ITP001 Axioms: ITP150^7.ax
%------------------------------------------------------------------------------
% File : ITP150^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : probability.ax [Gau19]
% : HL4150^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 214 ( 21 unt; 103 typ; 0 def)
% Number of atoms : 837 ( 110 equ; 14 cnn)
% Maximal formula atoms : 30 ( 3 avg)
% Number of connectives : 3786 ( 14 ~; 5 |; 158 &;3486 @)
% ( 21 <=>; 102 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 11 avg;3486 nst)
% Number of types : 5 ( 4 usr)
% Number of type conns : 1435 (1435 >; 0 *; 0 +; 0 <<)
% Number of symbols : 101 ( 99 usr; 6 con; 0-9 aty)
% Number of variables : 620 ( 29 ^ 493 !; 1 ?; 620 :)
% ( 97 !>; 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_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_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_2Emeasure_2EBorel,type,
c_2Emeasure_2EBorel: tyop_2Epair_2Eprod @ ( tyop_2Eextreal_2Eextreal > $o ) @ ( ( tyop_2Eextreal_2Eextreal > $o ) > $o ) ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > 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_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_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_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_2Eprobability_2Econditional__distribution,type,
c_2Eprobability_2Econditional__distribution:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > A_27b ) > ( A_27a > A_27c ) > ( A_27b > $o ) > ( A_27c > $o ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Eprobability_2Econditional__expectation,type,
c_2Eprobability_2Econditional__expectation:
!>[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 ) > ( ( A_27a > $o ) > $o ) > A_27a > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Eprobability_2Econditional__prob,type,
c_2Eprobability_2Econditional__prob:
!>[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 ) > ( ( A_27a > $o ) > $o ) > A_27a > tyop_2Eextreal_2Eextreal ) ).
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_2Eprobability_2Edistribution,type,
c_2Eprobability_2Edistribution:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27b > A_27a ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Eprobability_2Eevents,type,
c_2Eprobability_2Eevents:
!>[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_2Eprobability_2Eexpectation,type,
c_2Eprobability_2Eexpectation:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Eextreal_2Eextreal__mul,type,
c_2Eextreal_2Eextreal__mul: tyop_2Eextreal_2Eextreal > tyop_2Eextreal_2Eextreal > 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_2Eprobability_2Eindep,type,
c_2Eprobability_2Eindep:
!>[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 ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eprobability_2Eindep__families,type,
c_2Eprobability_2Eindep__families:
!>[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 ) > ( ( A_27a > $o ) > $o ) > $o ) ).
thf(c_2Eprobability_2Eindep__function,type,
c_2Eprobability_2Eindep__function:
!>[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 ) ) ) > ( A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) > $o ) ).
thf(c_2Eprobability_2Eindep__rv,type,
c_2Eprobability_2Eindep__rv:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > A_27b ) > ( A_27a > A_27c ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) ) > ( tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o ) ) > $o ) ).
thf(c_2Emeasure_2Eindicator__fn,type,
c_2Emeasure_2Eindicator__fn:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Elebesgue_2Eintegral,type,
c_2Elebesgue_2Eintegral:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > tyop_2Eextreal_2Eextreal ) ).
thf(c_2Eprobability_2Ejoint__distribution,type,
c_2Eprobability_2Ejoint__distribution:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27c > A_27a ) > ( A_27c > A_27b ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Eprobability_2Ejoint__distribution3,type,
c_2Eprobability_2Ejoint__distribution3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27d > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27d > $o ) > $o ) @ ( ( A_27d > $o ) > tyop_2Erealax_2Ereal ) ) ) > ( A_27d > A_27a ) > ( A_27d > A_27b ) > ( A_27d > A_27c ) > ( ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o ) > tyop_2Erealax_2Ereal ) ).
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__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_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_2Eprobability_2Ep__space,type,
c_2Eprobability_2Ep__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_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_2Eprobability_2Epossibly,type,
c_2Eprobability_2Epossibly:
!>[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_2Eprobability_2Eprob,type,
c_2Eprobability_2Eprob:
!>[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_2Eprobability_2Eprob__space,type,
c_2Eprobability_2Eprob__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_2Eprobability_2Eprobably,type,
c_2Eprobability_2Eprobably:
!>[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_2Eprobability_2Erandom__variable,type,
c_2Eprobability_2Erandom__variable:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( 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 ) @ ( ( A_27b > $o ) > $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_2Eprobability_2Ereal__random__variable,type,
c_2Eprobability_2Ereal__random__variable:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Eextreal_2Eextreal ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) > $o ) ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Eprobability_2Erv__conditional__expectation,type,
c_2Eprobability_2Erv__conditional__expectation:
!>[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 ) @ ( ( A_27b > $o ) > $o ) ) > ( A_27a > tyop_2Eextreal_2Eextreal ) > ( A_27a > A_27b ) > A_27a > tyop_2Eextreal_2Eextreal ) ).
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_2Espace,type,
c_2Emeasure_2Espace:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) ) > A_27a > $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_2Eprobability_2Euniform__distribution,type,
c_2Eprobability_2Euniform__distribution:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( A_27b > A_27c > ( tyop_2Epair_2Eprod @ ( A_27d > $o ) @ ( ( A_27d > $o ) > $o ) ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ).
