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5P: A_27b > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V6i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V6i @ V5P )
=> ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ ( V1f @ V6i ) @ V2s @ V3a @ ( V4x @ V6i ) ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V5P )
& ( (~)
@ ( V5P
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
=> ( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V7t: A_27a] :
( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V8i: A_27b] : ( V1f @ V8i @ V7t )
@ V5P )
@ V2s
@ V3a
@ ^ [V9i: tyop_2Enum_2Enum] :
( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b
@ ^ [V10j: A_27b] : ( V4x @ V10j @ V9i )
@ V5P ) )
& ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a
@ ^ [V11j: tyop_2Enum_2Enum] :
( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b
@ ^ [V12i: A_27b] : ( V4x @ V12i @ V11j )
@ V5P ) )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V13i: A_27b] : ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ ( V4x @ V13i ) )
@ V5P ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__sum__alt,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27b > A_27a > tyop_2Eextreal_2Eextreal,V2s: A_27b > tyop_2Enum_2Enum > $o,V3a: A_27b > tyop_2Enum_2Enum > A_27a > $o,V4x: A_27b > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5P: A_27b > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V6i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V6i @ V5P )
=> ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ ( V1f @ V6i ) @ ( V2s @ V6i ) @ ( V3a @ V6i ) @ ( V4x @ V6i ) ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V5P )
& ( (~)
@ ( V5P
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
=> ? [V7c: tyop_2Enum_2Enum > A_27a > $o,V8k: tyop_2Enum_2Enum > $o,V9z: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V10t: A_27a] :
( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V11i: A_27b] : ( V1f @ V11i @ V10t )
@ V5P )
@ V8k
@ V7c
@ V9z )
& ( ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V8k @ V7c @ V9z )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V12i: A_27b] : ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ ( V2s @ V12i ) @ ( V3a @ V12i ) @ ( V4x @ V12i ) )
@ V5P ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__sum,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27b > A_27a > tyop_2Eextreal_2Eextreal,V2a: A_27b > tyop_2Eextreal_2Eextreal,V3P: A_27b > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V4i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4i @ V3P )
=> ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( V2a @ V4i ) @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ ( V1f @ V4i ) ) ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V3P ) )
=> ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b @ V2a @ V3P )
@ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m
@ ^ [V5t: A_27a] :
( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V6i: A_27b] : ( V1f @ V6i @ V5t )
@ V3P ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__intro,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1a: A_27b > A_27a > $o,V2x: A_27b > tyop_2Erealax_2Ereal,V3P: A_27b > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V4i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4i @ V3P )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V1a @ V4i ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
& ! [V5i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5i @ V3P )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V2x @ V5i ) ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V3P ) )
=> ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal
@ ( c_2Eextreal_2ENormal
@ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b
@ ^ [V6i: A_27b] : ( c_2Erealax_2Ereal__mul @ ( V2x @ V6i ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( V1a @ V6i ) ) )
@ V3P ) )
@ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m
@ ^ [V7t: A_27a] :
( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V8i: A_27b] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ ( V2x @ V8i ) ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V1a @ V8i ) @ V7t ) )
@ V3P ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__simple__fn__integral__sub,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V5g: A_27a > tyop_2Eextreal_2Eextreal,V6s_27: tyop_2Enum_2Enum > $o,V7b: tyop_2Enum_2Enum > A_27a > $o,V8y: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V9x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( V5g @ V9x ) @ ( V1f @ V9x ) )
& ! [V10x: A_27a] :
( (~)
@ ( ( V5g @ V10x )
= c_2Eextreal_2EPosInf ) )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V5g @ V6s_27 @ V7b @ V8y ) )
=> ? [V11s_27_27: tyop_2Enum_2Enum > $o,V12c: tyop_2Enum_2Enum > A_27a > $o,V13z: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m
@ ^ [V14x: A_27a] : ( c_2Eextreal_2Eextreal__sub @ ( V1f @ V14x ) @ ( V5g @ V14x ) )
@ V11s_27_27
@ V12c
@ V13z )
& ( ( c_2Eextreal_2Eextreal__sub @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x ) @ ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V6s_27 @ V7b @ V8y ) )
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V11s_27_27 @ V12c @ V13z ) ) ) ) ).
