%------------------------------------------------------------------------------
% File     : ITP146^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : measure.ax [Gau19]
%          : HL4146^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  342 (  34 unt; 106 typ;   0 def)
%            Number of atoms       : 2435 ( 117 equ;  26 cnn)
%            Maximal formula atoms :   74 (   7 avg)
%            Number of connectives : 8660 (  26   ~;   1   |; 411   &;7936   @)
%                                         (  40 <=>; 246  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   27 (  12 avg;7936 nst)
%            Number of types       :    5 (   4 usr)
%            Number of type conns  : 2721 (2721   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  104 ( 102 usr;   6 con; 0-6 aty)
%            Number of variables   : 1141 ( 109   ^ 952   !;   7   ?;1141   :)
%                                         (  73  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Eextreal_2Eextreal,type,
    tyop_2Eextreal_2Eextreal: $tType ).

thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

thf(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: $tType > $tType > $tType ).

thf(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

thf(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: $tType > $tType > $tType ).

thf(tyop_2Erealax_2Ereal,type,
    tyop_2Erealax_2Ereal: $tType ).

thf(c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Epair_2E_2C,type,
    c_2Epair_2E_2C: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).

thf(c_2Eseq_2E_2D_2D_3E,type,
    c_2Eseq_2E_2D_2D_3E: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).

thf(c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $o > $o > $o ).

thf(c_2Enum_2E0,type,
    c_2Enum_2E0: tyop_2Enum_2Enum ).

thf(c_2Emin_2E_3D,type,
    c_2Emin_2E_3D: 
      !>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).

thf(c_2Emin_2E_3D_3D_3E,type,
    c_2Emin_2E_3D_3D_3E: $o > $o > $o ).

thf(c_2Ebool_2E_3F,type,
    c_2Ebool_2E_3F: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Epred__set_2EBIGINTER,type,
    c_2Epred__set_2EBIGINTER: 
      !>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EBIGUNION,type,
    c_2Epred__set_2EBIGUNION: 
      !>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).

thf(c_2Earithmetic_2EBIT1,type,
    c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2EBIT2,type,
    c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Emeasure_2EBorel,type,
    c_2Emeasure_2EBorel: tyop_2Epair_2Eprod @ ( tyop_2Eextreal_2Eextreal > $o ) @ ( ( tyop_2Eextreal_2Eextreal > $o ) > $o ) ).

thf(c_2Ebool_2ECOND,type,
    c_2Ebool_2ECOND: 
      !>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).

thf(c_2Epred__set_2ECROSS,type,
    c_2Epred__set_2ECROSS: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).

thf(c_2Epred__set_2EDIFF,type,
    c_2Epred__set_2EDIFF: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EDISJOINT,type,
    c_2Epred__set_2EDISJOINT: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Epred__set_2EEMPTY,type,
    c_2Epred__set_2EEMPTY: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Eextreal_2EEXTREAL__SUM__IMAGE,type,
    c_2Eextreal_2EEXTREAL__SUM__IMAGE: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Eextreal_2Eextreal ) > ( A_27a > $o ) > tyop_2Eextreal_2Eextreal ) ).

thf(c_2Epred__set_2EFINITE,type,
    c_2Epred__set_2EFINITE: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Epair_2EFST,type,
    c_2Epair_2EFST: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).

thf(c_2Epred__set_2EFUNSET,type,
    c_2Epred__set_2EFUNSET: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( A_27a > A_27b ) > $o ) ).

thf(c_2Epred__set_2EGSPEC,type,
    c_2Epred__set_2EGSPEC: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).

thf(c_2Ecombin_2EI,type,
    c_2Ecombin_2EI: 
      !>[A_27a: $tType] : ( A_27a > A_27a ) ).

thf(c_2Epred__set_2EIMAGE,type,
    c_2Epred__set_2EIMAGE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).

thf(c_2Ebool_2EIN,type,
    c_2Ebool_2EIN: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).

thf(c_2Epred__set_2EINSERT,type,
    c_2Epred__set_2EINSERT: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EINTER,type,
    c_2Epred__set_2EINTER: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Earithmetic_2ENUMERAL,type,
    c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Eextreal_2ENegInf,type,
    c_2Eextreal_2ENegInf: tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2ENormal,type,
    c_2Eextreal_2ENormal: tyop_2Erealax_2Ereal > tyop_2Eextreal_2Eextreal ).

thf(c_2Epred__set_2EPOW,type,
    c_2Epred__set_2EPOW: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Epred__set_2EPREIMAGE,type,
    c_2Epred__set_2EPREIMAGE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27b > $o ) > A_27a > $o ) ).

thf(c_2Eextreal_2EPosInf,type,
    c_2Eextreal_2EPosInf: tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2EQ__set,type,
    c_2Eextreal_2EQ__set: tyop_2Eextreal_2Eextreal > $o ).

thf(c_2Ereal__sigma_2EREAL__SUM__IMAGE,type,
    c_2Ereal__sigma_2EREAL__SUM__IMAGE: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Erealax_2Ereal ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ).

thf(c_2Epair_2ESND,type,
    c_2Epair_2ESND: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).

thf(c_2Epred__set_2ESUBSET,type,
    c_2Epred__set_2ESUBSET: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Enum_2ESUC,type,
    c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Epred__set_2EUNION,type,
    c_2Epred__set_2EUNION: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EUNIV,type,
    c_2Epred__set_2EUNIV: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Earithmetic_2EZERO,type,
    c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).

thf(c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $o > $o > $o ).

thf(c_2Emeasure_2Eadditive,type,
    c_2Emeasure_2Eadditive: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).

thf(c_2Emeasure_2Ealgebra,type,
    c_2Emeasure_2Ealgebra: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > $o ) ).

thf(c_2Emeasure_2Eclosed__cdi,type,
    c_2Emeasure_2Eclosed__cdi: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > $o ) ).

thf(c_2Epred__set_2Ecount,type,
    c_2Epred__set_2Ecount: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Epred__set_2Ecountable,type,
    c_2Epred__set_2Ecountable: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Emeasure_2Ecountably__additive,type,
    c_2Emeasure_2Ecountably__additive: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).

thf(c_2Emeasure_2Ecountably__subadditive,type,
    c_2Emeasure_2Ecountably__subadditive: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).

thf(c_2Eextreal_2Eext__suminf,type,
    c_2Eextreal_2Eext__suminf: ( tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__abs,type,
    c_2Eextreal_2Eextreal__abs: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__add,type,
    c_2Eextreal_2Eextreal__add: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__ainv,type,
    c_2Eextreal_2Eextreal__ainv: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__le,type,
    c_2Eextreal_2Eextreal__le: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > $o ).

thf(c_2Eextreal_2Eextreal__lt,type,
    c_2Eextreal_2Eextreal__lt: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > $o ).

thf(c_2Eextreal_2Eextreal__max,type,
    c_2Eextreal_2Eextreal__max: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__mul,type,
    c_2Eextreal_2Eextreal__mul: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__of__num,type,
    c_2Eextreal_2Eextreal__of__num: tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__pow,type,
    c_2Eextreal_2Eextreal__pow: tyop_2Eextreal_2Eextreal > tyop_2Enum_2Enum > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__sub,type,
    c_2Eextreal_2Eextreal__sub: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal ).

thf(c_2Eextreal_2Eextreal__sup,type,
    c_2Eextreal_2Eextreal__sup: ( tyop_2Eextreal_2Eextreal > $o ) > tyop_2Eextreal_2Eextreal ).

thf(c_2Emeasure_2Efn__abs,type,
    c_2Emeasure_2Efn__abs: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Eextreal_2Eextreal ) > A_27a > tyop_2Eextreal_2Eextreal ) ).

thf(c_2Emeasure_2Efn__minus,type,
    c_2Emeasure_2Efn__minus: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Eextreal_2Eextreal ) > A_27a > tyop_2Eextreal_2Eextreal ) ).

thf(c_2Emeasure_2Efn__plus,type,
    c_2Emeasure_2Efn__plus: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Eextreal_2Eextreal ) > A_27a > tyop_2Eextreal_2Eextreal ) ).

thf(c_2Emeasure_2Eincreasing,type,
    c_2Emeasure_2Eincreasing: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).

thf(c_2Emeasure_2Eindicator__fn,type,
    c_2Emeasure_2Eindicator__fn: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > tyop_2Eextreal_2Eextreal ) ).

thf(c_2Ereal_2Einf,type,
    c_2Ereal_2Einf: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).

thf(c_2Emeasure_2Einf__measure,type,
    c_2Emeasure_2Einf__measure: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ).

thf(c_2Emeasure_2Elambda__system,type,
    c_2Emeasure_2Elambda__system: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) > ( A_27a > $o ) > $o ) ).

thf(c_2Emeasure_2Em__space,type,
    c_2Emeasure_2Em__space: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > A_27a > $o ) ).

thf(c_2Emeasure_2Emeasurable,type,
    c_2Emeasure_2Emeasurable: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) ) > ( A_27a > A_27b ) > $o ) ).

thf(c_2Emeasure_2Emeasurable__sets,type,
    c_2Emeasure_2Emeasurable__sets: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > $o ) > $o ) ).

thf(c_2Emeasure_2Emeasure,type,
    c_2Emeasure_2Emeasure: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ).

thf(c_2Emeasure_2Emeasure__preserving,type,
    c_2Emeasure_2Emeasure__preserving: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > A_27b ) > $o ) ).

thf(c_2Emeasure_2Emeasure__space,type,
    c_2Emeasure_2Emeasure__space: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).

thf(c_2Emeasure_2Enull__set,type,
    c_2Emeasure_2Enull__set: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > $o ) > $o ) ).

thf(c_2Ecombin_2Eo,type,
    c_2Ecombin_2Eo: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).

thf(c_2Emeasure_2Eouter__measure__space,type,
    c_2Emeasure_2Eouter__measure__space: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).

thf(c_2Emeasure_2Epos__simple__fn,type,
    c_2Emeasure_2Epos__simple__fn: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > ( tyop_2Enum_2Enum > $o ) > ( tyop_2Enum_2Enum > A_27a > $o ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ) ).

thf(c_2Emeasure_2Epositive,type,
    c_2Emeasure_2Epositive: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).

thf(c_2Eutil__prob_2Eprod__sets,type,
    c_2Eutil__prob_2Eprod__sets: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( ( A_27a > $o ) > $o ) > ( ( A_27b > $o ) > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > $o ) ).

thf(c_2Erealax_2Ereal__add,type,
    c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Erealax_2Ereal__lt,type,
    c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).

thf(c_2Ereal_2Ereal__lte,type,
    c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).

thf(c_2Erealax_2Ereal__mul,type,
    c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Ereal_2Ereal__of__num,type,
    c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).

thf(c_2Ereal_2Ereal__sub,type,
    c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

thf(c_2Emeasure_2Esigma,type,
    c_2Emeasure_2Esigma: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( A_27a > $o ) > $o ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) ) ).