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_2Eprobability_2Ep__space__def,axiom,
! [A_27a: $tType] :
( ( c_2Eprobability_2Ep__space @ A_27a )
= ( c_2Emeasure_2Em__space @ A_27a ) ) ).
thf(thm_2Eprobability_2Eevents__def,axiom,
! [A_27a: $tType] :
( ( c_2Eprobability_2Eevents @ A_27a )
= ( c_2Emeasure_2Emeasurable__sets @ A_27a ) ) ).
thf(thm_2Eprobability_2Eprob__def,axiom,
! [A_27a: $tType] :
( ( c_2Eprobability_2Eprob @ A_27a )
= ( c_2Emeasure_2Emeasure @ A_27a ) ) ).
thf(thm_2Eprobability_2Eprob__space__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
<=> ( ( c_2Emeasure_2Emeasure__space @ A_27a @ V0p )
& ( ( c_2Emeasure_2Emeasure @ A_27a @ V0p @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Eprobability_2Eindep__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1a: A_27a > $o,V2b: A_27a > $o] :
( ( c_2Eprobability_2Eindep @ A_27a @ V0p @ V1a @ V2b )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1a @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2b @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINTER @ A_27a @ V1a @ V2b ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1a ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2b ) ) ) ) ) ).
thf(thm_2Eprobability_2Eindep__families__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1q: ( A_27a > $o ) > $o,V2r: ( A_27a > $o ) > $o] :
( ( c_2Eprobability_2Eindep__families @ A_27a @ V0p @ V1q @ V2r )
<=> ! [V3s: A_27a > $o,V4t: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ V1q )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4t @ V2r ) )
=> ( c_2Eprobability_2Eindep @ A_27a @ V0p @ V3s @ V4t ) ) ) ).
thf(thm_2Eprobability_2Eindep__function__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eindep__function @ A_27a @ A_27b @ V0p )
= ( c_2Epred__set_2EGSPEC @ ( A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )
@ ^ [V1f: A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27a )] : ( c_2Epair_2E_2C @ ( A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) @ $o @ V1f @ ( c_2Eprobability_2Eindep__families @ A_27a @ V0p @ ( c_2Epred__set_2EIMAGE @ ( A_27b > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2EFST @ A_27b @ A_27a ) @ V1f ) ) @ ( c_2Epred__set_2EUNIV @ ( A_27b > $o ) ) ) @ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ V1f ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ) ) ) ).
thf(thm_2Eprobability_2Eprobably__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1e: A_27a > $o] :
( ( c_2Eprobability_2Eprobably @ A_27a @ V0p @ V1e )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1e @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1e )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Eprobability_2Epossibly__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1e: A_27a > $o] :
( ( c_2Eprobability_2Epossibly @ A_27a @ V0p @ V1e )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1e @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( (~)
@ ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1e )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Eprobability_2Erandom__variable__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0X: A_27a > A_27b,V1p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2s: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o )] :
( ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V0X @ V1p @ V2s )
<=> ( ( c_2Eprobability_2Eprob__space @ A_27a @ V1p )
& ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0X @ ( c_2Emeasure_2Emeasurable @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V1p ) @ ( c_2Eprobability_2Eevents @ A_27a @ V1p ) ) @ V2s ) ) ) ) ).
thf(thm_2Eprobability_2Ereal__random__variable__def,axiom,
! [A_27a: $tType,V0X: A_27a > tyop_2Eextreal_2Eextreal,V1p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Ereal__random__variable @ A_27a @ V0X @ V1p )
<=> ( ( c_2Eprobability_2Eprob__space @ A_27a @ V1p )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eprobability_2Ep__space @ A_27a @ V1p ) )
=> ( ( (~)
@ ( ( V0X @ V2x )
= c_2Eextreal_2ENegInf ) )
& ( (~)
@ ( ( V0X @ V2x )
= c_2Eextreal_2EPosInf ) ) ) )
& ( c_2Ebool_2EIN @ ( A_27a > tyop_2Eextreal_2Eextreal ) @ V0X @ ( c_2Emeasure_2Emeasurable @ A_27a @ tyop_2Eextreal_2Eextreal @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V1p ) @ ( c_2Eprobability_2Eevents @ A_27a @ V1p ) ) @ c_2Emeasure_2EBorel ) ) ) ) ).
thf(thm_2Eprobability_2Edistribution__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27b > A_27a] :
( ( c_2Eprobability_2Edistribution @ A_27a @ A_27b @ V0p @ V1X )
= ( ^ [V2s: A_27a > $o] : ( c_2Eprobability_2Eprob @ A_27b @ V0p @ ( c_2Epred__set_2EINTER @ A_27b @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27a @ V1X @ V2s ) @ ( c_2Eprobability_2Ep__space @ A_27b @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27c > $o ) > $o ) @ ( ( A_27c > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27c > A_27a,V2Y: A_27c > A_27b] :
( ( c_2Eprobability_2Ejoint__distribution @ A_27a @ A_27b @ A_27c @ V0p @ V1X @ V2Y )
= ( ^ [V3a: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( c_2Eprobability_2Eprob @ A_27c @ V0p
@ ( c_2Epred__set_2EINTER @ A_27c
@ ( c_2Epred__set_2EPREIMAGE @ A_27c @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V4x: A_27c] : ( c_2Epair_2E_2C @ A_27a @ A_27b @ ( V1X @ V4x ) @ ( V2Y @ V4x ) )
@ V3a )
@ ( c_2Eprobability_2Ep__space @ A_27c @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution3__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27d > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27d > $o ) > $o ) @ ( ( A_27d > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27d > A_27a,V2Y: A_27d > A_27b,V3Z: A_27d > A_27c] :
( ( c_2Eprobability_2Ejoint__distribution3 @ A_27a @ A_27b @ A_27c @ A_27d @ V0p @ V1X @ V2Y @ V3Z )
= ( ^ [V4a: ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o] :
( c_2Eprobability_2Eprob @ A_27d @ V0p
@ ( c_2Epred__set_2EINTER @ A_27d
@ ( c_2Epred__set_2EPREIMAGE @ A_27d @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) )
@ ^ [V5x: A_27d] : ( c_2Epair_2E_2C @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( V1X @ V5x ) @ ( c_2Epair_2E_2C @ A_27b @ A_27c @ ( V2Y @ V5x ) @ ( V3Z @ V5x ) ) )
@ V4a )
@ ( c_2Eprobability_2Ep__space @ A_27d @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Econditional__distribution__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: A_27b > $o,V4b: A_27c > $o] :
( ( c_2Eprobability_2Econditional__distribution @ A_27a @ A_27b @ A_27c @ V0p @ V1X @ V2Y @ V3a @ V4b )
= ( c_2Ereal_2E_2F @ ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V3a @ V4b ) ) @ ( c_2Eprobability_2Edistribution @ A_27c @ A_27a @ V0p @ V2Y @ V4b ) ) ) ).
