ITP001 Axioms: ITP146^5.ax
%------------------------------------------------------------------------------
% File : ITP146^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : measure^2.ax [Gau20]
% : HL4146^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 289 ( 2 unt; 30 typ; 0 def)
% Number of atoms : 13563 ( 115 equ; 0 cnn)
% Maximal formula atoms : 302 ( 46 avg)
% Number of connectives : 20913 ( 25 ~; 0 |; 415 &;19642 @)
% ( 34 <=>; 797 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 19 avg;19642 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 31 ( 31 >; 0 *; 0 +; 0 <<)
% Number of symbols : 119 ( 118 usr; 89 con; 0-2 aty)
% Number of variables : 1083 ( 109 ^ 968 !; 6 ?;1083 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Emeasure_2EBorel,type,
c_2Emeasure_2EBorel: $i ).
thf(mem_c_2Emeasure_2EBorel,axiom,
mem @ c_2Emeasure_2EBorel @ ( ty_2Epair_2Eprod @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) @ ( arr @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Eadditive,type,
c_2Emeasure_2Eadditive: del > $i ).
thf(mem_c_2Emeasure_2Eadditive,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Eadditive @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Ealgebra,type,
c_2Emeasure_2Ealgebra: del > $i ).
thf(mem_c_2Emeasure_2Ealgebra,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Eclosed__cdi,type,
c_2Emeasure_2Eclosed__cdi: del > $i ).
thf(mem_c_2Emeasure_2Eclosed__cdi,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Ecountably__additive,type,
c_2Emeasure_2Ecountably__additive: del > $i ).
thf(mem_c_2Emeasure_2Ecountably__additive,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Ecountably__additive @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Ecountably__subadditive,type,
c_2Emeasure_2Ecountably__subadditive: del > $i ).
thf(mem_c_2Emeasure_2Ecountably__subadditive,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Ecountably__subadditive @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Efn__abs,type,
c_2Emeasure_2Efn__abs: del > $i ).
thf(mem_c_2Emeasure_2Efn__abs,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Efn__abs @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) ) ).
thf(tp_c_2Emeasure_2Efn__minus,type,
c_2Emeasure_2Efn__minus: del > $i ).
thf(mem_c_2Emeasure_2Efn__minus,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) ) ).
thf(tp_c_2Emeasure_2Efn__plus,type,
c_2Emeasure_2Efn__plus: del > $i ).
thf(mem_c_2Emeasure_2Efn__plus,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) ) ).
thf(tp_c_2Emeasure_2Eincreasing,type,
c_2Emeasure_2Eincreasing: del > $i ).
thf(mem_c_2Emeasure_2Eincreasing,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Eindicator__fn,type,
c_2Emeasure_2Eindicator__fn: del > $i ).
thf(mem_c_2Emeasure_2Eindicator__fn,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) ) ).
thf(tp_c_2Emeasure_2Einf__measure,type,
c_2Emeasure_2Einf__measure: del > $i ).
thf(mem_c_2Emeasure_2Einf__measure,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) ).
thf(tp_c_2Emeasure_2Elambda__system,type,
c_2Emeasure_2Elambda__system: del > $i ).
thf(mem_c_2Emeasure_2Elambda__system,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Emeasure_2Em__space,type,
c_2Emeasure_2Em__space: del > $i ).
thf(mem_c_2Emeasure_2Em__space,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Em__space @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Emeasure_2Emeasurable,type,
c_2Emeasure_2Emeasurable: del > del > $i ).
thf(mem_c_2Emeasure_2Emeasurable,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Emeasure_2Emeasurable__sets,type,
c_2Emeasure_2Emeasurable__sets: del > $i ).
thf(mem_c_2Emeasure_2Emeasurable__sets,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Emeasure_2Emeasure,type,
c_2Emeasure_2Emeasure: del > $i ).
thf(mem_c_2Emeasure_2Emeasure,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Emeasure @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) ).
thf(tp_c_2Emeasure_2Emeasure__preserving,type,
c_2Emeasure_2Emeasure__preserving: del > del > $i ).
thf(mem_c_2Emeasure_2Emeasure__preserving,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Emeasure_2Emeasure__space,type,
c_2Emeasure_2Emeasure__space: del > $i ).
thf(mem_c_2Emeasure_2Emeasure__space,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Enull__set,type,
c_2Emeasure_2Enull__set: del > $i ).
thf(mem_c_2Emeasure_2Enull__set,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Enull__set @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Emeasure_2Eouter__measure__space,type,
c_2Emeasure_2Eouter__measure__space: del > $i ).
thf(mem_c_2Emeasure_2Eouter__measure__space,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Eouter__measure__space @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Epos__simple__fn,type,
c_2Emeasure_2Epos__simple__fn: del > $i ).
thf(mem_c_2Emeasure_2Epos__simple__fn,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ) ) ) ) ) ).
thf(tp_c_2Emeasure_2Epositive,type,
c_2Emeasure_2Epositive: del > $i ).