thf(c_2Emeasure_2Esigma__algebra,type,
    c_2Emeasure_2Esigma__algebra: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > $o ) ).

thf(c_2Emeasure_2Esmallest__closed__cdi,type,
    c_2Emeasure_2Esmallest__closed__cdi: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) ) ).

thf(c_2Emeasure_2Espace,type,
    c_2Emeasure_2Espace: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > A_27a > $o ) ).

thf(c_2Emeasure_2Esubadditive,type,
    c_2Emeasure_2Esubadditive: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).

thf(c_2Emeasure_2Esubset__class,type,
    c_2Emeasure_2Esubset__class: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( A_27a > $o ) > $o ) > $o ) ).

thf(c_2Emeasure_2Esubsets,type,
    c_2Emeasure_2Esubsets: 
      !>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > ( A_27a > $o ) > $o ) ).

thf(c_2Ereal_2Esum,type,
    c_2Ereal_2Esum: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).

thf(c_2Eseq_2Esuminf,type,
    c_2Eseq_2Esuminf: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).

thf(c_2Eseq_2Esummable,type,
    c_2Eseq_2Esummable: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).

thf(c_2Eseq_2Esums,type,
    c_2Eseq_2Esums: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).

thf(c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $o > $o ).

thf(logicdef_2E_2F_5C,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
    <=> ( V0
        & V1 ) ) ).

thf(logicdef_2E_5C_2F,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
    <=> ( V0
        | V1 ) ) ).

thf(logicdef_2E_7E,axiom,
    ! [V0: $o] :
      ( ( c_2Ebool_2E_7E @ V0 )
    <=> ( (~) @ V0 ) ) ).

thf(logicdef_2E_3D_3D_3E,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
    <=> ( V0
       => V1 ) ) ).

thf(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0: A_27a,V1: A_27a] :
      ( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
    <=> ( V0 = V1 ) ) ).

thf(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_21 @ A_27a @ V0f )
    <=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_3F @ A_27a @ V0f )
    <=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Emeasure_2Espace__def,axiom,
    ! [A_27a: $tType,V0x: A_27a > $o,V1y: ( A_27a > $o ) > $o] :
      ( ( c_2Emeasure_2Espace @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0x @ V1y ) )
      = V0x ) ).

thf(thm_2Emeasure_2Esubsets__def,axiom,
    ! [A_27a: $tType,V0x: A_27a > $o,V1y: ( A_27a > $o ) > $o] :
      ( ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0x @ V1y ) )
      = V1y ) ).

thf(thm_2Emeasure_2Esubset__class__def,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1sts: ( A_27a > $o ) > $o] :
      ( ( c_2Emeasure_2Esubset__class @ A_27a @ V0sp @ V1sts )
    <=> ! [V2x: A_27a > $o] :
          ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2x @ V1sts )
         => ( c_2Epred__set_2ESUBSET @ A_27a @ V2x @ V0sp ) ) ) ).

thf(thm_2Emeasure_2Ealgebra__def,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
    <=> ( ( c_2Emeasure_2Esubset__class @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ! [V1s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
        & ! [V2s: A_27a > $o,V3t: A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
              & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNION @ A_27a @ V2s @ V3t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2Esigma__algebra__def,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
    <=> ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ! [V1c: ( A_27a > $o ) > $o] :
            ( ( ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V1c )
              & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1c @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1c ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2Esigma__def,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1st: ( A_27a > $o ) > $o] :
      ( ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V1st )
      = ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp
        @ ( c_2Epred__set_2EBIGINTER @ ( A_27a > $o )
          @ ( c_2Epred__set_2EGSPEC @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > $o )
            @ ^ [V2s: ( A_27a > $o ) > $o] : ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ $o @ V2s @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1st @ V2s ) @ ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V2s ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2Em__space__def,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1sts: ( A_27a > $o ) > $o,V2mu: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( c_2Emeasure_2Em__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0sp @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V1sts @ V2mu ) ) )
      = V0sp ) ).

thf(thm_2Emeasure_2Emeasurable__sets__def,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1sts: ( A_27a > $o ) > $o,V2mu: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0sp @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V1sts @ V2mu ) ) )
      = V1sts ) ).

thf(thm_2Emeasure_2Emeasure__def,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1sts: ( A_27a > $o ) > $o,V2mu: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( c_2Emeasure_2Emeasure @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0sp @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V1sts @ V2mu ) ) )
      = V2mu ) ).

thf(thm_2Emeasure_2Epositive__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Epositive @ A_27a @ V0m )
    <=> ( ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
          = ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
        & ! [V1s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
           => ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ) ) ).

thf(thm_2Emeasure_2Eadditive__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Eadditive @ A_27a @ V0m )
    <=> ! [V1s: A_27a > $o,V2t: A_27a > $o] :
          ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
            & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
            & ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t ) )
         => ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
            = ( c_2Erealax_2Ereal__add @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2t ) ) ) ) ) ).

thf(thm_2Emeasure_2Ecountably__additive__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0m )
    <=> ! [V1f: tyop_2Enum_2Enum > A_27a > $o] :
          ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
            & ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
                ( ( (~) @ ( V2m = V3n ) )
               => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V1f @ V2m ) @ ( V1f @ V3n ) ) )
            & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
         => ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2Esubadditive__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Esubadditive @ A_27a @ V0m )
    <=> ! [V1s: A_27a > $o,V2t: A_27a > $o] :
          ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
            & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
         => ( c_2Ereal_2Ereal__lte @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2t ) ) ) ) ) ).

thf(thm_2Emeasure_2Ecountably__subadditive__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Ecountably__subadditive @ A_27a @ V0m )
    <=> ! [V1f: tyop_2Enum_2Enum > A_27a > $o] :
          ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
            & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
            & ( c_2Eseq_2Esummable @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) ) )
         => ( c_2Ereal_2Ereal__lte @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) @ ( c_2Eseq_2Esuminf @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) ) ) ) ) ).

thf(thm_2Emeasure_2Eincreasing__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Eincreasing @ A_27a @ V0m )
    <=> ! [V1s: A_27a > $o,V2t: A_27a > $o] :
          ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
            & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
            & ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) )
         => ( c_2Ereal_2Ereal__lte @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2t ) ) ) ) ).

thf(thm_2Emeasure_2Emeasure__space__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2Elambda__system__def,axiom,
    ! [A_27a: $tType,V0gen: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( c_2Emeasure_2Elambda__system @ A_27a @ V0gen @ V1lam )
      = ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
        @ ^ [V2l: A_27a > $o] :
            ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V2l
            @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2l @ ( c_2Emeasure_2Esubsets @ A_27a @ V0gen ) )
              @ ( c_2Ebool_2E_21 @ ( A_27a > $o )
                @ ^ [V3s: A_27a > $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0gen ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ ( V1lam @ ( c_2Epred__set_2EINTER @ A_27a @ V2l @ V3s ) ) @ ( V1lam @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0gen ) @ V2l ) @ V3s ) ) ) @ ( V1lam @ V3s ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2Eouter__measure__space__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Eouter__measure__space @ A_27a @ V0m )
    <=> ( ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eincreasing @ A_27a @ V0m )
        & ( c_2Emeasure_2Ecountably__subadditive @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2Einf__measure__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
      ( ( c_2Emeasure_2Einf__measure @ A_27a @ V0m @ V1s )
      = ( c_2Ereal_2Einf
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
          @ ^ [V2r: tyop_2Erealax_2Ereal] :
              ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2r
              @ ( c_2Ebool_2E_3F @ ( tyop_2Enum_2Enum > A_27a > $o )
                @ ^ [V3f: tyop_2Enum_2Enum > A_27a > $o] :
                    ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V3f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
                    @ ( c_2Ebool_2E_2F_5C
                      @ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                        @ ^ [V4m: tyop_2Enum_2Enum] :
                            ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                            @ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V4m @ V5n ) ) @ ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V3f @ V4m ) @ ( V3f @ V5n ) ) ) ) )
                      @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V3f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) @ ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V3f ) @ V2r ) ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2Eclosed__cdi__def,axiom,
    ! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Eclosed__cdi @ A_27a @ V0p )
    <=> ( ( c_2Emeasure_2Esubset__class @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0p ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) )
        & ! [V1s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0p ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
        & ! [V2f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
              & ( ( V2f @ c_2Enum_2E0 )
                = ( c_2Epred__set_2EEMPTY @ A_27a ) )
              & ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V2f @ V3n ) @ ( V2f @ ( c_2Enum_2ESUC @ V3n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
        & ! [V4f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V4f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
              & ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
                  ( ( (~) @ ( V5m = V6n ) )
                 => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V4f @ V5m ) @ ( V4f @ V6n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) ) ) ) ).

thf(thm_2Emeasure_2Esmallest__closed__cdi__def,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a )
      = ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Espace @ A_27a @ V0a )
        @ ( c_2Epred__set_2EBIGINTER @ ( A_27a > $o )
          @ ( c_2Epred__set_2EGSPEC @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > $o )
            @ ^ [V1b: ( A_27a > $o ) > $o] : ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ $o @ V1b @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) @ V1b ) @ ( c_2Emeasure_2Eclosed__cdi @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V1b ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2Emeasurable__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o )] :
      ( ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V0a @ V1b )
      = ( c_2Epred__set_2EGSPEC @ ( A_27a > A_27b ) @ ( A_27a > A_27b )
        @ ^ [V2f: A_27a > A_27b] :
            ( c_2Epair_2E_2C @ ( A_27a > A_27b ) @ $o @ V2f
            @ ( c_2Ebool_2E_2F_5C @ ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
              @ ( c_2Ebool_2E_2F_5C @ ( c_2Emeasure_2Esigma__algebra @ A_27b @ V1b )
                @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Espace @ A_27b @ V1b ) ) )
                  @ ( c_2Ebool_2E_21 @ ( A_27b > $o )
                    @ ^ [V3s: A_27b > $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3s @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) ) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2Emeasure__preserving__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m2: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V0m1 @ V1m2 )
      = ( c_2Epred__set_2EGSPEC @ ( A_27a > A_27b ) @ ( A_27a > A_27b )
        @ ^ [V2f: A_27a > A_27b] :
            ( c_2Epair_2E_2C @ ( A_27a > A_27b ) @ $o @ V2f
            @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m1 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m1 ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27b @ V1m2 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m2 ) ) ) )
              @ ( c_2Ebool_2E_2F_5C @ ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m1 )
                @ ( c_2Ebool_2E_2F_5C @ ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m2 )
                  @ ( c_2Ebool_2E_21 @ ( A_27b > $o )
                    @ ^ [V3s: A_27b > $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3s @ ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m2 ) ) @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m1 @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m1 ) ) ) @ ( c_2Emeasure_2Emeasure @ A_27b @ V1m2 @ V3s ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2Eindicator__fn__def,axiom,
    ! [A_27a: $tType,V0s: A_27a > $o] :
      ( ( c_2Emeasure_2Eindicator__fn @ A_27a @ V0s )
      = ( ^ [V1x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) @ ( c_2Eextreal_2Eextreal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ) ).