thf(thm_2Eprobability_2Eexpectation__def,axiom,
! [A_27a: $tType] :
( ( c_2Eprobability_2Eexpectation @ A_27a )
= ( c_2Elebesgue_2Eintegral @ A_27a ) ) ).
thf(thm_2Eprobability_2Econditional__expectation__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > tyop_2Eextreal_2Eextreal,V2s: ( A_27a > $o ) > $o] :
( ( c_2Eprobability_2Econditional__expectation @ A_27a @ V0p @ V1X @ V2s )
= ( c_2Emin_2E_40 @ ( A_27a > tyop_2Eextreal_2Eextreal )
@ ^ [V3f: A_27a > tyop_2Eextreal_2Eextreal] :
( c_2Ebool_2E_2F_5C @ ( c_2Eprobability_2Ereal__random__variable @ A_27a @ V3f @ V0p )
@ ( c_2Ebool_2E_21 @ ( A_27a > $o )
@ ^ [V4g: A_27a > $o] :
( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4g @ V2s )
@ ( c_2Emin_2E_3D @ tyop_2Eextreal_2Eextreal
@ ( c_2Elebesgue_2Eintegral @ A_27a @ V0p
@ ^ [V5x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V3f @ V5x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V4g @ V5x ) ) )
@ ( c_2Elebesgue_2Eintegral @ A_27a @ V0p
@ ^ [V6x: A_27a] : ( c_2Eextreal_2Eextreal__mul @ ( V1X @ V6x ) @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V4g @ V6x ) ) ) ) ) ) ) ) ) ).
thf(thm_2Eprobability_2Econditional__prob__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1e1: A_27a > $o,V2e2: ( A_27a > $o ) > $o] :
( ( c_2Eprobability_2Econditional__prob @ A_27a @ V0p @ V1e1 @ V2e2 )
= ( c_2Eprobability_2Econditional__expectation @ A_27a @ V0p @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ V1e1 ) @ V2e2 ) ) ).
thf(thm_2Eprobability_2Erv__conditional__expectation__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V2X: A_27a > tyop_2Eextreal_2Eextreal,V3Y: A_27a > A_27b] :
( ( c_2Eprobability_2Erv__conditional__expectation @ A_27a @ A_27b @ V0p @ V1s @ V2X @ V3Y )
= ( c_2Eprobability_2Econditional__expectation @ A_27a @ V0p @ V2X
@ ( c_2Epred__set_2EIMAGE @ ( A_27b > $o ) @ ( A_27a > $o )
@ ^ [V4a: A_27b > $o] : ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V3Y @ V4a ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
@ ( c_2Emeasure_2Esubsets @ A_27b @ V1s ) ) ) ) ).
thf(thm_2Eprobability_2Eindep__rv__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3s: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V4t: tyop_2Epair_2Eprod @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o )] :
( ( c_2Eprobability_2Eindep__rv @ A_27a @ A_27b @ A_27c @ V0p @ V1X @ V2Y @ V3s @ V4t )
<=> ! [V5A: A_27b > $o,V6B: A_27c > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V5A @ ( c_2Emeasure_2Esubsets @ A_27b @ V3s ) )
& ( c_2Ebool_2EIN @ ( A_27c > $o ) @ V6B @ ( c_2Emeasure_2Esubsets @ A_27c @ V4t ) ) )
=> ( c_2Eprobability_2Eindep @ A_27a @ V0p @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V1X @ V5A ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27c @ V2Y @ V6B ) ) ) ) ).
thf(thm_2Eprobability_2Euniform__distribution__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0p: A_27b,V1X: A_27c,V2s: tyop_2Epair_2Eprod @ ( A_27d > $o ) @ ( ( A_27d > $o ) > $o )] :
( ( c_2Eprobability_2Euniform__distribution @ A_27a @ A_27b @ A_27c @ A_27d @ V0p @ V1X @ V2s )
= ( ^ [V3a: A_27a > $o] : ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V3a ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27d @ ( c_2Emeasure_2Espace @ A_27d @ V2s ) ) ) ) ) ) ).
thf(thm_2Eprobability_2EPOSITIVE__PROB,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Emeasure_2Epositive @ A_27a @ V0p )
<=> ( ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( 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_2Eprobability_2Eevents @ A_27a @ V0p ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ) ) ).