thf(mem_c_2Emeasure_2Epositive,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Esigma,type,
c_2Emeasure_2Esigma: del > $i ).
thf(mem_c_2Emeasure_2Esigma,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Esigma @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ) ).
thf(tp_c_2Emeasure_2Esigma__algebra,type,
c_2Emeasure_2Esigma__algebra: del > $i ).
thf(mem_c_2Emeasure_2Esigma__algebra,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Esmallest__closed__cdi,type,
c_2Emeasure_2Esmallest__closed__cdi: del > $i ).
thf(mem_c_2Emeasure_2Esmallest__closed__cdi,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Emeasure_2Espace,type,
c_2Emeasure_2Espace: del > $i ).
thf(mem_c_2Emeasure_2Espace,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Espace @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Emeasure_2Esubadditive,type,
c_2Emeasure_2Esubadditive: del > $i ).
thf(mem_c_2Emeasure_2Esubadditive,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Esubadditive @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Emeasure_2Esubset__class,type,
c_2Emeasure_2Esubset__class: del > $i ).
thf(mem_c_2Emeasure_2Esubset__class,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) ) ) ).
thf(tp_c_2Emeasure_2Esubsets,type,
c_2Emeasure_2Esubsets: del > $i ).
thf(mem_c_2Emeasure_2Esubsets,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(ax_thm_2Emeasure_2Espace__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0x ) @ V1y ) )
= V0x ) ) ) ).
thf(ax_thm_2Emeasure_2Esubsets__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0x ) @ V1y ) )
= V1y ) ) ) ).
thf(ax_thm_2Emeasure_2Esubset__class__def,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1sts: $i] :
( ( mem @ V1sts @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ V0sp ) @ V1sts ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2x ) @ V1sts ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2x ) @ V0sp ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Ealgebra__def,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2s ) @ V3t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Esigma__algebra__def,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V1c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V1c ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1c ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Esigma__def,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1st: $i] :
( ( mem @ V1st @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V1st )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp )
@ ( ap @ ( c_2Epred__set_2EBIGINTER @ ( arr @ A_27a @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ bool )
@ ^ [V2s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) @ V2s ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V1st ) @ V2s ) ) @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V2s ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Em__space__def,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1sts: $i] :
( ( mem @ V1sts @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2mu: $i] :
( ( mem @ V2mu @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V0sp ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ V1sts ) @ V2mu ) ) )
= V0sp ) ) ) ) ).
thf(ax_thm_2Emeasure_2Emeasurable__sets__def,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1sts: $i] :
( ( mem @ V1sts @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2mu: $i] :
( ( mem @ V2mu @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V0sp ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ V1sts ) @ V2mu ) ) )
= V1sts ) ) ) ) ).
thf(ax_thm_2Emeasure_2Emeasure__def,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1sts: $i] :
( ( mem @ V1sts @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2mu: $i] :
( ( mem @ V2mu @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V0sp ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ V1sts ) @ V2mu ) ) )
= V2mu ) ) ) ) ).
thf(ax_thm_2Emeasure_2Epositive__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
<=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Eadditive__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) )
<=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Ecountably__additive__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ecountably__additive @ A_27a ) @ V0m ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Esubadditive__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esubadditive @ A_27a ) @ V0m ) )
<=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Ecountably__subadditive__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ecountably__subadditive @ A_27a ) @ V0m ) )
<=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) @ ( ap @ c_2Eseq_2Esuminf @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Eincreasing__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) )
<=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2t ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Emeasure__space__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Ecountably__additive @ A_27a ) @ V0m ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Elambda__system__def,axiom,
! [A_27a: del,V0gen: $i] :
( ( mem @ V0gen @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0gen ) @ V1lam )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2l: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V2l )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2l ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0gen ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3s: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0gen ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V1lam @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2l ) @ V3s ) ) ) @ ( ap @ V1lam @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0gen ) ) @ V2l ) ) @ V3s ) ) ) ) @ ( ap @ V1lam @ V3s ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Eouter__measure__space__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Eouter__measure__space @ A_27a ) @ V0m ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Ecountably__subadditive @ A_27a ) @ V0m ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Einf__measure__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) @ V1s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Ereal_2Einf
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2r: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2r )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V3f: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4m: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V4m ) @ V5n ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V3f @ V4m ) ) @ ( ap @ V3f @ V5n ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V3f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V3f ) ) @ V2r ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Eclosed__cdi__def,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ V0p ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0p ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0p ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) )
& ( ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V2f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V4f @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) @ ( ap @ V4f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Esmallest__closed__cdi__def,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) )
@ ( ap @ ( c_2Epred__set_2EBIGINTER @ ( arr @ A_27a @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ bool )