thf(thm_2Emeasure_2Epos__simple__fn__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
      ( ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x )
    <=> ( ! [V5t: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V5t ) )
        & ! [V6t: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V6t @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
           => ( ( V1f @ V6t )
              = ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Enum_2Enum
                @ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ ( V4x @ V7i ) ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V3a @ V7i ) @ V6t ) )
                @ V2s ) ) )
        & ! [V8i: tyop_2Enum_2Enum] :
            ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V8i @ V2s )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V3a @ V8i ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V2s )
        & ! [V9i: tyop_2Enum_2Enum] :
            ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V9i @ V2s )
           => ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V4x @ V9i ) ) )
        & ! [V10i: tyop_2Enum_2Enum,V11j: tyop_2Enum_2Enum] :
            ( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V10i @ V2s )
              & ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V11j @ V2s )
              & ( (~) @ ( V10i = V11j ) ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V3a @ V10i ) @ ( V3a @ V11j ) ) )
        & ( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V3a @ V2s ) )
          = ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) ).

thf(thm_2Emeasure_2Enull__set__def,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
      ( ( c_2Emeasure_2Enull__set @ A_27a @ V0m @ V1s )
    <=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s )
          = ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).

thf(thm_2Emeasure_2EBorel__def,axiom,
    ( c_2Emeasure_2EBorel
    = ( c_2Emeasure_2Esigma @ tyop_2Eextreal_2Eextreal @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal )
      @ ( c_2Epred__set_2EIMAGE @ tyop_2Eextreal_2Eextreal @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ^ [V0a: tyop_2Eextreal_2Eextreal] :
            ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
            @ ^ [V1x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V1x @ ( c_2Eextreal_2Eextreal__lt @ V1x @ V0a ) ) )
        @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) ) ) ).

thf(thm_2Emeasure_2Efn__plus__def,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( c_2Emeasure_2Efn__plus @ A_27a @ V0f )
      = ( ^ [V1x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Eextreal_2Eextreal__lt @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V1x ) ) @ ( V0f @ V1x ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ) ).

thf(thm_2Emeasure_2Efn__minus__def,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( c_2Emeasure_2Efn__minus @ A_27a @ V0f )
      = ( ^ [V1x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Eextreal_2Eextreal @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V1x ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) @ ( c_2Eextreal_2Eextreal__ainv @ ( V0f @ V1x ) ) @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ) ).

thf(thm_2Emeasure_2Efn__abs__def,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( c_2Emeasure_2Efn__abs @ A_27a @ V0f )
      = ( ^ [V1x: A_27a] : ( c_2Eextreal_2Eextreal__abs @ ( V0f @ V1x ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1a: ( A_27a > $o ) > $o,V2p: ( A_27a > $o ) > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V1a ) )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1a @ V2p )
        & ( c_2Emeasure_2Eclosed__cdi @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V2p ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V1a ) ) @ V2p ) ) ).

thf(thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA2,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
     => ! [V1s: A_27a > $o,V2t: A_27a > $o] :
          ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) ) )
            & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) ) ) )
         => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA1,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
     => ! [V1s: A_27a > $o,V2t: A_27a > $o] :
          ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
            & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) ) ) )
         => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2ECLOSED__CDI__INCREASING,axiom,
    ! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Eclosed__cdi @ A_27a @ V0p )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
        & ( ( V1f @ c_2Enum_2E0 )
          = ( c_2Epred__set_2EEMPTY @ A_27a ) )
        & ! [V2n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V1f @ V2n ) @ ( V1f @ ( c_2Enum_2ESUC @ V2n ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) ) ).

thf(thm_2Emeasure_2ECLOSED__CDI__DISJOINT,axiom,
    ! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Eclosed__cdi @ A_27a @ V0p )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
        & ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
            ( ( (~) @ ( V2m = V3n ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V1f @ V2m ) @ ( V1f @ V3n ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) ) ).

thf(thm_2Emeasure_2ECLOSED__CDI__COMPL,axiom,
    ! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Eclosed__cdi @ A_27a @ V0p )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0p ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) ) ).

thf(thm_2Emeasure_2ECLOSED__CDI__DUNION,axiom,
    ! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1s: A_27a > $o,V2t: A_27a > $o] :
      ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) )
        & ( c_2Emeasure_2Eclosed__cdi @ A_27a @ V0p )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) )
        & ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) ) ).

thf(thm_2Emeasure_2ESMALLEST__CLOSED__CDI,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
     => ( ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) ) )
        & ( c_2Emeasure_2Eclosed__cdi @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) )
        & ( c_2Emeasure_2Esubset__class @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2ESPACE__SMALLEST__CLOSED__CDI,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Espace @ A_27a @ ( c_2Emeasure_2Esmallest__closed__cdi @ A_27a @ V0a ) )
      = ( c_2Emeasure_2Espace @ A_27a @ V0a ) ) ).

thf(thm_2Emeasure_2ESPACE,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
      = V0a ) ).

thf(thm_2Emeasure_2EALGEBRA__ALT__INTER,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
    <=> ( ( c_2Emeasure_2Esubset__class @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ! [V1s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
        & ! [V2s: A_27a > $o,V3t: A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
              & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V2s @ V3t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__EMPTY,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__SPACE,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__COMPL,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__UNION,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1s: A_27a > $o,V2t: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__INTER,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1s: A_27a > $o,V2t: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__DIFF,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1s: A_27a > $o,V2t: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__FINITE__UNION,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1c: ( A_27a > $o ) > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V1c )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1c @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1c ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__INTER__SPACE,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V1s )
          = V1s )
        & ( ( c_2Epred__set_2EINTER @ A_27a @ V1s @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
          = V1s ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__EMPTY,axiom,
    ! [A_27a: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0g0 )
        & ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__COMPL,axiom,
    ! [A_27a: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal,V2l: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0g0 )
        & ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2l @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ V2l ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__INTER,axiom,
    ! [A_27a: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal,V2l1: A_27a > $o,V3l2: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0g0 )
        & ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2l1 @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3l2 @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V2l1 @ V3l2 ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__ALGEBRA,axiom,
    ! [A_27a: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0g0 )
        & ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) ) )
     => ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__ADDITIVE,axiom,
    ! [A_27a: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal,V2g: A_27a > $o,V3l1: A_27a > $o,V4l2: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0g0 )
        & ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2g @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) )
        & ( c_2Epred__set_2EDISJOINT @ A_27a @ V3l1 @ V4l2 )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3l1 @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4l2 @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) ) )
     => ( ( V1lam @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V3l1 @ V4l2 ) @ V2g ) )
        = ( c_2Erealax_2Ereal__add @ ( V1lam @ ( c_2Epred__set_2EINTER @ A_27a @ V3l1 @ V2g ) ) @ ( V1lam @ ( c_2Epred__set_2EINTER @ A_27a @ V4l2 @ V2g ) ) ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__ADDITIVE,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal,V2l1: A_27b,V3l2: A_27c] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0g0 )
        & ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) ) )
     => ( c_2Emeasure_2Eadditive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) @ V1lam ) ) ) ) ).

thf(thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE__SUBADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Ecountably__subadditive @ A_27a @ V0m ) )
     => ( c_2Emeasure_2Esubadditive @ A_27a @ V0m ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__ALT,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
    <=> ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ! [V1f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__MONO,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
    <=> ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ! [V1f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
              & ( ( V1f @ c_2Enum_2E0 )
                = ( c_2Epred__set_2EEMPTY @ A_27a ) )
              & ! [V2n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V1f @ V2n ) @ ( V1f @ ( c_2Enum_2ESUC @ V2n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__DISJOINT,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
    <=> ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0a )
        & ! [V1f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
              & ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
                  ( ( (~) @ ( V2m = V3n ) )
                 => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V1f @ V2m ) @ ( V1f @ V3n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2ESUBADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2t: A_27a > $o,V3u: A_27a > $o] :
      ( ( ( c_2Emeasure_2Esubadditive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( V3u
          = ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) )
     => ( c_2Ereal_2Ereal__lte @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V3u ) @ ( c_2Erealax_2Ereal__add @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2t ) ) ) ) ).

thf(thm_2Emeasure_2EADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2t: A_27a > $o,V3u: A_27a > $o] :
      ( ( ( c_2Emeasure_2Eadditive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t )
        & ( V3u
          = ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) )
     => ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V3u )
        = ( c_2Erealax_2Ereal__add @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2t ) ) ) ) ).

thf(thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: tyop_2Enum_2Enum > A_27a > $o,V2s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ecountably__subadditive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Eseq_2Esummable @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) )
        & ( V2s
          = ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Ereal_2Ereal__lte @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2s ) @ ( c_2Eseq_2Esuminf @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) ) ) ) ).

thf(thm_2Emeasure_2EADDITIVE__SUM,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: tyop_2Enum_2Enum > A_27a > $o,V2n: tyop_2Enum_2Enum] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eadditive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
            ( ( (~) @ ( V3m = V4n ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V1f @ V3m ) @ ( V1f @ V4n ) ) ) )
     => ( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V2n ) @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) )
        = ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2Ecount @ V2n ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EINCREASING__ADDITIVE__SUMMABLE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eincreasing @ A_27a @ V0m )
        & ( c_2Emeasure_2Eadditive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
            ( ( (~) @ ( V2m = V3n ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V1f @ V2m ) @ ( V1f @ V3n ) ) ) )
     => ( c_2Eseq_2Esummable @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__POSITIVE,axiom,
    ! [A_27a: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) )
     => ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) @ V1lam ) ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__INCREASING,axiom,
    ! [A_27a: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( c_2Emeasure_2Eincreasing @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) )
     => ( c_2Emeasure_2Eincreasing @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) @ V1lam ) ) ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__SUM,axiom,
    ! [A_27a: $tType,V0g0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal,V2g: A_27a > $o,V3f: tyop_2Enum_2Enum > A_27a > $o,V4n: tyop_2Enum_2Enum] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ V0g0 )
        & ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0g0 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) @ V1lam ) ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2g @ ( c_2Emeasure_2Esubsets @ A_27a @ V0g0 ) )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V3f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0g0 @ V1lam ) ) )
        & ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
            ( ( (~) @ ( V5m = V6n ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V3f @ V5m ) @ ( V3f @ V6n ) ) ) )
     => ( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V4n )
          @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ V1lam
            @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( A_27a > $o )
              @ ^ [V7s: A_27a > $o] : ( c_2Epred__set_2EINTER @ A_27a @ V7s @ V2g )
              @ V3f ) ) )
        = ( V1lam @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V3f @ ( c_2Epred__set_2Ecount @ V4n ) ) ) @ V2g ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__ALGEBRA,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
     => ( c_2Emeasure_2Ealgebra @ A_27a @ V0a ) ) ).

thf(thm_2Emeasure_2EOUTER__MEASURE__SPACE__POSITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Eouter__measure__space @ A_27a @ V0m )
     => ( c_2Emeasure_2Epositive @ A_27a @ V0m ) ) ).