thf(thm_2Elebesgue_2Epsfis__sub,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3a: tyop_2Eextreal_2Eextreal,V4b: tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V5x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( V2g @ V5x ) @ ( V1f @ V5x ) )
& ! [V6x: A_27a] :
( (~)
@ ( ( V2g @ V6x )
= c_2Eextreal_2EPosInf ) )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V3a @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V1f ) )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V4b @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V2g ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( c_2Eextreal_2Eextreal__sub @ V3a @ V4b )
@ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m
@ ^ [V7x: A_27a] : ( c_2Eextreal_2Eextreal__sub @ ( V1f @ V7x ) @ ( V2g @ V7x ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__pos__simple__fn,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Epos__simple__fn__integral @ A_27a @ V0m @ V2s @ V3a @ V4x ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__mspace,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V3x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ V3x ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__zero,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V1x: A_27a] : ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) )
= ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__mono,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
( ! [V3x: A_27a] :
( ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) )
& ( c_2Eextreal_2Eextreal__le @ ( V1f @ V3x ) @ ( V2g @ V3x ) ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f ) @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V2g ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__mono__mspace,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( c_2Eextreal_2Eextreal__le @ ( V2g @ V3x ) @ ( V1f @ V3x ) ) )
& ! [V4x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V4x ) )
& ! [V5x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2g @ V5x ) ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V2g ) @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__pos,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__cmul,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2c ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V2c ) @ ( V1f @ V4x ) ) )
= ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V2c ) @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1s ) )
= ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__cmul__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2c ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V2c ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1s @ V3x ) ) )
= ( c_2Eextreal_2ENormal @ ( c_2Erealax_2Ereal__mul @ V2c @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__sum__cmul__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: tyop_2Enum_2Enum > $o,V2a: tyop_2Enum_2Enum > A_27a > $o,V3x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s )
& ! [V4i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4i @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V3x @ V4i ) ) )
& ! [V5i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5i @ V1s )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V2a @ V5i ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V6t: A_27a] :
( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ ( V3x @ V7i ) ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V2a @ V7i ) @ V6t ) )
@ V1s ) )
= ( c_2Eextreal_2ENormal
@ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ tyop_2Enum_2Enum
@ ^ [V8i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V3x @ V8i ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( V2a @ V8i ) ) )
@ V1s ) ) ) ) ).
thf(thm_2Elebesgue_2Elebesgue__monotone__convergence__lemma,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2fi: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V3g: A_27a > tyop_2Eextreal_2Eextreal,V4r_27: tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V5i: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V2fi @ V5i ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V6x: A_27a] :
( c_2Eextreal_2Eext__mono__increasing
@ ^ [V7i: tyop_2Enum_2Enum] : ( V2fi @ V7i @ V6x ) )
& ! [V8x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V8x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V9i: tyop_2Enum_2Enum] : ( V2fi @ V9i @ V8x )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( V1f @ V8x ) ) )
& ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V4r_27 @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ V3g ) )
& ! [V10x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( V3g @ V10x ) @ ( V1f @ V10x ) )
& ! [V11i: tyop_2Enum_2Enum,V12x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2fi @ V11i @ V12x ) ) )
=> ( c_2Eextreal_2Eextreal__le @ V4r_27
@ ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V13i: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( V2fi @ V13i ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Elebesgue_2Elebesgue__monotone__convergence,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2fi: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V3i: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V2fi @ V3i ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V4i: tyop_2Enum_2Enum,V5x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2fi @ V4i @ V5x ) )
& ! [V6x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V6x ) )
& ! [V7x: A_27a] :
( c_2Eextreal_2Eext__mono__increasing
@ ^ [V8i: tyop_2Enum_2Enum] : ( V2fi @ V8i @ V7x ) )
& ! [V9x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V9x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V10i: tyop_2Enum_2Enum] : ( V2fi @ V10i @ V9x )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( V1f @ V9x ) ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f )
= ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V11i: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( V2fi @ V11i ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Elebesgue_2Elebesgue__monotone__convergence__subset,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2fi: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V3A: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V4i: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V2fi @ V4i ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V5i: tyop_2Enum_2Enum,V6x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2fi @ V5i @ V6x ) )
& ! [V7x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V7x ) )
& ! [V8x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V8x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V9i: tyop_2Enum_2Enum] : ( V2fi @ V9i @ V8x )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( V1f @ V8x ) ) )
& ! [V10x: A_27a] :
( c_2Eextreal_2Eext__mono__increasing
@ ^ [V11i: tyop_2Enum_2Enum] : ( V2fi @ V11i @ V10x ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V12x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V12x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V3A @ V12x ) ) )
= ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V13i: tyop_2Enum_2Enum] :
( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V14x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V2fi @ V13i @ V14x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V3A @ V14x ) ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__1,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2n: tyop_2Enum_2Enum,V3x: A_27a,V4k: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4k @ ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) )
& ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__of__num @ V4k ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) @ ( V1f @ V3x ) )
& ( c_2Eextreal_2Eextreal__lt @ ( V1f @ V3x ) @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__add @ ( c_2Eextreal_2Eextreal__of__num @ V4k ) @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) ) )
=> ( ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V2n @ V3x )
= ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__of__num @ V4k ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__2,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2n: tyop_2Enum_2Enum,V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ ( V1f @ V3x ) ) )
=> ( ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V2n @ V3x )
= ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__3,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2n: tyop_2Enum_2Enum,V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) ) )
=> ( ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ ( V1f @ V3x ) )
| ? [V4k: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4k @ ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) )
& ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__of__num @ V4k ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) @ ( V1f @ V3x ) )
& ( c_2Eextreal_2Eextreal__lt @ ( V1f @ V3x ) @ ( c_2Eextreal_2Eextreal__div @ ( c_2Eextreal_2Eextreal__add @ ( c_2Eextreal_2Eextreal__of__num @ V4k ) @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eextreal_2Eextreal__pow @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__4,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2n: tyop_2Enum_2Enum,V3x: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) )
=> ( ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V2n @ V3x )
= ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__5,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2n: tyop_2Enum_2Enum,V3x: A_27a] :
( ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V2n @ V3x ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__mono__increasing,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2x: A_27a] :
( ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) )
=> ( c_2Eextreal_2Eext__mono__increasing
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V3n @ V2x ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__upper__bounded,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2n: tyop_2Enum_2Enum,V3x: A_27a] :
( ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V2n @ V3x ) @ ( V1f @ V3x ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__sup,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V3n @ V2x )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( V1f @ V2x ) ) ) ).