@ ^ [V1b: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) @ V1b ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) @ V1b ) ) @ ( ap @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V1b ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Emeasurable__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V0a ) @ V1b )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V2f: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ A_27b ) @ bool ) @ V2f )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27b ) @ V1b ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1b ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27b @ bool ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V3s: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Emeasure__preserving__def,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V2f: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ A_27b ) @ bool ) @ V2f )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m1 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m2 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m2 ) ) ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m1 ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m2 ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27b @ bool ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V3s: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m2 ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m1 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m1 ) ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27b ) @ V1m2 ) @ V3s ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Eindicator__fn__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V0s )
= ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Epos__simple__fn__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) )
<=> ( ! [V5t: $i] :
( ( mem @ V5t @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V5t ) ) ) )
& ! [V6t: $i] :
( ( mem @ V6t @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6t ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V6t ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ V4x @ V7i ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V3a @ V7i ) ) @ V6t ) ) ) )
@ V2s ) ) ) ) )
& ! [V8i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V8i ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V3a @ ( inj__ty_2Enum_2Enum @ V8i ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V2s ) )
& ! [V9i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V4x @ ( inj__ty_2Enum_2Enum @ V9i ) ) ) ) )
& ! [V10i: tp__ty_2Enum_2Enum,V11j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V10i ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V11j ) ) @ V2s ) )
& ( V10i != V11j ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V3a @ ( inj__ty_2Enum_2Enum @ V10i ) ) ) @ ( ap @ V3a @ ( inj__ty_2Enum_2Enum @ V11j ) ) ) ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V3a ) @ V2s ) )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Enull__set__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Enull__set @ A_27a ) @ V0m ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESPACE,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) )
= V0a ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__ALT__INTER,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2s ) @ V3t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__EMPTY,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__SPACE,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__COMPL,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__UNION,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__INTER,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__DIFF,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__FINITE__UNION,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V1c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V1c ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1c ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__INTER__SPACE,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V1s )
= V1s )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) )
= V1s ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__EMPTY,axiom,
! [A_27a: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0g0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__COMPL,axiom,
! [A_27a: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0g0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2l ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ V2l ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INTER,axiom,
! [A_27a: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( arr @ A_27a @ bool ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0g0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2l1 ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3l2 ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2l1 ) @ V3l2 ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ALGEBRA,axiom,
! [A_27a: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0g0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__ADDITIVE,axiom,
! [A_27a: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ bool ) )
=> ! [V3l1: $i] :
( ( mem @ V3l1 @ ( arr @ A_27a @ bool ) )
=> ! [V4l2: $i] :
( ( mem @ V4l2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0g0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2g ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V3l1 ) @ V4l2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3l1 ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4l2 ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1lam @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V3l1 ) @ V4l2 ) ) @ V2g ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V1lam @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V3l1 ) @ V2g ) ) ) @ ( ap @ V1lam @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V4l2 ) @ V2g ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ADDITIVE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ A_27b )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ A_27c )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0g0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) @ V1lam ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE__SUBADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Ecountably__subadditive @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Esubadditive @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__MONO,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
& ( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V1f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__DISJOINT,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESUBADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esubadditive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( V3u
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V3u ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) )
& ( V3u
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V3u ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ecountably__subadditive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) )
& ( V2s
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2s ) ) @ ( ap @ c_2Eseq_2Esuminf @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EADDITIVE__SUM,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINCREASING__ADDITIVE__SUMMABLE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
=> ( p @ ( ap @ c_2Eseq_2Esummable @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__POSITIVE,axiom,
! [A_27a: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) @ V1lam ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INCREASING,axiom,
! [A_27a: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) @ V1lam ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__SUM,axiom,
! [A_27a: del,V0g0: $i] :
( ( mem @ V0g0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ bool ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0g0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0g0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) @ V1lam ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2g ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0g0 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0g0 ) @ V1lam ) ) ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) @ ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
@ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ V1lam )
@ ( ap
@ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V7s: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V7s ) @ V2g ) ) )
@ V3f ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1lam @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V3f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) @ V2g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALGEBRA,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) ) ) ) ).