thf(thm_2Emeasure_2ELAMBDA__SYSTEM__CARATHEODORY,axiom,
    ! [A_27a: $tType,V0gsig: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0gsig )
        & ( c_2Emeasure_2Eouter__measure__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0gsig ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0gsig ) @ V1lam ) ) ) )
     => ! [V2f: tyop_2Enum_2Enum > A_27a > $o] :
          ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0gsig @ V1lam ) ) )
            & ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
                ( ( (~) @ ( V3m = V4n ) )
               => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V2f @ V3m ) @ ( V2f @ V4n ) ) ) )
         => ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0gsig @ V1lam ) )
            & ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ V1lam @ V2f ) @ ( V1lam @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2ECARATHEODORY__LEMMA,axiom,
    ! [A_27a: $tType,V0gsig: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1lam: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0gsig )
        & ( c_2Emeasure_2Eouter__measure__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0gsig ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0gsig ) @ V1lam ) ) ) )
     => ( c_2Emeasure_2Emeasure__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0gsig ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ V0gsig @ V1lam ) @ V1lam ) ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__NONEMPTY,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1g: A_27a > $o,V2s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Epred__set_2ESUBSET @ A_27a @ V1g @ V2s ) )
     => ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2s )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
          @ ^ [V3r: tyop_2Erealax_2Ereal] :
              ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3r
              @ ( c_2Ebool_2E_3F @ ( tyop_2Enum_2Enum > A_27a > $o )
                @ ^ [V4f: tyop_2Enum_2Enum > A_27a > $o] :
                    ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V4f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
                    @ ( c_2Ebool_2E_2F_5C
                      @ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                        @ ^ [V5m: tyop_2Enum_2Enum] :
                            ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                            @ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V5m @ V6n ) ) @ ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V4f @ V5m ) @ ( V4f @ V6n ) ) ) ) )
                      @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1g @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) @ ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V4f ) @ V3r ) ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__POS0,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1g: A_27a > $o,V2x: tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x
          @ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
            @ ^ [V3r: tyop_2Erealax_2Ereal] :
                ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3r
                @ ( c_2Ebool_2E_3F @ ( tyop_2Enum_2Enum > A_27a > $o )
                  @ ^ [V4f: tyop_2Enum_2Enum > A_27a > $o] :
                      ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V4f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
                      @ ( c_2Ebool_2E_2F_5C
                        @ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                          @ ^ [V5m: tyop_2Enum_2Enum] :
                              ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                              @ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V5m @ V6n ) ) @ ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V4f @ V5m ) @ ( V4f @ V6n ) ) ) ) )
                        @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1g @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) @ ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V4f ) @ V3r ) ) ) ) ) ) ) ) )
     => ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2x ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__POS,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1g: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Epred__set_2ESUBSET @ A_27a @ V1g @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) )
     => ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Emeasure_2Einf__measure @ A_27a @ V0m @ V1g ) ) ) ).

thf(thm_2Emeasure_2EINCREASING,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2t: A_27a > $o] :
      ( ( ( c_2Emeasure_2Eincreasing @ A_27a @ V0m )
        & ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Ereal_2Ereal__lte @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2t ) ) ) ).

thf(thm_2Emeasure_2EADDITIVE__INCREASING,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eadditive @ A_27a @ V0m ) )
     => ( c_2Emeasure_2Eincreasing @ A_27a @ V0m ) ) ).

thf(thm_2Emeasure_2ECOUNTABLY__ADDITIVE__ADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0m ) )
     => ( c_2Emeasure_2Eadditive @ A_27a @ V0m ) ) ).

thf(thm_2Emeasure_2ECOUNTABLY__ADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
            ( ( (~) @ ( V3m = V4n ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V2f @ V3m ) @ ( V2f @ V4n ) ) )
        & ( V1s
          = ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V2f ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__AGREES,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( ( c_2Emeasure_2Einf__measure @ A_27a @ V0m @ V1s )
        = ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__DOWN,axiom,
    ! [A_27a: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m1: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) ) )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m1 ) )
        & ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m0 )
          = ( c_2Emeasure_2Emeasure @ A_27a @ V1m1 ) )
        & ( c_2Emeasure_2Emeasure__space @ A_27a @ V1m1 ) )
     => ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m0 ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__SIGMA,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1sts: ( A_27a > $o ) > $o] :
      ( ( c_2Emeasure_2Esubset__class @ A_27a @ V0sp @ V1sts )
     => ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V1sts ) ) ) ).

thf(thm_2Emeasure_2EPOW__ALGEBRA,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o] : ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ ( c_2Epred__set_2EPOW @ A_27a @ V0sp ) ) ) ).

thf(thm_2Emeasure_2EPOW__SIGMA__ALGEBRA,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o] : ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ ( c_2Epred__set_2EPOW @ A_27a @ V0sp ) ) ) ).

thf(thm_2Emeasure_2EUNIV__SIGMA__ALGEBRA,axiom,
    ! [A_27a: $tType] : ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ( A_27a > $o ) ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__EMPTY,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m ) )
     => ( ( c_2Emeasure_2Einf__measure @ A_27a @ V0m @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
        = ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__POSITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m ) )
     => ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Emeasure_2Einf__measure @ A_27a @ V0m ) ) ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__INCREASING,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m ) )
     => ( c_2Emeasure_2Eincreasing @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Emeasure_2Einf__measure @ A_27a @ V0m ) ) ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__LE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2x: tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eincreasing @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x
          @ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
            @ ^ [V3r: tyop_2Erealax_2Ereal] :
                ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3r
                @ ( c_2Ebool_2E_3F @ ( tyop_2Enum_2Enum > A_27a > $o )
                  @ ^ [V4f: tyop_2Enum_2Enum > A_27a > $o] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V4f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) @ ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V4f ) @ V3r ) ) ) ) ) ) ) )
     => ( c_2Ereal_2Ereal__lte @ ( c_2Emeasure_2Einf__measure @ A_27a @ V0m @ V1s ) @ V2x ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__CLOSE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2e: tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
        & ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) )
     => ? [V3f: tyop_2Enum_2Enum > A_27a > $o,V4l: tyop_2Erealax_2Ereal] :
          ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V3f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
          & ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V3f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) )
          & ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
              ( ( (~) @ ( V5m = V6n ) )
             => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V3f @ V5m ) @ ( V3f @ V6n ) ) )
          & ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V3f ) @ V4l )
          & ( c_2Ereal_2Ereal__lte @ V4l @ ( c_2Erealax_2Ereal__add @ ( c_2Emeasure_2Einf__measure @ A_27a @ V0m @ V1s ) @ V2e ) ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__COUNTABLY__SUBADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eincreasing @ A_27a @ V0m ) )
     => ( c_2Emeasure_2Ecountably__subadditive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Emeasure_2Einf__measure @ A_27a @ V0m ) ) ) ) ) ).

thf(thm_2Emeasure_2EINF__MEASURE__OUTER,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eincreasing @ A_27a @ V0m ) )
     => ( c_2Emeasure_2Eouter__measure__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Emeasure_2Einf__measure @ A_27a @ V0m ) ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__SUBSET,axiom,
    ! [A_27a: $tType,V0a: ( A_27a > $o ) > $o,V1b: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1b )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V0a @ ( c_2Emeasure_2Esubsets @ A_27a @ V1b ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V1b ) @ V0a ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V1b ) ) ) ).

thf(thm_2Emeasure_2EALGEBRA__SUBSET__LAMBDA__SYSTEM,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eincreasing @ A_27a @ V0m )
        & ( c_2Emeasure_2Eadditive @ A_27a @ V0m ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) @ ( c_2Emeasure_2Elambda__system @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) @ ( c_2Emeasure_2Einf__measure @ A_27a @ V0m ) ) ) ) ).

thf(thm_2Emeasure_2ECARATHEODORY,axiom,
    ! [A_27a: $tType,V0m0: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m0 )
        & ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0m0 ) )
     => ? [V1m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
          ( ! [V2s: A_27a > $o] :
              ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) )
             => ( ( c_2Emeasure_2Emeasure @ A_27a @ V1m @ V2s )
                = ( c_2Emeasure_2Emeasure @ A_27a @ V0m0 @ V2s ) ) )
          & ( ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
            = ( c_2Emeasure_2Esigma @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m0 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m0 ) ) )
          & ( c_2Emeasure_2Emeasure__space @ A_27a @ V1m ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__SUBSET__SUBSETS,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1a: ( A_27a > $o ) > $o] : ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1a @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V1a ) ) ) ).

thf(thm_2Emeasure_2EIN__SIGMA,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1a: ( A_27a > $o ) > $o,V2x: A_27a > $o] :
      ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2x @ V1a )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2x @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA,axiom,
    ! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0p )
    <=> ( ( c_2Emeasure_2Esubset__class @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0p ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) )
        & ! [V1s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0p ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
        & ! [V2c: ( A_27a > $o ) > $o] :
            ( ( ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V2c )
              & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V2c @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V2c ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0p ) ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__COUNTABLE__UNION,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1c: ( A_27a > $o ) > $o] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V1c )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1c @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1c ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__ENUM,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__COMPL,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ V1s ) )
        = ( c_2Ereal_2Ereal__sub @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__PROPERTY,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1p: ( A_27a > $o ) > $o,V2a: ( A_27a > $o ) > $o] :
      ( ( ( c_2Emeasure_2Esubset__class @ A_27a @ V0sp @ V1p )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1p )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V2a @ V1p )
        & ! [V3s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ ( c_2Epred__set_2EINTER @ ( A_27a > $o ) @ V1p @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V0sp @ V3s ) @ V1p ) )
        & ! [V4c: ( A_27a > $o ) > $o] :
            ( ( ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V4c )
              & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V4c @ ( c_2Epred__set_2EINTER @ ( A_27a > $o ) @ V1p @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V4c ) @ V1p ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) @ V1p ) ) ).

thf(thm_2Emeasure_2EMEASURE__EMPTY,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
     => ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
        = ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__SUBSET__MEASURABLE__SETS,axiom,
    ! [A_27a: $tType,V0a: ( A_27a > $o ) > $o,V1m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V1m )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V0a @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) @ V0a ) ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__FN,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
    <=> ( ( c_2Emeasure_2Esubset__class @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ! [V1s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
        & ! [V2f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__FN__DISJOINT,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
    <=> ( ( c_2Emeasure_2Esubset__class @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ! [V1s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
        & ! [V2s: A_27a > $o,V3t: A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
              & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNION @ A_27a @ V2s @ V3t ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
        & ! [V4f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V4f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
              & ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
                  ( ( (~) @ ( V5m = V6n ) )
                 => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V4f @ V5m ) @ ( V4f @ V6n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__PROPERTY__ALT,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1p: ( A_27a > $o ) > $o,V2a: ( A_27a > $o ) > $o] :
      ( ( ( c_2Emeasure_2Esubset__class @ A_27a @ V0sp @ V1p )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1p )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V2a @ V1p )
        & ! [V3s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ ( c_2Epred__set_2EINTER @ ( A_27a > $o ) @ V1p @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V0sp @ V3s ) @ V1p ) )
        & ! [V4f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V4f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINTER @ ( A_27a > $o ) @ V1p @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ V1p ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) @ V1p ) ) ).