thf(thm_2Elebesgue_2Elemma__fn__in__psfis,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2n: tyop_2Enum_2Enum] :
( ( ! [V3x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) )
=> ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( c_2Elebesgue_2Efn__seq__integral @ A_27a @ V0m @ V1f @ V2n ) @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V2n ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__sequence,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ! [V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f )
= ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( c_2Elebesgue_2Efn__seq @ A_27a @ V0m @ V1f @ V3i ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Elebesgue_2Emeasurable__sequence,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) )
=> ( ? [V2fi: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V3ri: tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal] :
( ! [V4x: A_27a] :
( c_2Eextreal_2Eext__mono__increasing
@ ^ [V5i: tyop_2Enum_2Enum] : ( V2fi @ V5i @ V4x ) )
& ! [V6x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V6x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V7i: tyop_2Enum_2Enum] : ( V2fi @ V7i @ V6x )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( c_2Emeasure_2Efn__plus @ A_27a @ V1f @ V6x ) ) )
& ! [V8i: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( V3ri @ V8i ) @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ ( V2fi @ V8i ) ) )
& ! [V9i: tyop_2Enum_2Enum,V10x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( V2fi @ V9i @ V10x ) @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f @ V10x ) )
& ! [V11i: tyop_2Enum_2Enum,V12x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2fi @ V11i @ V12x ) )
& ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f ) )
= ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V13i: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( V2fi @ V13i ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
& ? [V14gi: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V15vi: tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal] :
( ! [V16x: A_27a] :
( c_2Eextreal_2Eext__mono__increasing
@ ^ [V17i: tyop_2Enum_2Enum] : ( V14gi @ V17i @ V16x ) )
& ! [V18x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V18x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V19i: tyop_2Enum_2Enum] : ( V14gi @ V19i @ V18x )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( c_2Emeasure_2Efn__minus @ A_27a @ V1f @ V18x ) ) )
& ! [V20i: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( V15vi @ V20i ) @ ( c_2Elebesgue_2Epsfis @ A_27a @ V0m @ ( V14gi @ V20i ) ) )
& ! [V21i: tyop_2Enum_2Enum,V22x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( V14gi @ V21i @ V22x ) @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f @ V22x ) )
& ! [V23i: tyop_2Enum_2Enum,V24x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V14gi @ V23i @ V24x ) )
& ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f ) )
= ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V25i: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( V14gi @ V25i ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__add,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V3x: A_27a] :
( ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) )
& ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2g @ V3x ) ) )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V1f @ V4x ) @ ( V2g @ V4x ) ) )
= ( c_2Eextreal_2Eextreal__add @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f ) @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V2g ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__sum,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27b > A_27a > tyop_2Eextreal_2Eextreal,V2s: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V3i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3i @ V2s )
=> ! [V4x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3i @ V4x ) ) )
& ! [V5i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5i @ V2s )
=> ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V1f @ V5i ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V6x: A_27a] :
( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V7i: A_27b] : ( V1f @ V7i @ V6x )
@ V2s ) )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V8i: A_27b] : ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( V1f @ V8i ) )
@ V2s ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__disjoint__sets,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V2s @ V3t )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V4x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V4x ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V5x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V2s @ V3t ) @ V5x ) ) )
= ( c_2Eextreal_2Eextreal__add
@ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V6x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V6x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V2s @ V6x ) ) )
@ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V7x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V7x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V3t @ V7x ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__disjoint__sets__sum,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: A_27b > $o,V3a: A_27b > A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V4i: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4i @ V2s )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V3a @ V4i ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
& ! [V5x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V5x ) )
& ! [V6i: A_27b,V7j: A_27b] :
( ( ( c_2Ebool_2EIN @ A_27b @ V6i @ V2s )
& ( c_2Ebool_2EIN @ A_27b @ V7j @ V2s )
& ( (~) @ ( V6i = V7j ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V3a @ V6i ) @ ( V3a @ V7j ) ) )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V8x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V8x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( A_27a > $o ) @ V3a @ V2s ) ) @ V8x ) ) )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
@ ^ [V9i: A_27b] :
( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V10x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V10x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V3a @ V9i ) @ V10x ) ) )
@ V2s ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__split,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ! [V3x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f )
= ( c_2Eextreal_2Eextreal__add
@ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V4x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V2s @ V4x ) ) )
@ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V5x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ V2s ) @ V5x ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__cmul__infty,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V2x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ c_2Eextreal_2EPosInf @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1s @ V2x ) ) )
= ( c_2Eextreal_2Eextreal__mul @ c_2Eextreal_2EPosInf @ ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ) ) ).