thf(conj_thm_2Emeasure_2EOUTER__MEASURE__SPACE__POSITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Eouter__measure__space @ A_27a ) @ V0m ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__CARATHEODORY,axiom,
! [A_27a: del,V0gsig: $i] :
( ( mem @ V0gsig @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0gsig ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eouter__measure__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0gsig ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0gsig ) ) @ V1lam ) ) ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0gsig ) @ V1lam ) ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0gsig ) @ V1lam ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ V1lam ) @ V2f ) ) @ ( ap @ V1lam @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECARATHEODORY__LEMMA,axiom,
! [A_27a: del,V0gsig: $i] :
( ( mem @ V0gsig @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0gsig ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eouter__measure__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0gsig ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0gsig ) ) @ V1lam ) ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0gsig ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ V0gsig ) @ V1lam ) ) @ V1lam ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__NONEMPTY,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1g ) @ V2s ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2s ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3r: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3r )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V4f: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5m: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V5m ) @ V6n ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V4f @ V5m ) ) @ ( ap @ V4f @ V6n ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1g ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V4f ) ) @ V3r ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__POS0,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3r: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3r )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V4f: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5m: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V5m ) @ V6n ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V4f @ V5m ) ) @ ( ap @ V4f @ V6n ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1g ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V4f ) ) @ V3r ) ) ) ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__POS,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1g ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINCREASING,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EADDITIVE__INCREASING,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Ecountably__additive @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ecountably__additive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
& ( V1s
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__AGREES,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Ecountably__additive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) @ V1s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__DOWN,axiom,
! [A_27a: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m1 ) ) )
& ( ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m0 )
= ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1m1 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1m1 ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m0 ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__SIGMA,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1sts: $i] :
( ( mem @ V1sts @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ V0sp ) @ V1sts ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V1sts ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EPOW__ALGEBRA,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0sp ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EPOW__SIGMA__ALGEBRA,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0sp ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EUNIV__SIGMA__ALGEBRA,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__EMPTY,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__POSITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__INCREASING,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__LE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3r: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3r )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) )
@ ^ [V4f: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V4f ) ) @ V3r ) ) ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) @ V1s ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__CLOSE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) )
=> ? [V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V4l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V3f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) @ ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V3f ) ) @ ( inj__ty_2Erealax_2Ereal @ V4l ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4l ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) @ V1s ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__COUNTABLY__SUBADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Ecountably__subadditive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINF__MEASURE__OUTER,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Eouter__measure__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__SUBSET,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V0a ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1b ) ) @ V0a ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EALGEBRA__SUBSET__LAMBDA__SYSTEM,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Elambda__system @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) @ ( ap @ ( c_2Emeasure_2Einf__measure @ A_27a ) @ V0m ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECARATHEODORY,axiom,
! [A_27a: del,V0m0: $i] :
( ( mem @ V0m0 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m0 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Ecountably__additive @ A_27a ) @ V0m0 ) ) )
=> ? [V1m: $i] :
( ( mem @ V1m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1m ) @ V2s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m0 ) @ V2s ) ) ) ) )
& ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) )
= ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m0 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m0 ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1m ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__SUBSET__SUBSETS,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V1a ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__SIGMA,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2x ) @ V1a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2x ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V1a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0p ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0p ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0p ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) )
& ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V2c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V2c ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V2c ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__COUNTABLE__UNION,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V1c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V1c ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1c ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ENUM,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__COMPL,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__PROPERTY,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ V0sp ) @ V1p ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1p ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V2a ) @ V1p ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ bool ) ) @ V1p ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0sp ) @ V3s ) ) @ V1p ) ) ) )
& ! [V4c: $i] :
( ( mem @ V4c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V4c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V4c ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ bool ) ) @ V1p ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V4c ) ) @ V1p ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) @ V1p ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__EMPTY,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__SUBSET__MEASURABLE__SETS,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1m ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V0a ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) @ V0a ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__DISJOINT,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2s ) @ V3t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V4f @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) @ ( ap @ V4f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__PROPERTY__ALT,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ V0sp ) @ V1p ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1p ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V2a ) @ V1p ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ bool ) ) @ V1p ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0sp ) @ V3s ) ) @ V1p ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ bool ) ) @ V1p ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ V1p ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) @ V1p ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESPACE__SMALLEST__CLOSED__CDI,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) )
= ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) ) ).
thf(conj_thm_2Emeasure_2ESMALLEST__CLOSED__CDI,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECLOSED__CDI__DUNION,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECLOSED__CDI__COMPL,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0p ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECLOSED__CDI__DISJOINT,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ECLOSED__CDI__INCREASING,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) )
& ( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V1f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA1,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA2,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ V0a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V1a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V1a ) @ V2p ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eclosed__cdi @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V2p ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V1a ) ) ) @ V2p ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__WEAK,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V2a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V2a ) @ V1p ) )
& ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ V0sp ) @ V1p ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ V1p ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0sp ) @ V3s ) ) @ V1p ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V1p ) ) )
& ( ( ap @ V4f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V5n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V4f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ ( ap @ V4f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ V1p ) ) ) )
& ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V6f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V1p ) ) )
& ! [V7m: tp__ty_2Enum_2Enum,V8n: tp__ty_2Enum_2Enum] :
( ( V7m != V8n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V6f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ V1p ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) @ V1p ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Ealgebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V2a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V2a ) @ V1p ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ bool ) ) @ V1p ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0sp ) @ V3s ) ) @ V1p ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ bool ) ) @ V1p ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) ) ) )
& ( ( ap @ V4f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V5n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V4f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ ( ap @ V4f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V4f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ V1p ) ) ) )
& ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V6f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ bool ) ) @ V1p ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) ) ) )
& ! [V7m: tp__ty_2Enum_2Enum,V8n: tp__ty_2Enum_2Enum] :
( ( V7m != V8n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V7m ) ) ) @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V6f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ V1p ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V2a ) ) ) @ V1p ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__COUNTABLY__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
& ( V1s
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) )
& ( V3u
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V3u ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__COUNTABLE__INCREASING,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V2f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ( V1s
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__REDUCE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) )
= V0m ) ) ).
thf(conj_thm_2Emeasure_2ESPACE__SIGMA,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V1a ) )
= V0sp ) ) ) ).