thf(thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__WEAK,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1p: ( A_27a > $o ) > $o,V2a: ( A_27a > $o ) > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V2a ) )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V2a @ V1p )
        & ( c_2Emeasure_2Esubset__class @ A_27a @ V0sp @ V1p )
        & ! [V3s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ V1p )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V0sp @ V3s ) @ V1p ) )
        & ! [V4f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V4f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V1p ) )
              & ( ( V4f @ c_2Enum_2E0 )
                = ( c_2Epred__set_2EEMPTY @ A_27a ) )
              & ! [V5n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V4f @ V5n ) @ ( V4f @ ( c_2Enum_2ESUC @ V5n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ V1p ) )
        & ! [V6f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V6f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V1p ) )
              & ! [V7m: tyop_2Enum_2Enum,V8n: tyop_2Enum_2Enum] :
                  ( ( (~) @ ( V7m = V8n ) )
                 => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V6f @ V7m ) @ ( V6f @ V8n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V6f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ V1p ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) @ V1p ) ) ).

thf(thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1p: ( A_27a > $o ) > $o,V2a: ( A_27a > $o ) > $o] :
      ( ( ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V2a ) )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V2a @ V1p )
        & ! [V3s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ ( c_2Epred__set_2EINTER @ ( A_27a > $o ) @ V1p @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V0sp @ V3s ) @ V1p ) )
        & ! [V4f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V4f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINTER @ ( A_27a > $o ) @ V1p @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) ) ) )
              & ( ( V4f @ c_2Enum_2E0 )
                = ( c_2Epred__set_2EEMPTY @ A_27a ) )
              & ! [V5n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V4f @ V5n ) @ ( V4f @ ( c_2Enum_2ESUC @ V5n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V4f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ V1p ) )
        & ! [V6f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V6f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EINTER @ ( A_27a > $o ) @ V1p @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) ) ) )
              & ! [V7m: tyop_2Enum_2Enum,V8n: tyop_2Enum_2Enum] :
                  ( ( (~) @ ( V7m = V8n ) )
                 => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V6f @ V7m ) @ ( V6f @ V8n ) ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V6f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ V1p ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V2a ) ) @ V1p ) ) ).

thf(thm_2Emeasure_2EMEASURE__COUNTABLY__ADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
            ( ( (~) @ ( V3m = V4n ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V2f @ V3m ) @ ( V2f @ V4n ) ) )
        & ( V1s
          = ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
     => ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V2f ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__ADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
     => ( c_2Emeasure_2Eadditive @ A_27a @ V0m ) ) ).

thf(thm_2Emeasure_2EMEASURE__ADDITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2t: A_27a > $o,V3u: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t )
        & ( V3u
          = ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) )
     => ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V3u )
        = ( c_2Erealax_2Ereal__add @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2t ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__COUNTABLE__INCREASING,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( ( V2f @ c_2Enum_2E0 )
          = ( c_2Epred__set_2EEMPTY @ A_27a ) )
        & ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V2f @ V3n ) @ ( V2f @ ( c_2Enum_2ESUC @ V3n ) ) )
        & ( V1s
          = ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
     => ( c_2Eseq_2E_2D_2D_3E @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V2f ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__REDUCE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) ) )
      = V0m ) ).

thf(thm_2Emeasure_2ESPACE__SIGMA,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1a: ( A_27a > $o ) > $o] :
      ( ( c_2Emeasure_2Espace @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V1a ) )
      = V0sp ) ).

thf(thm_2Emeasure_2EMONOTONE__CONVERGENCE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V2f @ V3n ) @ ( V2f @ ( c_2Enum_2ESUC @ V3n ) ) )
        & ( V1s
          = ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
     => ( c_2Eseq_2E_2D_2D_3E @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V2f ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__REDUCE,axiom,
    ! [A_27a: $tType,V0sp: A_27a > $o,V1a: ( A_27a > $o ) > $o] :
      ( ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V1a ) ) )
      = ( c_2Emeasure_2Esigma @ A_27a @ V0sp @ V1a ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__SETS__SUBSET__SPACE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__DIFF__PROPERTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V2f: A_27a > A_27b] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Emeasure_2Esigma__algebra @ A_27b @ V1b )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Espace @ A_27b @ V1b ) ) )
        & ! [V3s: A_27b > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3s @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) )
     => ! [V4s: A_27b > $o] :
          ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V4s @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) )
         => ( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ ( c_2Epred__set_2EDIFF @ A_27b @ ( c_2Emeasure_2Espace @ A_27b @ V1b ) @ V4s ) )
            = ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V4s ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__BIGUNION__PROPERTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V2f: A_27a > A_27b] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Emeasure_2Esigma__algebra @ A_27b @ V1b )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Espace @ A_27b @ V1b ) ) )
        & ! [V3s: A_27b > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3s @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) )
     => ! [V4c: ( A_27b > $o ) > $o] :
          ( ( c_2Epred__set_2ESUBSET @ ( A_27b > $o ) @ V4c @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) )
         => ( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ ( c_2Epred__set_2EBIGUNION @ A_27b @ V4c ) )
            = ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( A_27b > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f ) @ V4c ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASUBABLE__BIGUNION__LEMMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V2f: A_27a > A_27b] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Emeasure_2Esigma__algebra @ A_27b @ V1b )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Espace @ A_27b @ V1b ) ) )
        & ! [V3s: A_27b > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3s @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) )
     => ! [V4c: ( A_27a > $o ) > $o] :
          ( ( ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V4c )
            & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V4c @ ( c_2Epred__set_2EIMAGE @ ( A_27b > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) ) ) )
         => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V4c ) @ ( c_2Epred__set_2EIMAGE @ ( A_27b > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__SIGMA__PREIMAGES,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V2f: A_27a > A_27b] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Emeasure_2Esigma__algebra @ A_27b @ V1b )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Espace @ A_27b @ V1b ) ) )
        & ! [V3s: A_27b > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3s @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) )
     => ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Epred__set_2EIMAGE @ ( A_27b > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V2f: A_27a > A_27b] :
      ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V0a @ V1b ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Emeasure_2Esigma__algebra @ A_27b @ V1b )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Espace @ A_27b @ V1b ) ) )
        & ! [V3s: A_27b > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3s @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__SIGMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V2b: ( A_27b > $o ) > $o,V3sp: A_27b > $o] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Emeasure_2Esubset__class @ A_27b @ V3sp @ V2b )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ V3sp ) )
        & ! [V4s: A_27b > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V4s @ V2b )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V4s ) @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V1a @ ( c_2Emeasure_2Esigma @ A_27b @ V3sp @ V2b ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__SUBSET,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o )] : ( c_2Epred__set_2ESUBSET @ ( A_27a > A_27b ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V0a @ V1b ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V0a @ ( c_2Emeasure_2Esigma @ A_27b @ ( c_2Emeasure_2Espace @ A_27b @ V1b ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V1b ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__LIFT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V2b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V1a @ V2b ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V1a @ ( c_2Emeasure_2Esigma @ A_27b @ ( c_2Emeasure_2Espace @ A_27b @ V2b ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V2b ) ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURE__PRESERVING,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m2: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2f: A_27a > A_27b] :
      ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V0m1 @ V1m2 ) )
    <=> ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m1 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m1 ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27b @ V1m2 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m2 ) ) ) )
        & ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m1 )
        & ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m2 )
        & ! [V3s: A_27b > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3s @ ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m2 ) )
           => ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m1 @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m1 ) ) )
              = ( c_2Emeasure_2Emeasure @ A_27b @ V1m2 @ V3s ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__PRESERVING__LIFT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m2: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2a: ( A_27b > $o ) > $o,V3f: A_27a > A_27b] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m1 )
        & ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m2 )
        & ( ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m2 )
          = ( c_2Emeasure_2Esubsets @ A_27b @ ( c_2Emeasure_2Esigma @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m2 ) @ V2a ) ) )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V3f @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V0m1 @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27b @ V1m2 ) @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ V2a @ ( c_2Emeasure_2Emeasure @ A_27b @ V1m2 ) ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V3f @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V0m1 @ V1m2 ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__PRESERVING__SUBSET,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m2: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2a: ( A_27b > $o ) > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m1 )
        & ( c_2Emeasure_2Emeasure__space @ A_27b @ V1m2 )
        & ( ( c_2Emeasure_2Emeasurable__sets @ A_27b @ V1m2 )
          = ( c_2Emeasure_2Esubsets @ A_27b @ ( c_2Emeasure_2Esigma @ A_27b @ ( c_2Emeasure_2Em__space @ A_27b @ V1m2 ) @ V2a ) ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > A_27b ) @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V0m1 @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27b @ V1m2 ) @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ V2a @ ( c_2Emeasure_2Emeasure @ A_27b @ V1m2 ) ) ) ) @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V0m1 @ V1m2 ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__I,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27a @ V0a @ V0a ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__COMP,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27b > A_27c,V2a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V3b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V4c: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o )] :
      ( ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V2a @ V3b ) )
        & ( c_2Ebool_2EIN @ ( A_27b > A_27c ) @ V1g @ ( c_2Emeasure_2Emeasurable @ A_27b @ A_27c @ V3b @ V4c ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27c ) @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1g @ V0f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27c @ V2a @ V4c ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__COMP__STRONG,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27b > A_27c,V2a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V3b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V4c: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o )] :
      ( ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V2a @ V3b ) )
        & ( c_2Emeasure_2Esigma__algebra @ A_27c @ V4c )
        & ( c_2Ebool_2EIN @ ( A_27b > A_27c ) @ V1g @ ( c_2Epred__set_2EFUNSET @ A_27b @ A_27c @ ( c_2Emeasure_2Espace @ A_27b @ V3b ) @ ( c_2Emeasure_2Espace @ A_27c @ V4c ) ) )
        & ! [V5x: A_27c > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27c > $o ) @ V5x @ ( c_2Emeasure_2Esubsets @ A_27c @ V4c ) )
           => ( c_2Ebool_2EIN @ ( A_27b > $o ) @ ( c_2Epred__set_2EINTER @ A_27b @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27c @ V1g @ V5x ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Emeasure_2Espace @ A_27a @ V2a ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V3b ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27c ) @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1g @ V0f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27c @ V2a @ V4c ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__COMP__STRONGER,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27b > A_27c,V2a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V3b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V4c: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o ),V5t: A_27b > $o] :
      ( ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V2a @ V3b ) )
        & ( c_2Emeasure_2Esigma__algebra @ A_27c @ V4c )
        & ( c_2Ebool_2EIN @ ( A_27b > A_27c ) @ V1g @ ( c_2Epred__set_2EFUNSET @ A_27b @ A_27c @ ( c_2Emeasure_2Espace @ A_27b @ V3b ) @ ( c_2Emeasure_2Espace @ A_27c @ V4c ) ) )
        & ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Emeasure_2Espace @ A_27a @ V2a ) ) @ V5t )
        & ! [V6s: A_27c > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27c > $o ) @ V6s @ ( c_2Emeasure_2Esubsets @ A_27c @ V4c ) )
           => ( c_2Ebool_2EIN @ ( A_27b > $o ) @ ( c_2Epred__set_2EINTER @ A_27b @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27c @ V1g @ V6s ) @ V5t ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V3b ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27c ) @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1g @ V0f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27c @ V2a @ V4c ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__UP__LIFT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0sp: A_27a > $o,V1a: ( A_27a > $o ) > $o,V2b: ( A_27a > $o ) > $o,V3c: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V4f: A_27a > A_27b] :
      ( ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V4f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V1a ) @ V3c ) )
        & ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V2b ) )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1a @ V2b ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V4f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V2b ) @ V3c ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__UP__SUBSET,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0sp: A_27a > $o,V1a: ( A_27a > $o ) > $o,V2b: ( A_27a > $o ) > $o,V3c: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o )] :
      ( ( ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1a @ V2b )
        & ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V2b ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > A_27b ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V1a ) @ V3c ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0sp @ V2b ) @ V3c ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__UP__SIGMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1b: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o )] : ( c_2Epred__set_2ESUBSET @ ( A_27a > A_27b ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V0a @ V1b ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Emeasure_2Esigma @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) @ V1b ) ) ).