thf(thm_2Elebesgue_2Epos__fn__integral__suminf,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V2i: tyop_2Enum_2Enum,V3x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2i @ V3x ) )
& ! [V4i: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V1f @ V4i ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V5x: A_27a] :
( c_2Eextreal_2Eext__suminf
@ ^ [V6i: tyop_2Enum_2Enum] : ( V1f @ V6i @ V5x ) ) )
= ( c_2Eextreal_2Eext__suminf
@ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ ( V1f @ V7i ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__pos__fn,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__pos,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ( ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ( (~)
@ ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f )
= c_2Eextreal_2EPosInf ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__infty,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( V1f @ V3x )
= c_2Eextreal_2EPosInf ) ) )
=> ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2s )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__infty__null,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f ) )
=> ( c_2Emeasure_2Enull__set @ A_27a @ V0m
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V2x ) @ c_2Eextreal_2EPosInf ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__bounded,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V3x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__abs @ ( V2g @ V3x ) ) @ ( V1f @ V3x ) ) )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V2g ) ) ).
thf(thm_2Elebesgue_2Eintegrable__fn__plus,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f ) )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__fn__minus,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f ) )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__const,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1c: tyop_2Erealax_2Ereal] :
( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V2x: A_27a] : ( c_2Eextreal_2ENormal @ V1c ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__zero,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1c: A_27b] :
( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V2x: A_27a] : ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__plus__minus,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
=> ( ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f ) )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__add__pos,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V2g )
& ! [V3x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) )
& ! [V4x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2g @ V4x ) ) )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V1f @ V5x ) @ ( V2g @ V5x ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__add__lemma,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V2g ) )
=> ( ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f @ V3x ) @ ( c_2Emeasure_2Efn__plus @ A_27a @ V2g @ V3x ) ) )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f @ V4x ) @ ( c_2Emeasure_2Efn__minus @ A_27a @ V2g @ V4x ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__add,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f1: A_27a > tyop_2Eextreal_2Eextreal,V2f2: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f1 )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V2f2 ) )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V1f1 @ V3x ) @ ( V2f2 @ V3x ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__cmul,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f ) )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V2c ) @ ( V1f @ V3x ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__sub,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f1: A_27a > tyop_2Eextreal_2Eextreal,V2f2: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f1 )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V2f2 ) )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__sub @ ( V1f1 @ V3x ) @ ( V2f2 @ V3x ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1s ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__not__infty,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ! [V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ? [V3g: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V3g )
& ! [V4x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V3g @ V4x ) )
& ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( (~)
@ ( ( V3g @ V5x )
= c_2Eextreal_2EPosInf ) ) )
& ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V3g ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__not__infty__alt,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ! [V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ( ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V3x ) @ c_2Eextreal_2EPosInf ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) ) )
& ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Eintegral @ A_27a @ V0m
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V4x ) @ c_2Eextreal_2EPosInf ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V4x ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__not__infty__alt2,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ! [V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ( ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V3x ) @ c_2Eextreal_2EPosInf ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) ) )
& ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V4x ) @ c_2Eextreal_2EPosInf ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V4x ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegrable__not__infty__alt3,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f ) )
=> ( ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m
@ ^ [V2x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V2x ) @ c_2Eextreal_2ENegInf ) @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V2x ) @ c_2Eextreal_2EPosInf ) ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
& ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Eintegral @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V3x ) @ c_2Eextreal_2ENegInf ) @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V1f @ V3x ) @ c_2Eextreal_2EPosInf ) ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V3x ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1s ) )
= ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__add__lemma,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2f1: A_27a > tyop_2Eextreal_2Eextreal,V3f2: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V2f1 )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V3f2 )
& ( V1f
= ( ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__sub @ ( V2f1 @ V4x ) @ ( V3f2 @ V4x ) ) ) )
& ! [V5x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2f1 @ V5x ) )
& ! [V6x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V3f2 @ V6x ) ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Eextreal_2Eextreal__sub @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V2f1 ) @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m @ V3f2 ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__add,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V2g ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V1f @ V3x ) @ ( V2g @ V3x ) ) )
= ( c_2Eextreal_2Eextreal__add @ ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f ) @ ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V2g ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__cmul,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Elebesgue_2Eintegrable @ A_27a @ V0m @ V1f ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V2c ) @ ( V1f @ V3x ) ) )
= ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V2c ) @ ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__cmul__indicator,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2c: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V2c ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1s @ V3x ) ) )
= ( c_2Eextreal_2ENormal @ ( c_2Erealax_2Ereal__mul @ V2c @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__zero,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m
@ ^ [V1x: A_27a] : ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) )
= ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__mspace,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Eintegral @ A_27a @ V0m
@ ^ [V2x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V2x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ V2x ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eintegral__mono,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f1: A_27a > tyop_2Eextreal_2Eextreal,V2f2: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ! [V3t: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3t @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( c_2Eextreal_2Eextreal__le @ ( V1f1 @ V3t ) @ ( V2f2 @ V3t ) ) ) )
=> ( c_2Eextreal_2Eextreal__le @ ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f1 ) @ ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V2f2 ) ) ) ).