thf(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V2f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ( V1s
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__REDUCE,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V1a ) ) )
= ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0sp ) @ V1a ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__SETS__SUBSET__SPACE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__DIFF__PROPERTY,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27b ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1b ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V4s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1b ) ) @ V4s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V4s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__BIGUNION__PROPERTY,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27b ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1b ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) )
=> ! [V4c: $i] :
( ( mem @ V4c @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27b @ bool ) ) @ V4c ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ V4c ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27b @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) ) @ V4c ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASUBABLE__BIGUNION__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27b ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1b ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) )
=> ! [V4c: $i] :
( ( mem @ V4c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V4c ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V4c ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27b @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V4c ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27b @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__SIGMA__PREIMAGES,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27b ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1b ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27b @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V0a ) @ V1b ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27b ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1b ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__SIGMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) )
=> ! [V3sp: $i] :
( ( mem @ V3sp @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27b ) @ V3sp ) @ V2b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ V3sp ) ) )
& ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V4s ) @ V2b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V4s ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V1a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27b ) @ V3sp ) @ V2b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V0a ) @ V1b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V0a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1b ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__LIFT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V1a ) @ V2b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V1a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27b ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V2b ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V2b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURE__PRESERVING,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m1 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m1 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m2 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m2 ) ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m1 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m2 ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V3s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m2 ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m1 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m1 ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27b ) @ V1m2 ) @ V3s ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__PRESERVING__LIFT,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m1 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m2 ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m2 )
= ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m2 ) ) @ V2a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V0m1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) @ V2a ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27b ) @ V1m2 ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__PRESERVING__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m1 ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27b ) @ V1m2 ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27b ) @ V1m2 )
= ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27b ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m2 ) ) @ V2a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V0m1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27b ) @ V1m2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) @ V2a ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27b ) @ V1m2 ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__I,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27a ) ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27a ) @ V0a ) @ V0a ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__COMP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V4c: $i] :
( ( mem @ V4c @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V2a ) @ V3b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ A_27c ) ) @ V1g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27b @ A_27c ) @ V3b ) @ V4c ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1g ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27c ) @ V2a ) @ V4c ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V4c: $i] :
( ( mem @ V4c @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V2a ) @ V3b ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27c ) @ V4c ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ A_27c ) ) @ V1g ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V3b ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27c ) @ V4c ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ ( arr @ A_27c @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27c @ bool ) ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27c ) @ V4c ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27c ) @ V1g ) @ V5x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V2a ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V3b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1g ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27c ) @ V2a ) @ V4c ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONGER,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V4c: $i] :
( ( mem @ V4c @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) ) )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V2a ) @ V3b ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27c ) @ V4c ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ A_27c ) ) @ V1g ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V3b ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27c ) @ V4c ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V2a ) ) ) @ V5t ) )
& ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ A_27c @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27c @ bool ) ) @ V6s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27c ) @ V4c ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27c ) @ V1g ) @ V6s ) ) @ V5t ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V3b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1g ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27c ) @ V2a ) @ V4c ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__UP__LIFT,axiom,
! [A_27a: del,A_27b: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V1a ) ) @ V3c ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V2b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V1a ) @ V2b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V2b ) ) @ V3c ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__UP__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V1a ) @ V2b ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V2b ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V1a ) ) @ V3c ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0sp ) @ V2b ) ) @ V3c ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__UP__SIGMA,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V0a ) @ V1b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__LIFT,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1m1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m1 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ V0a ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1m1 ) ) ) ) @ V2m2 ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m1 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m1 ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V0a ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m1 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V1m1 ) @ V2m2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1m1: $i] :
( ( mem @ V1m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2m2: $i] :
( ( mem @ V2m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1m1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V0a ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m1 ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m1 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m1 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m1 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ V0a ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1m1 ) ) ) ) @ V2m2 ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V1m1 ) @ V2m2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SIGMA,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m1 ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m1 )
= ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m1 ) ) @ V2a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m1 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ V2a ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m1 ) ) ) ) @ V1m2 ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__PROD__SIGMA,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) )
=> ! [V2a2: $i] :
( ( mem @ V2a2 @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ bool ) @ ( arr @ ( arr @ A_27c @ bool ) @ bool ) ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epair_2EFST @ A_27b @ A_27c ) ) @ V3f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ V0a ) @ V1a1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epair_2ESND @ A_27b @ A_27c ) ) @ V3f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27c ) @ V0a ) @ V2a2 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ V3f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V0a ) @ ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27b ) @ V1a1 ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27c ) @ V2a2 ) ) ) @ ( ap @ ( ap @ ( c_2Eutil__prob_2Eprod__sets @ A_27b @ A_27c ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27b ) @ V1a1 ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27c ) @ V2a2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__RANGE__REDUCE,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) @ V2s ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ V2s ) ) ) ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V2s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V2s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m )
= ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m )
= ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27a @ bool ) )
=> ! [V2m: $i] :
( ( mem @ V2m @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s_27 ) @ V0s ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s ) ) @ V2m ) ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V1s_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V1s_27 ) ) @ V2m ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESTRONG__MEASURE__SPACE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s_27 ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0s ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V1s_27 ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0s ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V1s_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V1s_27 ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__REAL__SUM__IMAGE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) )
@ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__POW,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ A_27a ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0s ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0s ) ) @ V1m ) ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0s ) ) @ V1m ) ) ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0s ) ) @ V1m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces2,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( c_2Emeasure_2Eadditive @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIMAGE__SING,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ ( ap @ V0f @ V1x ) ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESUBSET__BIGINTER,axiom,
! [A_27a: del,V0X: $i] :
( ( mem @ V0X @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0X ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1P ) ) )
<=> ! [V2Y: $i] :