thf(thm_2Emeasure_2EMEASURE__PRESERVING__UP__LIFT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: ( A_27a > $o ) > $o,V1m1: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2m2: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V3f: A_27a > A_27b] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V1m1 )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V3f @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V1m1 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V0a @ ( c_2Emeasure_2Emeasure @ A_27a @ V1m1 ) ) ) @ V2m2 ) )
        & ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V1m1 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m1 ) ) )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V0a @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m1 ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V3f @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V1m1 @ V2m2 ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__PRESERVING__UP__SUBSET,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: ( A_27a > $o ) > $o,V1m1: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2m2: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V1m1 )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V0a @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m1 ) )
        & ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V1m1 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m1 ) ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > A_27b ) @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V1m1 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V0a @ ( c_2Emeasure_2Emeasure @ A_27a @ V1m1 ) ) ) @ V2m2 ) @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V1m1 @ V2m2 ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__PRESERVING__UP__SIGMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1m2: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V2a: ( A_27a > $o ) > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m1 )
        & ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m1 )
          = ( c_2Emeasure_2Esubsets @ A_27a @ ( c_2Emeasure_2Esigma @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m1 ) @ V2a ) ) ) )
     => ( c_2Epred__set_2ESUBSET @ ( A_27a > A_27b ) @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m1 ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V2a @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m1 ) ) ) @ V1m2 ) @ ( c_2Emeasure_2Emeasure__preserving @ A_27a @ A_27b @ V0m1 @ V1m2 ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__PROD__SIGMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1a1: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V2a2: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o ),V3f: A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27c )] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epair_2EFST @ A_27b @ A_27c ) @ V3f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ V0a @ V1a1 ) )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27c ) @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epair_2ESND @ A_27b @ A_27c ) @ V3f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27c @ V0a @ V2a2 ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V3f @ ( c_2Emeasure_2Emeasurable @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V0a @ ( c_2Emeasure_2Esigma @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Emeasure_2Espace @ A_27b @ V1a1 ) @ ( c_2Emeasure_2Espace @ A_27c @ V2a2 ) ) @ ( c_2Eutil__prob_2Eprod__sets @ A_27b @ A_27c @ ( c_2Emeasure_2Esubsets @ A_27b @ V1a1 ) @ ( c_2Emeasure_2Esubsets @ A_27c @ V2a2 ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__RANGE__REDUCE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > A_27b,V2s: A_27b > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ V2s @ ( c_2Epred__set_2EPOW @ A_27b @ V2s ) ) ) )
        & ( (~)
          @ ( V2s
            = ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ ( c_2Epred__set_2EINTER @ A_27b @ V2s @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) @ ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Epred__set_2EINTER @ A_27b @ V2s @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__POW__TO__POW,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > A_27b] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m )
          = ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ ( c_2Epred__set_2EUNIV @ A_27b ) @ ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURABLE__POW__TO__POW__IMAGE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > A_27b] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m )
          = ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) @ ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__SUBSET,axiom,
    ! [A_27a: $tType,V0s: A_27a > $o,V1s_27: A_27a > $o,V2m: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s_27 @ V0s )
        & ( c_2Emeasure_2Emeasure__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0s @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s ) @ V2m ) ) ) )
     => ( c_2Emeasure_2Emeasure__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V1s_27 @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ V1s_27 ) @ V2m ) ) ) ) ).

thf(thm_2Emeasure_2ESTRONG__MEASURE__SPACE__SUBSET,axiom,
    ! [A_27a: $tType,V0s: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s_27: A_27a > $o] :
      ( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s_27 @ ( c_2Emeasure_2Em__space @ A_27a @ V0s ) )
        & ( c_2Emeasure_2Emeasure__space @ A_27a @ V0s )
        & ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Epred__set_2EPOW @ A_27a @ V1s_27 ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0s ) ) )
     => ( c_2Emeasure_2Emeasure__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V1s_27 @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ A_27a @ V1s_27 ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0s ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__REAL__SUM__IMAGE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ! [V2x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
     => ( ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s )
        = ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
          @ ^ [V3x: A_27a] : ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EINSERT @ A_27a @ V3x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
          @ V1s ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__POW,axiom,
    ! [A_27a: $tType,V0s: A_27a > $o] :
      ( ( c_2Emeasure_2Esigma @ A_27a @ V0s @ ( c_2Epred__set_2EPOW @ A_27a @ V0s ) )
      = ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ V0s @ ( c_2Epred__set_2EPOW @ A_27a @ V0s ) ) ) ).

thf(thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces,axiom,
    ! [A_27a: $tType,V0s: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1m: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0s )
        & ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0s ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0s ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0s ) @ V1m ) ) )
        & ( c_2Emeasure_2Eadditive @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0s ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0s ) @ V1m ) ) ) )
     => ( c_2Emeasure_2Emeasure__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27a @ V0s ) @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0s ) @ V1m ) ) ) ) ).

thf(thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces2,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) )
        & ( c_2Emeasure_2Epositive @ A_27a @ V0m )
        & ( c_2Emeasure_2Eadditive @ A_27a @ V0m ) )
     => ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m ) ) ).

thf(thm_2Emeasure_2EIMAGE__SING,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: A_27a] :
      ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
      = ( c_2Epred__set_2EINSERT @ A_27b @ ( V0f @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).

thf(thm_2Emeasure_2ESUBSET__BIGINTER,axiom,
    ! [A_27a: $tType,V0X: A_27a > $o,V1P: ( A_27a > $o ) > $o] :
      ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0X @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1P ) )
    <=> ! [V2Y: A_27a > $o] :
          ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2Y @ V1P )
         => ( c_2Epred__set_2ESUBSET @ A_27a @ V0X @ V2Y ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__INCREASING,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
     => ( c_2Emeasure_2Eincreasing @ A_27a @ V0m ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__POSITIVE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
     => ( c_2Emeasure_2Epositive @ A_27a @ V0m ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__INTER,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2t: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__UNION,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2t: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__DIFF,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2t: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__COMPL__SUBSET,axiom,
    ! [A_27a: $tType,V0t: A_27a > $o,V1m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V1m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V0t @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ( c_2Epred__set_2ESUBSET @ A_27a @ V0t @ V2s ) )
     => ( ( c_2Emeasure_2Emeasure @ A_27a @ V1m @ ( c_2Epred__set_2EDIFF @ A_27a @ V2s @ V0t ) )
        = ( c_2Ereal_2Ereal__sub @ ( c_2Emeasure_2Emeasure @ A_27a @ V1m @ V2s ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V1m @ V0t ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__BIGUNION,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ! [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V1s @ V2n ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1s @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__IN__MSPACE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1A: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ! [V2x: A_27a] :
          ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1A )
         => ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__SUBSET__MSPACE,axiom,
    ! [A_27a: $tType,V0A: A_27a > $o,V1m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V1m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V0A @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) ) )
     => ( c_2Epred__set_2ESUBSET @ A_27a @ V0A @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__EMPTY__MEASURABLE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__MSPACE__MEASURABLE,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__FN__BIGINTER,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
     => ( ( c_2Emeasure_2Esubset__class @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ! [V1s: A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V1s ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
        & ! [V2f: tyop_2Enum_2Enum > A_27a > $o] :
            ( ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__BIGINTER,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ! [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V1s @ V2n ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1s @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) ) ).

thf(thm_2Emeasure_2EMONOTONE__CONVERGENCE2,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ! [V2n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V1f @ V2n ) @ ( V1f @ ( c_2Enum_2ESUC @ V2n ) ) ) )
     => ( c_2Eseq_2E_2D_2D_3E @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V2f @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V2f @ V3n ) )
        & ( V1s
          = ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) )
     => ( c_2Eseq_2E_2D_2D_3E @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V2f ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V1s ) ) ) ).

thf(thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER2,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
        & ! [V2n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V1f @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V1f @ V2n ) ) )
     => ( c_2Eseq_2E_2D_2D_3E @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) @ V1f ) @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__RESTRICTED,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) )
     => ( c_2Emeasure_2Emeasure__space @ A_27a
        @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V1s
          @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal )
            @ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27a > $o )
              @ ^ [V2t: A_27a > $o] : ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t )
              @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) )
            @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m ) ) ) ) ) ).

thf(thm_2Emeasure_2EMEASURE__SPACE__CMUL,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1c: tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1c ) )
     => ( c_2Emeasure_2Emeasure__space @ A_27a
        @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m )
          @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m )
            @ ^ [V2a: A_27a > $o] : ( c_2Erealax_2Ereal__mul @ V1c @ ( c_2Emeasure_2Emeasure @ A_27a @ V0m @ V2a ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EBIGUNION__IMAGE__Q,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: tyop_2Eextreal_2Eextreal > A_27a > $o] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > A_27a > $o ) @ V1f @ ( c_2Epred__set_2EFUNSET @ tyop_2Eextreal_2Eextreal @ ( A_27a > $o ) @ c_2Eextreal_2EQ__set @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Eextreal_2Eextreal @ ( A_27a > $o ) @ V1f @ c_2Eextreal_2EQ__set ) ) @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) ) ).

thf(thm_2Emeasure_2ESIGMA__ALGEBRA__BOREL,axiom,
    c_2Emeasure_2Esigma__algebra @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ).