thf(thm_2Eprobability_2EINCREASING__PROB,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Emeasure_2Eincreasing @ A_27a @ V0p )
<=> ! [V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t ) ) ) ) ).
thf(thm_2Eprobability_2EADDITIVE__PROB,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Emeasure_2Eadditive @ A_27a @ V0p )
<=> ! [V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t ) ) ) ) ) ).
thf(thm_2Eprobability_2ECOUNTABLY__ADDITIVE__PROB,axiom,
! [A_27a: $tType,V0p: 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 @ V0p )
<=> ! [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_2Eprobability_2Eevents @ 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_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Eseq_2Esums @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p ) @ V1f ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( 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_2Eprobability_2EPROB__SPACE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
<=> ( ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ( c_2Emeasure_2Epositive @ A_27a @ V0p )
& ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Eprobability_2EEVENTS__SIGMA__ALGEBRA,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( c_2Emeasure_2Esigma__algebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ) ).
thf(thm_2Eprobability_2EEVENTS__ALGEBRA,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( c_2Emeasure_2Ealgebra @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( A_27a > $o ) > $o ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__EMPTY,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eprobability_2EPROB__UNIV,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__SPACE__POSITIVE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( c_2Emeasure_2Epositive @ A_27a @ V0p ) ) ).
thf(thm_2Eprobability_2EPROB__SPACE__COUNTABLY__ADDITIVE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( c_2Emeasure_2Ecountably__additive @ A_27a @ V0p ) ) ).
thf(thm_2Eprobability_2EPROB__SPACE__ADDITIVE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( c_2Emeasure_2Eadditive @ A_27a @ V0p ) ) ).
thf(thm_2Eprobability_2EPROB__SPACE__INCREASING,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( c_2Emeasure_2Eincreasing @ A_27a @ V0p ) ) ).
thf(thm_2Eprobability_2EPROB__POSITIVE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ).
thf(thm_2Eprobability_2EPROB__INCREASING,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t ) ) ) ).
thf(thm_2Eprobability_2EPROB__ADDITIVE,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t )
& ( V3u
= ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V3u )
= ( c_2Erealax_2Ereal__add @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__COUNTABLY__ADDITIVE,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( 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_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ! [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_2Eprobability_2Eprob @ A_27a @ V0p ) @ V2f ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ).
thf(thm_2Eprobability_2EPROB__COMPL,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ V1s ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__INDEP,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eindep @ A_27a @ V0p @ V1s @ V2t )
& ( V3u
= ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V3u )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t ) ) ) ) ).
thf(thm_2Eprobability_2EPSPACE,axiom,
! [A_27a: $tType,V0a: A_27a > $o,V1b: ( A_27a > $o ) > $o,V2c: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
( ( c_2Eprobability_2Ep__space @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0a @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V1b @ V2c ) ) )
= V0a ) ).
thf(thm_2Eprobability_2EEVENTS,axiom,
! [A_27a: $tType,V0a: A_27a > $o,V1b: ( A_27a > $o ) > $o,V2c: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
( ( c_2Eprobability_2Eevents @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0a @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V1b @ V2c ) ) )
= V1b ) ).
thf(thm_2Eprobability_2EPROB,axiom,
! [A_27a: $tType,V0a: A_27a > $o,V1b: ( A_27a > $o ) > $o,V2c: ( A_27a > $o ) > tyop_2Erealax_2Ereal] :
( ( c_2Eprobability_2Eprob @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) @ V0a @ ( c_2Epair_2E_2C @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) @ V1b @ V2c ) ) )
= V2c ) ).
thf(thm_2Eprobability_2EEVENTS__INTER,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ).
thf(thm_2Eprobability_2EEVENTS__EMPTY,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ).
thf(thm_2Eprobability_2EEVENTS__SPACE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ).
thf(thm_2Eprobability_2EEVENTS__UNION,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ).
thf(thm_2Eprobability_2EINDEP__EMPTY,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Eprobability_2Eindep @ A_27a @ V0p @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1s ) ) ).
thf(thm_2Eprobability_2EINTER__PSPACE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ V1s )
= V1s ) ) ).
thf(thm_2Eprobability_2EINDEP__SPACE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Eprobability_2Eindep @ A_27a @ V0p @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ V1s ) ) ).
thf(thm_2Eprobability_2EEVENTS__DIFF,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ).
thf(thm_2Eprobability_2EEVENTS__COMPL,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ V1s ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ).
thf(thm_2Eprobability_2EEVENTS__COUNTABLE__UNION,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1c: ( A_27a > $o ) > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1c @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V1c ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1c ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ).
thf(thm_2Eprobability_2EPROB__ZERO__UNION,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
= ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ).
thf(thm_2Eprobability_2EPROB__EQ__COMPL,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ V1s ) )
= ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ V2t ) ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s )
= ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t ) ) ) ).
thf(thm_2Eprobability_2EPROB__ONE__INTER,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) )
= ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ).
thf(thm_2Eprobability_2EEVENTS__COUNTABLE__INTER,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1c: ( A_27a > $o ) > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1c @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V1c )
& ( (~)
@ ( V1c
= ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1c ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ).
thf(thm_2Eprobability_2EABS__PROB,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( ( c_2Ereal_2Eabs @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) )
= ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ).