( ( mem @ V2Y @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2Y ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0X ) @ V2Y ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__INCREASING,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Eincreasing @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__POSITIVE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( p @ ( ap @ ( c_2Emeasure_2Epositive @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__INTER,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__UNION,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__DIFF,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__COMPL__SUBSET,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V0t ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0t ) @ V2s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V2s ) @ V0t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1m ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V1m ) @ V0t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGUNION,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__IN__MSPACE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1A ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET__MSPACE,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V1m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V0A ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0A ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__EMPTY__MEASURABLE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__MSPACE__MEASURABLE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__BIGINTER,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emeasure_2Esubset__class @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V1s ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGINTER,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE2,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V1f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V2f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ( V1s
= ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER2,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V1f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__RESTRICTED,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) )
=> ( p
@ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ V1s )
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2t: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EMEASURE__SPACE__CMUL,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) )
=> ( p
@ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2a: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V0m ) @ V2a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EBIGUNION__IMAGE__Q,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Eextreal_2Eextreal @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ ( arr @ A_27a @ bool ) ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ ty_2Eextreal_2Eextreal @ ( arr @ A_27a @ bool ) ) @ c_2Eextreal_2EQ__set ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Eextreal_2Eextreal @ ( arr @ A_27a @ bool ) ) @ V1f ) @ c_2Eextreal_2EQ__set ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2EBorel__def,axiom,
( c_2Emeasure_2EBorel
= ( ap @ ( ap @ ( c_2Emeasure_2Esigma @ ty_2Eextreal_2Eextreal ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Eextreal_2Eextreal @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V0a: $i] :
( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V1x ) @ V0a ) ) ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__BOREL,axiom,
p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ).
thf(conj_thm_2Emeasure_2EMEASURABLE__BOREL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0f )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V3x ) @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__NEGINF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V0f @ V2x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT1,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) @ ( ap @ V0f @ V3x ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT2,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT3,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) @ ( ap @ V0f @ V3x ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT4,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal,V3d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) @ ( ap @ V0f @ V4x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V4x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V3d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT5,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal,V3d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) @ ( ap @ V0f @ V4x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V4x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V3d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT6,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal,V3d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) @ ( ap @ V0f @ V4x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V4x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V3d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT7,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal,V3d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) @ ( ap @ V0f @ V4x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V4x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V3d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT8,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT9,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
=> ( ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V4c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V4c ) ) @ ( ap @ V0f @ V5x ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V6c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V7x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V6c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V8c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V9x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V9x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V8c ) ) @ ( ap @ V0f @ V9x ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V10c: tp__ty_2Eextreal_2Eextreal,V11d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V12x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V12x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V10c ) ) @ ( ap @ V0f @ V12x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V12x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V11d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V13c: tp__ty_2Eextreal_2Eextreal,V14d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V15x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V15x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V13c ) ) @ ( ap @ V0f @ V15x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V15x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V14d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V16c: tp__ty_2Eextreal_2Eextreal,V17d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V18x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V18x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V16c ) ) @ ( ap @ V0f @ V18x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V18x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V17d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V19c: tp__ty_2Eextreal_2Eextreal,V20d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V21x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V21x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V19c ) ) @ ( ap @ V0f @ V21x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V21x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V20d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V22c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V23x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V23x ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V0f @ V23x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V22c ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) )
& ! [V24c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V25x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V25x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V0f @ V25x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V24c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL__MEASURE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) ) @ c_2Emeasure_2EBorel ) ) )
=> ( ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V4c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V4c ) ) @ ( ap @ V0f @ V5x ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V6c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V7x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V6c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V8c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V9x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V9x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V8c ) ) @ ( ap @ V0f @ V9x ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V10c: tp__ty_2Eextreal_2Eextreal,V11d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V12x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V12x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V10c ) ) @ ( ap @ V0f @ V12x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V12x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V11d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V13c: tp__ty_2Eextreal_2Eextreal,V14d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V15x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V15x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V13c ) ) @ ( ap @ V0f @ V15x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V15x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V14d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V16c: tp__ty_2Eextreal_2Eextreal,V17d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V18x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V18x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V16c ) ) @ ( ap @ V0f @ V18x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V18x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V17d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V19c: tp__ty_2Eextreal_2Eextreal,V20d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V21x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V21x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V19c ) ) @ ( ap @ V0f @ V21x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V21x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V20d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V22c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V23x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V23x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V0f @ V23x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V22c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) )
& ! [V24c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V25x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V25x ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ ( ap @ V0f @ V25x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V24c ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V1m ) ) )
@ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V1m ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V2a ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V2a ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V2a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V2a ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V2a ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V2a ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V2a ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2ESPACE__BOREL,axiom,
( ( ap @ ( c_2Emeasure_2Espace @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel )
= ( c_2Epred__set_2EUNIV @ ty_2Eextreal_2Eextreal ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS1,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V1x ) @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS2,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) @ V1x ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS3,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ V1x ) @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS4,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) @ V1x ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS5,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) @ V2x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V2x ) @ ( inj__ty_2Eextreal_2Eextreal @ V1d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS6,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) @ V2x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ V2x ) @ ( inj__ty_2Eextreal_2Eextreal @ V1d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS7,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) @ V2x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ V2x ) @ ( inj__ty_2Eextreal_2Eextreal @ V1d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS8,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) @ V2x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V2x ) @ ( inj__ty_2Eextreal_2Eextreal @ V1d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS9,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Eextreal_2Eextreal ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS10,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V1x ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ V1x ) @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ).