thf(thm_2Elebesgue_2Efinite__space__integral__reduce,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( (~)
@ ( ( V1f @ V2x )
= c_2Eextreal_2ENegInf ) )
& ( (~)
@ ( ( V1f @ V2x )
= c_2Eextreal_2EPosInf ) ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Efinite__space__integral @ A_27a @ V0m @ V1f ) ) ) ).
thf(thm_2Elebesgue_2Efinite__support__integral__reduce,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( (~)
@ ( ( V1f @ V2x )
= c_2Eextreal_2ENegInf ) )
& ( (~)
@ ( ( V1f @ V2x )
= c_2Eextreal_2EPosInf ) ) ) )
& ( c_2Epred__set_2EFINITE @ tyop_2Eextreal_2Eextreal @ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1f @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Elebesgue_2Efinite__space__integral @ A_27a @ V0m @ V1f ) ) ) ).
thf(thm_2Elebesgue_2Efinite__space__POW__integral__reduce,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
=> ( ( (~)
@ ( ( V1f @ V2x )
= c_2Eextreal_2ENegInf ) )
& ( (~)
@ ( ( V1f @ V2x )
= c_2Eextreal_2EPosInf ) ) ) ) )
=> ( ( c_2Elebesgue_2Eintegral @ A_27a @ V0m @ V1f )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V3x ) @ ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EINSERT @ A_27a @ V3x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) )
@ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) ).
thf(thm_2Elebesgue_2Efinite__POW__prod__measure__reduce,axiom,
! [A_27a: $tType,A_27b: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m0 )
& ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m1 )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
& ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) )
& ( ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) ) )
=> ! [V2a0: A_27a > $o,V3a1: A_27b > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2a0 @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) )
& ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3a1 @ ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) ) )
=> ( ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b @ V0m0 @ V1m1 @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V2a0 @ V3a1 ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m0 @ V2a0 ) @ ( c_2Emeasure_2Emeasure @ A_27b @ V1m1 @ V3a1 ) ) ) ) ) ).
thf(thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW1,axiom,
! [A_27a: $tType,A_27b: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m0 )
& ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m1 )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
& ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) )
& ( ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) ) )
=> ( c_2Emeasure_2Emeasure__space @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Elebesgue_2Eprod__measure__space @ A_27a @ A_27b @ V0m0 @ V1m1 ) ) ) ).
thf(thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW2,axiom,
! [A_27a: $tType,A_27b: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m0 )
& ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m1 )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
& ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) )
& ( ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) ) )
=> ( c_2Emeasure_2Emeasure__space @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) ) @ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) ) ) @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b @ V0m0 @ V1m1 ) ) ) ) ) ).
thf(thm_2Elebesgue_2Efinite__prod__measure__space__POW,axiom,
! [A_27a: $tType,A_27b: $tType,V0s1: A_27a > $o,V1s2: A_27b > $o,V2u: ( A_27a > $o ) > tyop_2Erealax_2Ereal,V3v: ( A_27b > $o ) > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s1 )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1s2 ) )
=> ( ( c_2Elebesgue_2Eprod__measure__space @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0s1 @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s1 ) @ V2u ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ V1s2 @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27b @ V1s2 ) @ V3v ) ) )
= ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s1 @ V1s2 ) @ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s1 @ V1s2 ) ) @ ( c_2Elebesgue_2Eprod__measure @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0s1 @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s1 ) @ V2u ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ V1s2 @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27b @ V1s2 ) @ V3v ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Efinite__POW__prod__measure__reduce3,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2m2: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m0 )
& ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m1 )
& ( c_2Emeasure_2Emeasure__space @ A_27c @ V2m2 )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
& ( c_2Epred__set_2EFINITE @ A_27c @ ( c_2Emeasure_2Em__space @ A_27c @ V2m2 ) )
& ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) )
& ( ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) )
& ( ( c_2Epred__set_2EPOW @ A_27c @ ( c_2Emeasure_2Em__space @ A_27c @ V2m2 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27c @ V2m2 ) ) )
=> ! [V3a0: A_27a > $o,V4a1: A_27b > $o,V5a2: A_27c > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3a0 @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) )
& ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V4a1 @ ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) )
& ( c_2Ebool_2EIN @ ( A_27c > $o ) @ V5a2 @ ( c_2Emeasure_2Emeasurable__sets @ A_27c @ V2m2 ) ) )
=> ( ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c @ V0m0 @ V1m1 @ V2m2 @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V3a0 @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V4a1 @ V5a2 ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__mul @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m0 @ V3a0 ) @ ( c_2Emeasure_2Emeasure @ A_27b @ V1m1 @ V4a1 ) ) @ ( c_2Emeasure_2Emeasure @ A_27c @ V2m2 @ V5a2 ) ) ) ) ) ).