thf(thm_2Eprobability_2EPROB__COMPL__LE1,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o,V2r: tyop_2Erealax_2Ereal] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) @ V1s ) ) @ V2r )
= ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2r ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__LE__1,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__EQ__BIGUNION__IMAGE,axiom,
! [A_27a: $tType,V0g: tyop_2Enum_2Enum > A_27a > $o,V1f: tyop_2Enum_2Enum > A_27a > $o,V2p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V2p )
& ( 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_2Eprobability_2Eevents @ A_27a @ V2p ) ) )
& ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V0g @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Eprobability_2Eevents @ A_27a @ V2p ) ) )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( (~) @ ( V3m = V4n ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V1f @ V3m ) @ ( V1f @ V4n ) ) )
& ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( (~) @ ( V5m = V6n ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V0g @ V5m ) @ ( V0g @ V6n ) ) )
& ! [V7n: tyop_2Enum_2Enum] :
( ( c_2Eprobability_2Eprob @ A_27a @ V2p @ ( V1f @ V7n ) )
= ( c_2Eprobability_2Eprob @ A_27a @ V2p @ ( V0g @ V7n ) ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V2p @ ( 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_2Eprobability_2Eprob @ A_27a @ V2p @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V0g @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__FINITELY__ADDITIVE,axiom,
! [A_27a: $tType,V0p: 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,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( tyop_2Enum_2Enum > A_27a > $o ) @ V2f @ ( c_2Epred__set_2EFUNSET @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ ( c_2Epred__set_2Ecount @ V3n ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ! [V4a: tyop_2Enum_2Enum,V5b: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V4a @ V3n )
& ( c_2Eprim__rec_2E_3C @ V5b @ V3n )
& ( (~) @ ( V4a = V5b ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V2f @ V4a ) @ ( V2f @ V5b ) ) )
& ( V1s
= ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V2f @ ( c_2Epred__set_2Ecount @ V3n ) ) ) ) )
=> ( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V3n ) @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p ) @ V2f ) )
= ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ).
thf(thm_2Eprobability_2EABS__1__MINUS__PROB,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( (~)
@ ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__INCREASING__UNION,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( 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_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ! [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_2Eprobability_2Eprob @ A_27a @ V0p ) @ V2f ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) ) ) ).
thf(thm_2Eprobability_2EPROB__SUBADDITIVE,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3u @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( V1s
= ( c_2Epred__set_2EUNION @ A_27a @ V2t @ V3u ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s ) @ ( c_2Erealax_2Ereal__add @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2t ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V3u ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__COUNTABLY__SUBADDITIVE,axiom,
! [A_27a: $tType,V0p: 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_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Eseq_2Esummable @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ tyop_2Erealax_2Ereal @ ( A_27a > $o ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p ) @ V1f ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( 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_2Eprobability_2Eprob @ A_27a @ V0p ) @ V1f ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__COUNTABLY__ZERO,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1c: ( A_27a > $o ) > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V1c )
& ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V1c @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ! [V2x: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2x @ V1c )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1c ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eprobability_2EINDEP__SYM,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1a: A_27a > $o,V2b: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Eprobability_2Eindep @ A_27a @ V0p @ V1a @ V2b ) )
=> ( c_2Eprobability_2Eindep @ A_27a @ V0p @ V2b @ V1a ) ) ).
thf(thm_2Eprobability_2EINDEP__REFL,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1a: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1a @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
=> ( ( c_2Eprobability_2Eindep @ A_27a @ V0p @ V1a @ V1a )
<=> ( ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1a )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1a )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__REAL__SUM__IMAGE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ! [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_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINSERT @ A_27a @ V3x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
@ V1s ) ) ) ).
thf(thm_2Eprobability_2EPROB__EQUIPROBABLE__FINITE__UNIONS,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s: A_27a > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ! [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_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V1s ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINSERT @ A_27a @ V3x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINSERT @ A_27a @ V4y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V1s )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINSERT @ A_27a @ ( c_2Epred__set_2ECHOICE @ A_27a @ V1s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__REAL__SUM__IMAGE__FN,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1f: A_27b > A_27a > $o,V2e: A_27a > $o,V3s: A_27b > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2e @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ! [V4x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4x @ V3s )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V2e @ ( V1f @ V4x ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V3s )
& ! [V5x: A_27b,V6y: A_27b] :
( ( ( c_2Ebool_2EIN @ A_27b @ V5x @ V3s )
& ( c_2Ebool_2EIN @ A_27b @ V6y @ V3s )
& ( (~) @ ( V5x = V6y ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V1f @ V5x ) @ ( V1f @ V6y ) ) )
& ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( A_27a > $o ) @ V1f @ V3s ) ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
= ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ V2e )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b
@ ^ [V7x: A_27b] : ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINTER @ A_27a @ V2e @ ( V1f @ V7x ) ) )
@ V3s ) ) ) ).
thf(thm_2Eprobability_2EPROB__REAL__SUM__IMAGE__SPACE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ! [V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V2x: A_27a] : ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
@ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__DISCRETE__EVENTS,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ! [V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) )
=> ! [V2s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V2s @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ) ).
thf(thm_2Eprobability_2EPROB__DISCRETE__EVENTS__COUNTABLE,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) )] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ! [V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) )
=> ! [V2s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V2s @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) ) ) ).