thf(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS,axiom,
( ! [V0c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V1x ) @ ( inj__ty_2Eextreal_2Eextreal @ V0c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V2c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V2c ) ) @ V3x ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V4c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V4c ) ) @ V5x ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V6c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ V7x ) @ ( inj__ty_2Eextreal_2Eextreal @ V6c ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V8c: tp__ty_2Eextreal_2Eextreal,V9d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V10x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V10x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V8c ) ) @ V10x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V10x ) @ ( inj__ty_2Eextreal_2Eextreal @ V9d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V11c: tp__ty_2Eextreal_2Eextreal,V12d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V13x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V13x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V11c ) ) @ V13x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ V13x ) @ ( inj__ty_2Eextreal_2Eextreal @ V12d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V14c: tp__ty_2Eextreal_2Eextreal,V15d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V16x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V16x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( inj__ty_2Eextreal_2Eextreal @ V14c ) ) @ V16x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ V16x ) @ ( inj__ty_2Eextreal_2Eextreal @ V15d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V17c: tp__ty_2Eextreal_2Eextreal,V18d: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V19x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V19x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V17c ) ) @ V19x ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ V19x ) @ ( inj__ty_2Eextreal_2Eextreal @ V18d ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V20c: tp__ty_2Eextreal_2Eextreal] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V20c ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Eextreal_2Eextreal ) ) ) @ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) )
& ! [V21c: tp__ty_2Eextreal_2Eextreal] :
( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V22x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V22x ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Eextreal_2Eextreal ) @ V22x ) @ ( inj__ty_2Eextreal_2Eextreal @ V21c ) ) ) ) ) ) )
@ ( ap @ ( c_2Emeasure_2Esubsets @ ty_2Eextreal_2Eextreal ) @ c_2Emeasure_2EBorel ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CONST,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1k: tp__ty_2Eextreal_2Eextreal,V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2f @ V3x ) )
= V1k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__INDICATOR,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1A ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2f @ V3x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1A ) @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CMUL,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2g @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ABS,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2g @ V3x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__abs @ ( ap @ V1f @ V3x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SQR,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2g @ V3x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ V1f @ V3x ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ADD,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V3h @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V3h ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SUB,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V3h @ V4x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__sub @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V3h ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V4x ) )
!= fo__c_2Eextreal_2ENegInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V4x ) )
!= fo__c_2Eextreal_2EPosInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2g @ V4x ) )
!= fo__c_2Eextreal_2ENegInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2g @ V4x ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V3h @ V5x ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V5x ) ) @ ( ap @ V2g @ V5x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V3h ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SUM,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3s ) )
& ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ! [V4i: $i] :
( ( mem @ V4i @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4i ) @ V3s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V1f @ V4i ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2g @ V5x ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b )
@ ( lam @ A_27b
@ ^ [V6i: $i] : ( ap @ ( ap @ V1f @ V6i ) @ V5x ) ) )
@ V3s ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CMUL__INDICATOR,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1z: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V2s ) @ V3x ) ) ) )
@ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL__INDICATOR,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ ( ap @ ( c_2Emeasure_2Esubsets @ A_27a ) @ V0a ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V2s ) @ V3x ) ) ) )
@ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL__INDICATOR__EQ,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ V1f @ V2x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V0a ) ) @ V2x ) ) ) )
@ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__POS__SIMPLE__FN,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ? [V2s: $i] :
( ( mem @ V2s @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
& ? [V4x: $i] :
( ( mem @ V4x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emeasure_2Epos__simple__fn @ A_27a ) @ V0m ) @ V1f ) @ V2s ) @ V3a ) @ V4x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V0m ) ) ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__POW,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V1a ) ) )
=> ( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2f @ V3x ) )
!= fo__c_2Eextreal_2ENegInf )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V2f @ V3x ) )
!= fo__c_2Eextreal_2EPosInf ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__pow @ ( ap @ V2f @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
@ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V1a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MAX,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V2g ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__max @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) )
@ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MONO__SUP,axiom,
! [A_27a: del,V0fn: $i] :
( ( mem @ V0fn @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Esigma__algebra @ A_27a ) @ V2a ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ V0fn @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V2a ) @ c_2Emeasure_2EBorel ) ) )
& ! [V4n: tp__ty_2Enum_2Enum,V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V2a ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ V0fn @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V5x ) ) @ ( ap @ ( ap @ V0fn @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V5x ) ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Emeasure_2Espace @ A_27a ) @ V2a ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ V1f @ V6x ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ V0fn @ V7n ) @ V6x ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V2a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Efn__plus__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V0f )
= ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V1x ) ) ) @ ( ap @ V0f @ V1x ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Efn__minus__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V0f )
= ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__lt @ ( ap @ V0f @ V1x ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__ainv @ ( ap @ V0f @ V1x ) ) ) @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(ax_thm_2Emeasure_2Efn__abs__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__abs @ A_27a ) @ V0f )
= ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ c_2Eextreal_2Eextreal__abs @ ( ap @ V0f @ V1x ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EFN__PLUS__POS,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V0g ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EFN__MINUS__POS,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V0g ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EFN__PLUS__POS__ID,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0g @ V1x ) ) ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V0g )
= V0g ) ) ) ).