thf(thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW3,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2m2: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m0 )
& ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m1 )
& ( c_2Emeasure_2Emeasure__space @ A_27c @ V2m2 )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
& ( c_2Epred__set_2EFINITE @ A_27c @ ( c_2Emeasure_2Em__space @ A_27c @ V2m2 ) )
& ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) )
& ( ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m1 ) )
& ( ( c_2Epred__set_2EPOW @ A_27c @ ( c_2Emeasure_2Em__space @ A_27c @ V2m2 ) )
= ( c_2Emeasure_2Emeasurable__sets @ A_27c @ V2m2 ) ) )
=> ( c_2Emeasure_2Emeasure__space @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) @ ( c_2Emeasure_2Em__space @ A_27c @ V2m2 ) ) ) @ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Emeasure_2Em__space @ A_27b @ V1m1 ) @ ( c_2Emeasure_2Em__space @ A_27c @ V2m2 ) ) ) ) @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c @ V0m0 @ V1m1 @ V2m2 ) ) ) ) ) ).
thf(thm_2Elebesgue_2Efinite__prod__measure__space__POW3,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s1: A_27a > $o,V1s2: A_27b > $o,V2s3: A_27c > $o,V3u: ( A_27a > $o ) > tyop_2Erealax_2Ereal,V4v: ( A_27b > $o ) > tyop_2Erealax_2Ereal,V5w: ( A_27c > $o ) > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s1 )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1s2 )
& ( c_2Epred__set_2EFINITE @ A_27c @ V2s3 ) )
=> ( ( c_2Elebesgue_2Eprod__measure__space3 @ A_27a @ A_27b @ A_27c @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0s1 @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s1 ) @ V3u ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ V1s2 @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27b @ V1s2 ) @ V4v ) ) @ ( c_2Epair_2E_2C @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) ) @ V2s3 @ ( c_2Epair_2E_2C @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27c @ V2s3 ) @ V5w ) ) )
= ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V0s1 @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V1s2 @ V2s3 ) ) @ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V0s1 @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V1s2 @ V2s3 ) ) ) @ ( c_2Elebesgue_2Eprod__measure3 @ A_27a @ A_27b @ A_27c @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0s1 @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s1 ) @ V3u ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ V1s2 @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27b @ V1s2 ) @ V4v ) ) @ ( c_2Epair_2E_2C @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) ) @ V2s3 @ ( c_2Epair_2E_2C @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27c @ V2s3 ) @ V5w ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2Eseq__sup__def__compute,axiom,
( ! [V0P: tyop_2Eextreal_2Eextreal > $o] :
( ( c_2Elebesgue_2Eseq__sup @ V0P @ c_2Enum_2E0 )
= ( c_2Emin_2E_40 @ tyop_2Eextreal_2Eextreal
@ ^ [V1r: tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V1r @ V0P ) @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Eextreal_2Eextreal__sup @ V0P ) @ ( c_2Eextreal_2Eextreal__add @ V1r @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ! [V2P: tyop_2Eextreal_2Eextreal > $o,V3n: tyop_2Enum_2Enum] :
( ( c_2Elebesgue_2Eseq__sup @ V2P @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) )
= ( c_2Emin_2E_40 @ tyop_2Eextreal_2Eextreal
@ ^ [V4r: tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V4r @ V2P ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Eextreal_2Eextreal__sup @ V2P ) @ ( c_2Eextreal_2Eextreal__add @ V4r @ ( c_2Eextreal_2ENormal @ ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) ) ) ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Elebesgue_2Eseq__sup @ V2P @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V4r ) @ ( c_2Eextreal_2Eextreal__lt @ V4r @ ( c_2Eextreal_2Eextreal__sup @ V2P ) ) ) ) ) ) )
& ! [V5P: tyop_2Eextreal_2Eextreal > $o,V6n: tyop_2Enum_2Enum] :
( ( c_2Elebesgue_2Eseq__sup @ V5P @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V6n ) ) )
= ( c_2Emin_2E_40 @ tyop_2Eextreal_2Eextreal
@ ^ [V7r: tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ V7r @ V5P ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Eextreal_2Eextreal__sup @ V5P ) @ ( c_2Eextreal_2Eextreal__add @ V7r @ ( c_2Eextreal_2ENormal @ ( c_2Ereal_2Epow @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V6n ) ) ) ) ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Elebesgue_2Eseq__sup @ V5P @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6n ) ) ) @ V7r ) @ ( c_2Eextreal_2Eextreal__lt @ V7r @ ( c_2Eextreal_2Eextreal__sup @ V5P ) ) ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2EEXTREAL__SUP__SEQ,axiom,
! [V0P: tyop_2Eextreal_2Eextreal > $o] :
( ( ? [V1x: tyop_2Eextreal_2Eextreal] : ( V0P @ V1x )
& ? [V2z: tyop_2Eextreal_2Eextreal] :
( ( (~) @ ( V2z = c_2Eextreal_2EPosInf ) )
& ! [V3x: tyop_2Eextreal_2Eextreal] :
( ( V0P @ V3x )
=> ( c_2Eextreal_2Eextreal__le @ V3x @ V2z ) ) ) )
=> ? [V4x: tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal] :
( ! [V5n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Eextreal_2Eextreal @ ( V4x @ V5n ) @ V0P )
& ! [V6n: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__le @ ( V4x @ V6n ) @ ( V4x @ ( c_2Enum_2ESUC @ V6n ) ) )
& ( ( c_2Eextreal_2Eextreal__sup @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal @ V4x @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( c_2Eextreal_2Eextreal__sup @ V0P ) ) ) ) ).