thf(thm_2Eprobability_2Emarginal__joint__zero,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3s: A_27b > $o,V4t: A_27c > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
& ( ( ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V0p @ V1X @ V3s )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( ( c_2Eprobability_2Edistribution @ A_27c @ A_27a @ V0p @ V2Y @ V4t )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V3s @ V4t ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eprobability_2Emarginal__joint__zero3,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3Z: A_27a > A_27d,V4s: A_27b > $o,V5t: A_27c > $o,V6u: A_27d > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
& ( ( ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V0p @ V1X @ V4s )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( ( c_2Eprobability_2Edistribution @ A_27c @ A_27a @ V0p @ V2Y @ V5t )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( ( c_2Eprobability_2Edistribution @ A_27d @ A_27a @ V0p @ V3Z @ V6u )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eprobability_2Ejoint__distribution3 @ A_27b @ A_27c @ A_27d @ A_27a @ V0p @ V1X @ V2Y @ V3Z @ ( c_2Epred__set_2ECROSS @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) @ V4s @ ( c_2Epred__set_2ECROSS @ A_27c @ A_27d @ V5t @ V6u ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eprobability_2Edistribution__prob__space,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2s: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o )] :
( ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V1X @ V0p @ V2s )
=> ( c_2Eprobability_2Eprob__space @ A_27b @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27b @ V2s ) @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V2s ) @ ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V0p @ V1X ) ) ) ) ) ).
thf(thm_2Eprobability_2Euniform__distribution__prob__space,axiom,
! [A_27a: $tType,A_27b: $tType,V0X: A_27a > A_27b,V1p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2s: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o )] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V1p ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Emeasure_2Espace @ A_27b @ V2s ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V0X @ V1p @ V2s ) )
=> ( c_2Eprobability_2Eprob__space @ A_27b @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27b @ V2s ) @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V2s ) @ ( c_2Eprobability_2Euniform__distribution @ A_27b @ ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ) ) @ ( A_27a > A_27b ) @ A_27b @ V1p @ V0X @ V2s ) ) ) ) ) ).
thf(thm_2Eprobability_2Edistribution__partition,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ V0p @ c_2Emeasure_2EBorel ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ tyop_2Eextreal_2Eextreal
@ ^ [V3x: tyop_2Eextreal_2Eextreal] : ( c_2Eprobability_2Edistribution @ tyop_2Eextreal_2Eextreal @ A_27a @ V0p @ V1X @ ( c_2Epred__set_2EINSERT @ tyop_2Eextreal_2Eextreal @ V3x @ ( c_2Epred__set_2EEMPTY @ tyop_2Eextreal_2Eextreal ) ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eprobability_2Edistribution__lebesgue__thm1,axiom,
! [A_27a: $tType,A_27b: $tType,V0X: A_27a > A_27b,V1p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2s: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V3A: A_27b > $o] :
( ( ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V0X @ V1p @ V2s )
& ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3A @ ( c_2Emeasure_2Esubsets @ A_27b @ V2s ) ) )
=> ( ( c_2Eextreal_2ENormal @ ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V1p @ V0X @ V3A ) )
= ( c_2Elebesgue_2Eintegral @ A_27a @ V1p @ ( c_2Emeasure_2Eindicator__fn @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0X @ V3A ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V1p ) ) ) ) ) ) ).
thf(thm_2Eprobability_2Edistribution__lebesgue__thm2,axiom,
! [A_27a: $tType,A_27b: $tType,V0X: A_27a > A_27b,V1p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2s: tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ),V3A: A_27b > $o] :
( ( ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V0X @ V1p @ V2s )
& ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V3A @ ( c_2Emeasure_2Esubsets @ A_27b @ V2s ) ) )
=> ( ( c_2Eextreal_2ENormal @ ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V1p @ V0X @ V3A ) )
= ( c_2Elebesgue_2Eintegral @ A_27b @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Emeasure_2Espace @ A_27b @ V2s ) @ ( c_2Epair_2E_2C @ ( ( A_27b > $o ) > $o ) @ ( ( A_27b > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Emeasure_2Esubsets @ A_27b @ V2s ) @ ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V1p @ V0X ) ) ) @ ( c_2Emeasure_2Eindicator__fn @ A_27b @ V3A ) ) ) ) ).