thf(thm_2Emeasure_2EMEASURABLE__BOREL,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V0f
                @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
                  @ ^ [V3x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V3x @ ( c_2Eextreal_2Eextreal__lt @ V3x @ V2c ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V3x ) @ V2c ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__NEGINF,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o )
        @ ( c_2Epred__set_2EINTER @ A_27a
          @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
            @ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V0f @ V2x ) @ c_2Eextreal_2ENegInf ) ) )
          @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
        @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT1,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Eextreal_2Eextreal__le @ V2c @ ( V0f @ V3x ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT2,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V3x ) @ V2c ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT3,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Eextreal_2Eextreal__lt @ V2c @ ( V0f @ V3x ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT4,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal,V3d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V2c @ ( V0f @ V4x ) ) @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V4x ) @ V3d ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT5,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal,V3d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V2c @ ( V0f @ V4x ) ) @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V4x ) @ V3d ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT6,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal,V3d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V2c @ ( V0f @ V4x ) ) @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V4x ) @ V3d ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT7,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
     => ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal,V3d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V2c @ ( V0f @ V4x ) ) @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V4x ) @ V3d ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT8,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
     => ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V0f @ V3x ) @ V2c ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT9,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
     => ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) @ ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) )
        & ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V0f @ V3x ) @ V2c ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
     => ( ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V3x ) @ V2c ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V4c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Eextreal_2Eextreal__le @ V4c @ ( V0f @ V5x ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V6c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V7x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V7x @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V7x ) @ V6c ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V8c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V9x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V9x @ ( c_2Eextreal_2Eextreal__lt @ V8c @ ( V0f @ V9x ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V10c: tyop_2Eextreal_2Eextreal,V11d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V12x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V12x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V10c @ ( V0f @ V12x ) ) @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V12x ) @ V11d ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V13c: tyop_2Eextreal_2Eextreal,V14d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V15x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V15x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V13c @ ( V0f @ V15x ) ) @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V15x ) @ V14d ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V16c: tyop_2Eextreal_2Eextreal,V17d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V18x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V18x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V16c @ ( V0f @ V18x ) ) @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V18x ) @ V17d ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V19c: tyop_2Eextreal_2Eextreal,V20d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V21x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V21x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V19c @ ( V0f @ V21x ) ) @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V21x ) @ V20d ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V22c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V23x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V23x @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V0f @ V23x ) @ V22c ) ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) )
        & ! [V24c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V25x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V25x @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V0f @ V25x ) @ V24c ) ) )
              @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
            @ ( c_2Emeasure_2Esubsets @ A_27a @ V1a ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL__MEASURE,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) ) @ c_2Emeasure_2EBorel ) )
     => ( ! [V2c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V3x ) @ V2c ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V4c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Eextreal_2Eextreal__le @ V4c @ ( V0f @ V5x ) ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V6c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V7x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V7x @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V7x ) @ V6c ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V8c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V9x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V9x @ ( c_2Eextreal_2Eextreal__lt @ V8c @ ( V0f @ V9x ) ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V10c: tyop_2Eextreal_2Eextreal,V11d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V12x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V12x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V10c @ ( V0f @ V12x ) ) @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V12x ) @ V11d ) ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V13c: tyop_2Eextreal_2Eextreal,V14d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V15x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V15x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V13c @ ( V0f @ V15x ) ) @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V15x ) @ V14d ) ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V16c: tyop_2Eextreal_2Eextreal,V17d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V18x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V18x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V16c @ ( V0f @ V18x ) ) @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V18x ) @ V17d ) ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V19c: tyop_2Eextreal_2Eextreal,V20d: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V21x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V21x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V19c @ ( V0f @ V21x ) ) @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V21x ) @ V20d ) ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V22c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V23x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V23x @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V0f @ V23x ) @ V22c ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) )
        & ! [V24c: tyop_2Eextreal_2Eextreal] :
            ( c_2Ebool_2EIN @ ( A_27a > $o )
            @ ( c_2Epred__set_2EINTER @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
                @ ^ [V25x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V25x @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ ( V0f @ V25x ) @ V24c ) ) ) )
              @ ( c_2Emeasure_2Em__space @ A_27a @ V1m ) )
            @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V1m ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__LT,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1g: A_27a > tyop_2Eextreal_2Eextreal,V2a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V2a @ c_2Emeasure_2EBorel ) )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V2a @ c_2Emeasure_2EBorel ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o )
        @ ( c_2Epred__set_2EINTER @ A_27a
          @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
            @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Eextreal_2Eextreal__lt @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) ) )
          @ ( c_2Emeasure_2Espace @ A_27a @ V2a ) )
        @ ( c_2Emeasure_2Esubsets @ A_27a @ V2a ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__LE,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1g: A_27a > tyop_2Eextreal_2Eextreal,V2a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V2a @ c_2Emeasure_2EBorel ) )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V2a @ c_2Emeasure_2EBorel ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > $o )
        @ ( c_2Epred__set_2EINTER @ A_27a
          @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
            @ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Eextreal_2Eextreal__le @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) ) )
          @ ( c_2Emeasure_2Espace @ A_27a @ V2a ) )
        @ ( c_2Emeasure_2Esubsets @ A_27a @ V2a ) ) ) ).

thf(thm_2Emeasure_2ESPACE__BOREL,axiom,
    ( ( c_2Emeasure_2Espace @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel )
    = ( c_2Epred__set_2EUNIV @ tyop_2Eextreal_2Eextreal ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS1,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V1x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V1x @ ( c_2Eextreal_2Eextreal__lt @ V1x @ V0c ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS2,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V1x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V1x @ ( c_2Eextreal_2Eextreal__le @ V0c @ V1x ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS3,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V1x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V1x @ ( c_2Eextreal_2Eextreal__le @ V1x @ V0c ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS4,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V1x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V1x @ ( c_2Eextreal_2Eextreal__lt @ V0c @ V1x ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS5,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal,V1d: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V2x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V0c @ V2x ) @ ( c_2Eextreal_2Eextreal__lt @ V2x @ V1d ) ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS6,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal,V1d: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V2x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V0c @ V2x ) @ ( c_2Eextreal_2Eextreal__le @ V2x @ V1d ) ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS7,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal,V1d: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V2x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V0c @ V2x ) @ ( c_2Eextreal_2Eextreal__le @ V2x @ V1d ) ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS8,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal,V1d: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V2x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V0c @ V2x ) @ ( c_2Eextreal_2Eextreal__lt @ V2x @ V1d ) ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS9,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Eextreal_2Eextreal @ V0c @ ( c_2Epred__set_2EEMPTY @ tyop_2Eextreal_2Eextreal ) ) @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS10,axiom,
    ! [V0c: tyop_2Eextreal_2Eextreal] :
      ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
      @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
        @ ^ [V1x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V1x @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ V1x @ V0c ) ) ) )
      @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ).

thf(thm_2Emeasure_2EBOREL__MEASURABLE__SETS,axiom,
    ( ! [V0c: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V1x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V1x @ ( c_2Eextreal_2Eextreal__lt @ V1x @ V0c ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V2c: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V3x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V3x @ ( c_2Eextreal_2Eextreal__le @ V2c @ V3x ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V4c: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V5x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V5x @ ( c_2Eextreal_2Eextreal__lt @ V4c @ V5x ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V6c: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V7x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V7x @ ( c_2Eextreal_2Eextreal__le @ V7x @ V6c ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V8c: tyop_2Eextreal_2Eextreal,V9d: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V10x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V10x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V8c @ V10x ) @ ( c_2Eextreal_2Eextreal__lt @ V10x @ V9d ) ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V11c: tyop_2Eextreal_2Eextreal,V12d: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V13x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V13x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V11c @ V13x ) @ ( c_2Eextreal_2Eextreal__lt @ V13x @ V12d ) ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V14c: tyop_2Eextreal_2Eextreal,V15d: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V16x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V16x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__lt @ V14c @ V16x ) @ ( c_2Eextreal_2Eextreal__le @ V16x @ V15d ) ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V17c: tyop_2Eextreal_2Eextreal,V18d: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V19x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V19x @ ( c_2Ebool_2E_2F_5C @ ( c_2Eextreal_2Eextreal__le @ V17c @ V19x ) @ ( c_2Eextreal_2Eextreal__le @ V19x @ V18d ) ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V20c: tyop_2Eextreal_2Eextreal] : ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o ) @ ( c_2Epred__set_2EINSERT @ tyop_2Eextreal_2Eextreal @ V20c @ ( c_2Epred__set_2EEMPTY @ tyop_2Eextreal_2Eextreal ) ) @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) )
    & ! [V21c: tyop_2Eextreal_2Eextreal] :
        ( c_2Ebool_2EIN @ ( tyop_2Eextreal_2Eextreal > $o )
        @ ( c_2Epred__set_2EGSPEC @ tyop_2Eextreal_2Eextreal @ tyop_2Eextreal_2Eextreal
          @ ^ [V22x: tyop_2Eextreal_2Eextreal] : ( c_2Epair_2E_2C @ tyop_2Eextreal_2Eextreal @ $o @ V22x @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal @ V22x @ V21c ) ) ) )
        @ ( c_2Emeasure_2Esubsets @ tyop_2Eextreal_2Eextreal @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__CONST,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1k: tyop_2Eextreal_2Eextreal,V2f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ! [V3x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V2f @ V3x )
              = V1k ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__INDICATOR,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1A: A_27a > $o,V2f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1A @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) )
        & ! [V3x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V2f @ V3x )
              = ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1A @ V3x ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__CMUL,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3z: tyop_2Erealax_2Ereal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ! [V4x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V2g @ V4x )
              = ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V3z ) @ ( V1f @ V4x ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ABS,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ! [V3x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V2g @ V3x )
              = ( c_2Eextreal_2Eextreal__abs @ ( V1f @ V3x ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__SQR,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ! [V3x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V2g @ V3x )
              = ( c_2Eextreal_2Eextreal__pow @ ( V1f @ V3x ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__ADD,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3h: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ! [V4x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V3h @ V4x )
              = ( c_2Eextreal_2Eextreal__add @ ( V1f @ V4x ) @ ( V2g @ V4x ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V3h @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__SUB,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3h: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ! [V4x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V3h @ V4x )
              = ( c_2Eextreal_2Eextreal__sub @ ( V1f @ V4x ) @ ( V2g @ V4x ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V3h @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3h: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ! [V4x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( (~)
                @ ( ( V1f @ V4x )
                  = c_2Eextreal_2ENegInf ) )
              & ( (~)
                @ ( ( V1f @ V4x )
                  = c_2Eextreal_2EPosInf ) )
              & ( (~)
                @ ( ( V2g @ V4x )
                  = c_2Eextreal_2ENegInf ) )
              & ( (~)
                @ ( ( V2g @ V4x )
                  = c_2Eextreal_2EPosInf ) ) ) )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ! [V5x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V3h @ V5x )
              = ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V5x ) @ ( V2g @ V5x ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V3h @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__SUM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27b > A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal,V3s: A_27b > $o] :
      ( ( ( c_2Epred__set_2EFINITE @ A_27b @ V3s )
        & ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ! [V4i: A_27b] :
            ( ( c_2Ebool_2EIN @ A_27b @ V4i @ V3s )
           => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V1f @ V4i ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) )
        & ! [V5x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) )
           => ( ( V2g @ V5x )
              = ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ A_27b
                @ ^ [V6i: A_27b] : ( V1f @ V6i @ V5x )
                @ V3s ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__CMUL__INDICATOR,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1z: tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal )
        @ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V1z ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V2s @ V3x ) )
        @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL__INDICATOR,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2s: A_27a > $o] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Emeasure_2Esubsets @ A_27a @ V0a ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal )
        @ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V3x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V2s @ V3x ) )
        @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL__INDICATOR__EQ,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
     => ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        = ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal )
          @ ^ [V2x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1f @ V2x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Emeasure_2Espace @ A_27a @ V0a ) @ V2x ) )
          @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__POS__SIMPLE__FN,axiom,
    ! [A_27a: $tType,V0m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0m )
        & ? [V2s: tyop_2Enum_2Enum > $o,V3a: tyop_2Enum_2Enum > A_27a > $o,V4x: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] : ( c_2Emeasure_2Epos__simple__fn @ A_27a @ V0m @ V1f @ V2s @ V3a @ V4x ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Emeasure_2Em__space @ A_27a @ V0m ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V0m ) ) @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__POW,axiom,
    ! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V2f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V1a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) )
        & ! [V3x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Emeasure_2Espace @ A_27a @ V1a ) )
           => ( ( (~)
                @ ( ( V2f @ V3x )
                  = c_2Eextreal_2ENegInf ) )
              & ( (~)
                @ ( ( V2f @ V3x )
                  = c_2Eextreal_2EPosInf ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal )
        @ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__pow @ ( V2f @ V4x ) @ V0n )
        @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__MAX,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal,V2g: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V0a )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V2g @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal )
        @ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__max @ ( V1f @ V3x ) @ ( V2g @ V3x ) )
        @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__MONO__SUP,axiom,
    ! [A_27a: $tType,V0fn: tyop_2Enum_2Enum > A_27a > tyop_2Eextreal_2Eextreal,V1f: A_27a > tyop_2Eextreal_2Eextreal,V2a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o )] :
      ( ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ V2a )
        & ! [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( V0fn @ V3n ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V2a @ c_2Emeasure_2EBorel ) )
        & ! [V4n: tyop_2Enum_2Enum,V5x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Emeasure_2Espace @ A_27a @ V2a ) )
           => ( c_2Eextreal_2Eextreal__le @ ( V0fn @ V4n @ V5x ) @ ( V0fn @ ( c_2Enum_2ESUC @ V4n ) @ V5x ) ) )
        & ! [V6x: A_27a] :
            ( ( c_2Ebool_2EIN @ A_27a @ V6x @ ( c_2Emeasure_2Espace @ A_27a @ V2a ) )
           => ( ( V1f @ V6x )
              = ( c_2Eextreal_2Eextreal__sup
                @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Eextreal_2Eextreal
                  @ ^ [V7n: tyop_2Enum_2Enum] : ( V0fn @ V7n @ V6x )
                  @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V2a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EFN__PLUS__POS,axiom,
    ! [A_27a: $tType,V0g: A_27a > tyop_2Eextreal_2Eextreal,V1x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( c_2Emeasure_2Efn__plus @ A_27a @ V0g @ V1x ) ) ).