thf(thm_2Elebesgue_2EEXTREAL__SUP__FUN__SEQ__IMAGE,axiom,
! [A_27a: $tType,V0P: tyop_2Eextreal_2Eextreal > $o,V1P_27: ( A_27a > tyop_2Eextreal_2Eextreal ) > $o,V2f: ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal] :
( ( ? [V3x: tyop_2Eextreal_2Eextreal] : ( V0P @ V3x )
& ? [V4z: tyop_2Eextreal_2Eextreal] :
( ( (~) @ ( V4z = c_2Eextreal_2EPosInf ) )
& ! [V5x: tyop_2Eextreal_2Eextreal] :
( ( V0P @ V5x )
=> ( c_2Eextreal_2Eextreal__le @ V5x @ V4z ) ) )
& ( V0P
= ( c_2Epred__set_2EIMAGE @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ tyop_2Eextreal_2Eextreal @ V2f @ V1P_27 ) ) )
=> ? [V6g: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal] :
( ! [V7n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V6g @ V7n ) @ V1P_27 )
& ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V8n: tyop_2Enum_2Enum] : ( V2f @ ( V6g @ V8n ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( c_2Eextreal_2Eextreal__sup @ V0P ) ) ) ) ).
thf(thm_2Elebesgue_2Emax__fn__seq__def__compute,axiom,
! [A_27a: $tType] :
( ! [V0g: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V1x: A_27a] :
( ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V0g @ c_2Enum_2E0 @ V1x )
= ( V0g @ c_2Enum_2E0 @ V1x ) )
& ! [V2g: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V3n: tyop_2Enum_2Enum,V4x: A_27a] :
( ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V2g @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) @ V4x )
= ( c_2Eextreal_2Eextreal__max @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V2g @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V4x ) @ ( V2g @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3n ) ) @ V4x ) ) )
& ! [V5g: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V6n: tyop_2Enum_2Enum,V7x: A_27a] :
( ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V5g @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V6n ) ) @ V7x )
= ( c_2Eextreal_2Eextreal__max @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V5g @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6n ) ) @ V7x ) @ ( V5g @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V6n ) ) @ V7x ) ) ) ) ).
thf(thm_2Elebesgue_2Emax__fn__seq__mono,axiom,
! [A_27a: $tType,V0g: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V1n: tyop_2Enum_2Enum,V2x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V0g @ V1n @ V2x ) @ ( c_2Elebesgue_2Emax__fn__seq @ A_27a @ V0g @ ( c_2Enum_2ESUC @ V1n ) @ V2x ) ) ).
thf(thm_2Elebesgue_2EEXTREAL__SUP__FUN__SEQ__MONO__IMAGE,axiom,
! [A_27a: $tType,V0f: ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal,V1P: tyop_2Eextreal_2Eextreal > $o,V2P_27: ( A_27a > tyop_2Eextreal_2Eextreal ) > $o] :
( ( ? [V3x: tyop_2Eextreal_2Eextreal] : ( V1P @ V3x )
& ? [V4z: tyop_2Eextreal_2Eextreal] :
( ( (~) @ ( V4z = c_2Eextreal_2EPosInf ) )
& ! [V5x: tyop_2Eextreal_2Eextreal] :
( ( V1P @ V5x )
=> ( c_2Eextreal_2Eextreal__le @ V5x @ V4z ) ) )
& ( V1P
= ( c_2Epred__set_2EIMAGE @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ tyop_2Eextreal_2Eextreal @ V0f @ V2P_27 ) )
& ! [V6g1: A_27a > tyop_2Eextreal_2Eextreal,V7g2: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V6g1 @ V2P_27 )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V7g2 @ V2P_27 )
& ! [V8x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( V6g1 @ V8x ) @ ( V7g2 @ V8x ) ) )
=> ( c_2Eextreal_2Eextreal__le @ ( V0f @ V6g1 ) @ ( V0f @ V7g2 ) ) )
& ! [V9g1: A_27a > tyop_2Eextreal_2Eextreal,V10g2: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V9g1 @ V2P_27 )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V10g2 @ V2P_27 ) )
=> ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal )
@ ^ [V11x: A_27a] : ( c_2Eextreal_2Eextreal__max @ ( V9g1 @ V11x ) @ ( V10g2 @ V11x ) )
@ V2P_27 ) ) )
=> ? [V12g: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal] :
( ! [V13n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V12g @ V13n ) @ V2P_27 )
& ! [V14x: A_27a,V15n: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__le @ ( V12g @ V15n @ V14x ) @ ( V12g @ ( c_2Enum_2ESUC @ V15n ) @ V14x ) )
& ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V16n: tyop_2Enum_2Enum] : ( V0f @ ( V12g @ V16n ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( c_2Eextreal_2Eextreal__sup @ V1P ) ) ) ) ).