thf(conj_thm_2Emeasure_2EFN__MINUS__POS__ZERO,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0g @ V1x ) ) ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V0g )
= ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EFN__PLUS__CMUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) @ ( ap @ V0f @ V2x ) ) ) )
= ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V0f ) @ V3x ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) @ ( ap @ V0f @ V4x ) ) ) )
= ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2Eextreal__ainv @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V0f ) @ V5x ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EFN__MINUS__CMUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) @ ( ap @ V0f @ V2x ) ) ) )
= ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V0f ) @ V3x ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) @ ( ap @ V0f @ V4x ) ) ) )
= ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2Eextreal__ainv @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V0f ) @ V5x ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EFN__PLUS__ADD__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p
@ ( ap
@ ( ap @ c_2Eextreal_2Eextreal__le
@ ( ap
@ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2x ) )
@ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V0f ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1g ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EFN__MINUS__ADD__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p
@ ( ap
@ ( ap @ c_2Eextreal_2Eextreal__le
@ ( ap
@ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2x ) )
@ ( ap @ ( ap @ c_2Eextreal_2Eextreal__add @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V0f ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1g ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__FN__PLUS,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__FN__MINUS,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__PLUS__MINUS,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ ( c_2Emeasure_2Efn__plus @ A_27a ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) ) @ ( ap @ ( c_2Emeasure_2Efn__minus @ A_27a ) @ V1f ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasurable @ A_27a @ ty_2Eextreal_2Eextreal ) @ V0a ) @ c_2Emeasure_2EBorel ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2EINDICATOR__FN__MUL__INTER,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ( ( lam @ A_27a
@ ^ [V2t: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V0A ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V1B ) @ V2t ) ) )
= ( lam @ A_27a
@ ^ [V3t: $i] : ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0A ) @ V1B ) ) @ V3t ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2Eindicator__fn__split,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0r ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V2b ) @ V0r ) )
= V1s )
& ! [V3i: tp__ty_2Enum_2Enum,V4j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4j ) ) @ V0r ) )
& ( V3i != V4j ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V2b @ ( inj__ty_2Enum_2Enum @ V3i ) ) ) @ ( ap @ V2b @ ( inj__ty_2Enum_2Enum @ V4j ) ) ) ) ) )
=> ! [V5a: $i] :
( ( mem @ V5a @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V5a ) @ V1s ) )
=> ( ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ V5a )
= ( lam @ A_27a
@ ^ [V6x: $i] :
( ap
@ ( ap @ ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V5a ) @ ( ap @ V2b @ V7i ) ) ) @ V6x ) ) )
@ V0r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2Eindicator__fn__suminf,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V0a @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V0a @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eext__suminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ V0a @ V4i ) ) @ V1x ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Emeasure_2Eindicator__fn @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V0a ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Emeasure_2Emeasure__split,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2m: $i] :
( ( mem @ V2m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V2m ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0r ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V1b ) @ V0r ) )
= ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V2m ) )
& ! [V3i: tp__ty_2Enum_2Enum,V4j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4j ) ) @ V0r ) )
& ( V3i != V4j ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V1b @ ( inj__ty_2Enum_2Enum @ V3i ) ) ) @ ( ap @ V1b @ ( inj__ty_2Enum_2Enum @ V4j ) ) ) ) )
& ! [V5i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5i ) ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ V1b @ ( inj__ty_2Enum_2Enum @ V5i ) ) ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V2m ) ) ) ) )
=> ! [V6a: $i] :
( ( mem @ V6a @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V6a ) @ ( ap @ ( c_2Emeasure_2Emeasurable__sets @ A_27a ) @ V2m ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V2m ) @ V6a ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ ( c_2Emeasure_2Emeasure @ A_27a ) @ V2m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V6a ) @ ( ap @ V1b @ V7i ) ) ) ) )
@ V0r ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------