thf(thm_2Emeasure_2EFN__MINUS__POS,axiom,
    ! [A_27a: $tType,V0g: A_27a > tyop_2Eextreal_2Eextreal,V1x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( c_2Emeasure_2Efn__minus @ A_27a @ V0g @ V1x ) ) ).

thf(thm_2Emeasure_2EFN__PLUS__POS__ID,axiom,
    ! [A_27a: $tType,V0g: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ! [V1x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V0g @ V1x ) )
     => ( ( c_2Emeasure_2Efn__plus @ A_27a @ V0g )
        = V0g ) ) ).

thf(thm_2Emeasure_2EFN__MINUS__POS__ZERO,axiom,
    ! [A_27a: $tType,V0g: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ! [V1x: A_27a] : ( c_2Eextreal_2Eextreal__le @ ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) @ ( V0g @ V1x ) )
     => ( ( c_2Emeasure_2Efn__minus @ A_27a @ V0g )
        = ( ^ [V2x: A_27a] : ( c_2Eextreal_2Eextreal__of__num @ c_2Enum_2E0 ) ) ) ) ).

thf(thm_2Emeasure_2EFN__PLUS__CMUL,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1c: tyop_2Erealax_2Ereal] :
      ( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1c )
       => ( ( c_2Emeasure_2Efn__plus @ A_27a
            @ ^ [V2x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V1c ) @ ( V0f @ V2x ) ) )
          = ( ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V1c ) @ ( c_2Emeasure_2Efn__plus @ A_27a @ V0f @ V3x ) ) ) ) )
      & ( ( c_2Ereal_2Ereal__lte @ V1c @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
       => ( ( c_2Emeasure_2Efn__plus @ A_27a
            @ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V1c ) @ ( V0f @ V4x ) ) )
          = ( ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2Eextreal__ainv @ ( c_2Eextreal_2ENormal @ V1c ) ) @ ( c_2Emeasure_2Efn__minus @ A_27a @ V0f @ V5x ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EFN__MINUS__CMUL,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1c: tyop_2Erealax_2Ereal] :
      ( ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1c )
       => ( ( c_2Emeasure_2Efn__minus @ A_27a
            @ ^ [V2x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V1c ) @ ( V0f @ V2x ) ) )
          = ( ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V1c ) @ ( c_2Emeasure_2Efn__minus @ A_27a @ V0f @ V3x ) ) ) ) )
      & ( ( c_2Ereal_2Ereal__lte @ V1c @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
       => ( ( c_2Emeasure_2Efn__minus @ A_27a
            @ ^ [V4x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2ENormal @ V1c ) @ ( V0f @ V4x ) ) )
          = ( ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Eextreal_2Eextreal__ainv @ ( c_2Eextreal_2ENormal @ V1c ) ) @ ( c_2Emeasure_2Efn__plus @ A_27a @ V0f @ V5x ) ) ) ) ) ) ).

thf(thm_2Emeasure_2EFN__PLUS__ADD__LE,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1g: A_27a > tyop_2Eextreal_2Eextreal,V2x: A_27a] :
      ( c_2Eextreal_2Eextreal__le
      @ ( c_2Emeasure_2Efn__plus @ A_27a
        @ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
        @ V2x )
      @ ( c_2Eextreal_2Eextreal__add @ ( c_2Emeasure_2Efn__plus @ A_27a @ V0f @ V2x ) @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1g @ V2x ) ) ) ).

thf(thm_2Emeasure_2EFN__MINUS__ADD__LE,axiom,
    ! [A_27a: $tType,V0f: A_27a > tyop_2Eextreal_2Eextreal,V1g: A_27a > tyop_2Eextreal_2Eextreal,V2x: A_27a] :
      ( c_2Eextreal_2Eextreal__le
      @ ( c_2Emeasure_2Efn__minus @ A_27a
        @ ^ [V3x: A_27a] : ( c_2Eextreal_2Eextreal__add @ ( V0f @ V3x ) @ ( V1g @ V3x ) )
        @ V2x )
      @ ( c_2Eextreal_2Eextreal__add @ ( c_2Emeasure_2Efn__minus @ A_27a @ V0f @ V2x ) @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1g @ V2x ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__FN__PLUS,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__FN__MINUS,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
     => ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ).

thf(thm_2Emeasure_2EIN__MEASURABLE__BOREL__PLUS__MINUS,axiom,
    ! [A_27a: $tType,V0a: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ),V1f: A_27a > tyop_2Eextreal_2Eextreal] :
      ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V1f @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
    <=> ( ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( c_2Emeasure_2Efn__plus @ A_27a @ V1f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) )
        & ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ ( c_2Emeasure_2Efn__minus @ A_27a @ V1f ) @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ V0a @ c_2Emeasure_2EBorel ) ) ) ) ).

thf(thm_2Emeasure_2EINDICATOR__FN__MUL__INTER,axiom,
    ! [A_27a: $tType,V0A: A_27a > $o,V1B: A_27a > $o] :
      ( ( ^ [V2t: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V0A @ V2t ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1B @ V2t ) ) )
      = ( ^ [V3t: A_27a] : ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0A @ V1B ) @ V3t ) ) ) ).

thf(thm_2Emeasure_2Eindicator__fn__split,axiom,
    ! [A_27a: $tType,V0r: tyop_2Enum_2Enum > $o,V1s: A_27a > $o,V2b: tyop_2Enum_2Enum > A_27a > $o] :
      ( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0r )
        & ( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2b @ V0r ) )
          = V1s )
        & ! [V3i: tyop_2Enum_2Enum,V4j: tyop_2Enum_2Enum] :
            ( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3i @ V0r )
              & ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4j @ V0r )
              & ( (~) @ ( V3i = V4j ) ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V2b @ V3i ) @ ( V2b @ V4j ) ) ) )
     => ! [V5a: A_27a > $o] :
          ( ( c_2Epred__set_2ESUBSET @ A_27a @ V5a @ V1s )
         => ( ( c_2Emeasure_2Eindicator__fn @ A_27a @ V5a )
            = ( ^ [V6x: A_27a] :
                  ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Enum_2Enum
                  @ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V5a @ ( V2b @ V7i ) ) @ V6x )
                  @ V0r ) ) ) ) ) ).

thf(thm_2Emeasure_2Eindicator__fn__suminf,axiom,
    ! [A_27a: $tType,V0a: tyop_2Enum_2Enum > A_27a > $o,V1x: A_27a] :
      ( ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
          ( ( (~) @ ( V2m = V3n ) )
         => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V0a @ V2m ) @ ( V0a @ V3n ) ) )
     => ( ( c_2Eextreal_2Eext__suminf
          @ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( V0a @ V4i ) @ V1x ) )
        = ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V0a @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) @ V1x ) ) ) ).

thf(thm_2Emeasure_2Emeasure__split,axiom,
    ! [A_27a: $tType,V0r: tyop_2Enum_2Enum > $o,V1b: tyop_2Enum_2Enum > A_27a > $o,V2m: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
      ( ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V2m )
        & ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0r )
        & ( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1b @ V0r ) )
          = ( c_2Emeasure_2Em__space @ A_27a @ V2m ) )
        & ! [V3i: tyop_2Enum_2Enum,V4j: tyop_2Enum_2Enum] :
            ( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3i @ V0r )
              & ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4j @ V0r )
              & ( (~) @ ( V3i = V4j ) ) )
           => ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V1b @ V3i ) @ ( V1b @ V4j ) ) )
        & ! [V5i: tyop_2Enum_2Enum] :
            ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5i @ V0r )
           => ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( V1b @ V5i ) @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V2m ) ) ) )
     => ! [V6a: A_27a > $o] :
          ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V6a @ ( c_2Emeasure_2Emeasurable__sets @ A_27a @ V2m ) )
         => ( ( c_2Emeasure_2Emeasure @ A_27a @ V2m @ V6a )
            = ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ tyop_2Enum_2Enum
              @ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Emeasure_2Emeasure @ A_27a @ V2m @ ( c_2Epred__set_2EINTER @ A_27a @ V6a @ ( V1b @ V7i ) ) )
              @ V0r ) ) ) ) ).

%------------------------------------------------------------------------------