thf(thm_2Eprobability_2Efinite__expectation1,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( c_2Eprobability_2Ereal__random__variable @ A_27a @ V1X @ V0p ) )
=> ( ( c_2Eprobability_2Eexpectation @ A_27a @ V0p @ V1X )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Eextreal_2Eextreal
@ ^ [V2r: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul @ V2r @ ( c_2Eextreal_2ENormal @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Epred__set_2EINSERT @ tyop_2Eextreal_2Eextreal @ V2r @ ( c_2Epred__set_2EEMPTY @ tyop_2Eextreal_2Eextreal ) ) ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Efinite__expectation2,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Eextreal_2Eextreal @ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
& ( c_2Eprobability_2Ereal__random__variable @ A_27a @ V1X @ V0p ) )
=> ( ( c_2Eprobability_2Eexpectation @ A_27a @ V0p @ V1X )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Eextreal_2Eextreal
@ ^ [V2r: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul @ V2r @ ( c_2Eextreal_2ENormal @ ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Epred__set_2EINSERT @ tyop_2Eextreal_2Eextreal @ V2r @ ( c_2Epred__set_2EEMPTY @ tyop_2Eextreal_2Eextreal ) ) ) @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Efinite__expectation,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( c_2Eprobability_2Ereal__random__variable @ A_27a @ V1X @ V0p ) )
=> ( ( c_2Eprobability_2Eexpectation @ A_27a @ V0p @ V1X )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Eextreal_2Eextreal
@ ^ [V2r: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul @ V2r @ ( c_2Eextreal_2ENormal @ ( c_2Eprobability_2Edistribution @ tyop_2Eextreal_2Eextreal @ A_27a @ V0p @ V1X @ ( c_2Epred__set_2EINSERT @ tyop_2Eextreal_2Eextreal @ V2r @ ( c_2Epred__set_2EEMPTY @ tyop_2Eextreal_2Eextreal ) ) ) ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Efinite__support__expectation,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > tyop_2Eextreal_2Eextreal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Eextreal_2Eextreal @ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
& ( c_2Eprobability_2Ereal__random__variable @ A_27a @ V1X @ V0p ) )
=> ( ( c_2Eprobability_2Eexpectation @ A_27a @ V0p @ V1X )
= ( c_2Eextreal_2EEXTREAL__SUM__IMAGE @ tyop_2Eextreal_2Eextreal
@ ^ [V2r: tyop_2Eextreal_2Eextreal] : ( c_2Eextreal_2Eextreal__mul @ V2r @ ( c_2Eextreal_2ENormal @ ( c_2Eprobability_2Edistribution @ tyop_2Eextreal_2Eextreal @ A_27a @ V0p @ V1X @ ( c_2Epred__set_2EINSERT @ tyop_2Eextreal_2Eextreal @ V2r @ ( c_2Epred__set_2EEMPTY @ tyop_2Eextreal_2Eextreal ) ) ) ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Eextreal_2Eextreal @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Efinite__marginal__product__space__POW,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c] :
( ( ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
= ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V1X @ V0p @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) @ ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27c @ V2Y @ V0p @ ( c_2Epair_2E_2C @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ V2Y @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) @ ( c_2Epred__set_2EPOW @ A_27c @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ V2Y @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
=> ( c_2Emeasure_2Emeasure__space @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ V2Y @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) )
@ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ V2Y @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
@ ^ [V3a: ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o] :
( c_2Eprobability_2Eprob @ A_27a @ V0p
@ ( c_2Epred__set_2EINTER @ A_27a
@ ( c_2Epred__set_2EPREIMAGE @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27b @ A_27c @ ( V1X @ V4x ) @ ( V2Y @ V4x ) )
@ V3a )
@ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) ) ) ) ).
thf(thm_2Eprobability_2Efinite__marginal__product__space__POW2,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s1: A_27b > $o,V2s2: A_27c > $o,V3X: A_27a > A_27b,V4Y: A_27a > A_27c] :
( ( ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
= ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V3X @ V0p @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ V1s1 @ ( c_2Epred__set_2EPOW @ A_27b @ V1s1 ) ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27c @ V4Y @ V0p @ ( c_2Epair_2E_2C @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o ) @ V2s2 @ ( c_2Epred__set_2EPOW @ A_27c @ V2s2 ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1s1 )
& ( c_2Epred__set_2EFINITE @ A_27c @ V2s2 ) )
=> ( c_2Emeasure_2Emeasure__space @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V1s1 @ V2s2 ) @ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V1s1 @ V2s2 ) ) @ ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V3X @ V4Y ) ) ) ) ) ).
thf(thm_2Eprobability_2Efinite__marginal__product__space__POW3,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1s1: A_27b > $o,V2s2: A_27c > $o,V3s3: A_27d > $o,V4X: A_27a > A_27b,V5Y: A_27a > A_27c,V6Z: A_27a > A_27d] :
( ( ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
= ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V4X @ V0p @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ V1s1 @ ( c_2Epred__set_2EPOW @ A_27b @ V1s1 ) ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27c @ V5Y @ V0p @ ( c_2Epair_2E_2C @ ( A_27c > $o ) @ ( ( A_27c > $o ) > $o ) @ V2s2 @ ( c_2Epred__set_2EPOW @ A_27c @ V2s2 ) ) )
& ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27d @ V6Z @ V0p @ ( c_2Epair_2E_2C @ ( A_27d > $o ) @ ( ( A_27d > $o ) > $o ) @ V3s3 @ ( c_2Epred__set_2EPOW @ A_27d @ V3s3 ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1s1 )
& ( c_2Epred__set_2EFINITE @ A_27c @ V2s2 )
& ( c_2Epred__set_2EFINITE @ A_27d @ V3s3 ) )
=> ( c_2Emeasure_2Emeasure__space @ ( tyop_2Epair_2Eprod @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) @ ( c_2Epair_2E_2C @ ( ( tyop_2Epair_2Eprod @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) > $o ) @ ( tyop_2Epair_2Eprod @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) > $o ) > tyop_2Erealax_2Ereal ) ) @ ( c_2Epred__set_2ECROSS @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) @ V1s1 @ ( c_2Epred__set_2ECROSS @ A_27c @ A_27d @ V2s2 @ V3s3 ) ) @ ( c_2Epair_2E_2C @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) > $o ) > $o ) @ ( ( ( tyop_2Epair_2Eprod @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) > $o ) > tyop_2Erealax_2Ereal ) @ ( c_2Epred__set_2EPOW @ ( tyop_2Epair_2Eprod @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) @ ( c_2Epred__set_2ECROSS @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) @ V1s1 @ ( c_2Epred__set_2ECROSS @ A_27c @ A_27d @ V2s2 @ V3s3 ) ) ) @ ( c_2Eprobability_2Ejoint__distribution3 @ A_27b @ A_27c @ A_27d @ A_27a @ V0p @ V4X @ V5Y @ V6Z ) ) ) ) ) ).