thf(thm_2Elebesgue_2Elemma__radon__max__in__F,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1g: A_27a > tyop_2Eextreal_2Eextreal,V2m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V3v: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V2m )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V3v )
& ( ( c_2Emeasure_2Em__space @ A_27a @ V3v )
= ( c_2Emeasure_2Em__space @ A_27a @ V2m ) )
& ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V3v )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V2m ) )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Elebesgue_2ERADON__F @ A_27a @ V2m @ V3v ) )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1g @ ( c_2Elebesgue_2ERADON__F @ A_27a @ V2m @ V3v ) ) )
=> ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal )
@ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__max @ ( V0f @ V4x ) @ ( V1g @ V4x ) )
@ ( c_2Elebesgue_2ERADON__F @ A_27a @ V2m @ V3v ) ) ) ).
thf(thm_2Elebesgue_2Elemma__radon__seq__conv__sup,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a,V1m: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2v: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m )
& ( c_2Emeasure_2Emeasure__space @ A_27b @ V2v )
& ( ( c_2Emeasure_2Em__space @ A_27b @ V2v )
= ( c_2Emeasure_2Em__space @ A_27b @ V1m ) )
& ( ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V2v )
= ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m ) ) )
=> ? [V3f__n: tyop_2Enum_2Enum > A_27b > tyop_2Eextreal_2Eextreal] :
( ! [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27b > tyop_2Eextreal_2Eextreal ) @ ( V3f__n @ V4n ) @ ( c_2Elebesgue_2ERADON__F @ A_27b @ V1m @ V2v ) )
& ! [V5x: A_27b,V6n: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__le @ ( V3f__n @ V6n @ V5x ) @ ( V3f__n @ ( c_2Enum_2ESUC @ V6n ) @ V5x ) )
& ( ( c_2Eextreal_2Eextreal__sup
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Elebesgue_2Epos__fn__integral @ A_27b @ V1m @ ( V3f__n @ V7n ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( c_2Eextreal_2Eextreal__sup @ ( c_2Elebesgue_2ERADON__F__integrals @ A_27b @ V1m @ V2v ) ) ) ) ) ).
thf(thm_2Elebesgue_2ERN__lemma1,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1v: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2e: tyop_2Erealax_2Ereal] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V1v )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
& ( ( c_2Emeasure_2Em__space @ A_27a @ V1v )
= ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1v )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ? [V3A: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__sub @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V3A ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ V3A ) ) )
& ! [V4B: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4B @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V4B @ V3A ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Erealax_2Ereal__neg @ V2e ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V4B ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ V4B ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2ERN__lemma2,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1v: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V1v )
& ( ( c_2Emeasure_2Em__space @ A_27a @ V1v )
= ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1v )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
=> ? [V2A: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__sub @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2A ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ V2A ) ) )
& ! [V3B: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3B @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3B @ V2A ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V3B ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ V3B ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2ERadon__Nikodym,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1v: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V1v )
& ( ( c_2Emeasure_2Em__space @ A_27a @ V1v )
= ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1v )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Elebesgue_2Emeasure__absolutely__continuous @ A_27a @ V1v @ V0m ) )
=> ? [V2f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V3x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2f @ V3x ) )
& ! [V4A: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V2f @ V5x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V4A @ V5x ) ) )
= ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ V4A ) ) ) ) ) ) ).
thf(thm_2Elebesgue_2ERadon__Nikodym2,axiom,
! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1v: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
& ( c_2Emeasure_2Emeasure__space @ A_27a @ V1v )
& ( ( c_2Emeasure_2Em__space @ A_27a @ V1v )
= ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
& ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1v )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
& ( c_2Elebesgue_2Emeasure__absolutely__continuous @ A_27a @ V1v @ V0m ) )
=> ? [V2f: A_27a > tyop_2Eextreal_2Eextreal] :
( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) )
& ! [V3x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V2f @ V3x ) )
& ! [V4x: A_27a] :
( (~)
@ ( ( V2f @ V4x )
= c_2Eextreal_2EPosInf ) )
& ! [V5A: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V5A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
=> ( ( c_2Elebesgue_2Epos__fn__integral @ A_27a @ V0m
@ ^ [V6x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V2f @ V6x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V5A @ V6x ) ) )
= ( c_2Eextreal_2ENormal @ ( c_2Emeasure_2Emeasure @ A_27a @ V1v @ V5A ) ) ) ) ) ) ).
%------------------------------------------------------------------------------