thf(thm_2Eprobability_2Eprob__x__eq__1__imp__prob__y__eq__0,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1x: A_27a] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ! [V2y: A_27a] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V2y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Eprobability_2Eevents @ A_27a @ V0p ) )
& ( (~) @ ( V2y = V1x ) ) )
=> ( ( c_2Eprobability_2Eprob @ A_27a @ V0p @ ( c_2Epred__set_2EINSERT @ A_27a @ V2y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eprobability_2Edistribution__x__eq__1__imp__distribution__y__eq__0,axiom,
! [A_27a: $tType,A_27b: $tType,V0X: A_27a > A_27b,V1p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V2x: A_27b] :
( ( ( c_2Eprobability_2Erandom__variable @ A_27a @ A_27b @ V0X @ V1p @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( ( A_27b > $o ) > $o ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0X @ ( c_2Eprobability_2Ep__space @ A_27a @ V1p ) ) @ ( c_2Epred__set_2EPOW @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0X @ ( c_2Eprobability_2Ep__space @ A_27a @ V1p ) ) ) ) )
& ( ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V1p @ V0X @ ( c_2Epred__set_2EINSERT @ A_27b @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ! [V3y: A_27b] :
( ( (~) @ ( V3y = V2x ) )
=> ( ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V1p @ V0X @ ( c_2Epred__set_2EINSERT @ A_27b @ V3y @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution__sym,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: A_27b > $o,V4b: A_27c > $o] :
( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
=> ( ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V3a @ V4b ) )
= ( c_2Eprobability_2Ejoint__distribution @ A_27c @ A_27b @ A_27a @ V0p @ V2Y @ V1X @ ( c_2Epred__set_2ECROSS @ A_27c @ A_27b @ V4b @ V3a ) ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution__pos,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ V3a ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution__le__1,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ V3a ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution__le,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: A_27b > $o,V4b: A_27c > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V3a @ V4b ) ) @ ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V0p @ V1X @ V3a ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution__le2,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: A_27b > $o,V4b: A_27c > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V3a @ V4b ) ) @ ( c_2Eprobability_2Edistribution @ A_27c @ A_27a @ V0p @ V2Y @ V4b ) ) ) ).
thf(thm_2Eprobability_2Ejoint__conditional,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: A_27b > $o,V4b: A_27c > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V3a @ V4b ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eprobability_2Econditional__distribution @ A_27a @ A_27c @ A_27b @ V0p @ V2Y @ V1X @ V4b @ V3a ) @ ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V0p @ V1X @ V3a ) ) ) ) ).
thf(thm_2Eprobability_2Edistribution__pos,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2a: A_27b > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V0p @ V1X @ V2a ) ) ) ).
thf(thm_2Eprobability_2Econditional__distribution__pos,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: A_27b > $o,V4b: A_27c > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eprobability_2Econditional__distribution @ A_27a @ A_27b @ A_27c @ V0p @ V1X @ V2Y @ V3a @ V4b ) ) ) ).
thf(thm_2Eprobability_2Econditional__distribution__le__1,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: A_27b > $o,V4b: A_27c > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eprobability_2Econditional__distribution @ A_27a @ A_27b @ A_27c @ V0p @ V1X @ V2Y @ V3a @ V4b ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eprobability_2Emarginal__distribution1,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3a: A_27b > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V0p @ V1X @ V3a )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27c
@ ^ [V4x: A_27c] : ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V3a @ ( c_2Epred__set_2EINSERT @ A_27c @ V4x @ ( c_2Epred__set_2EEMPTY @ A_27c ) ) ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ V2Y @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Emarginal__distribution2,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3b: A_27c > $o] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( ( c_2Eprobability_2Edistribution @ A_27c @ A_27a @ V0p @ V2Y @ V3b )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b
@ ^ [V4x: A_27b] : ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Epred__set_2EINSERT @ A_27b @ V4x @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) @ V3b ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution__sums__1,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ tyop_2Erealax_2Ereal
@ ^ [V3x: A_27b,V4y: A_27c] : ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2EINSERT @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epair_2E_2C @ A_27b @ A_27c @ V3x @ V4y ) @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) ) ) )
@ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ V2Y @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eprobability_2Ejoint__distribution__sum__mul1,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( tyop_2Epair_2Eprod @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ),V1X: A_27a > A_27b,V2Y: A_27a > A_27c,V3f: A_27b > tyop_2Erealax_2Ereal] :
( ( ( c_2Eprobability_2Eprob__space @ A_27a @ V0p )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) )
& ( ( c_2Eprobability_2Eevents @ A_27a @ V0p )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c )
@ ( c_2Epair_2EUNCURRY @ A_27b @ A_27c @ tyop_2Erealax_2Ereal
@ ^ [V4x: A_27b,V5y: A_27c] : ( c_2Erealax_2Ereal__mul @ ( c_2Eprobability_2Ejoint__distribution @ A_27b @ A_27c @ A_27a @ V0p @ V1X @ V2Y @ ( c_2Epred__set_2EINSERT @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epair_2E_2C @ A_27b @ A_27c @ V4x @ V5y ) @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) ) ) @ ( V3f @ V4x ) ) )
@ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ V2Y @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b
@ ^ [V6x: A_27b] : ( c_2Erealax_2Ereal__mul @ ( c_2Eprobability_2Edistribution @ A_27b @ A_27a @ V0p @ V1X @ ( c_2Epred__set_2EINSERT @ A_27b @ V6x @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) @ ( V3f @ V6x ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1X @ ( c_2Eprobability_2Ep__space @ A_27a @ V0p ) ) ) ) ) ).
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