ITP001 Axioms: ITP146+5.ax
%------------------------------------------------------------------------------
% File : ITP146+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : measure+2.ax [Gau20]
% : HL4146+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 349 ( 5 unt; 0 def)
% Number of atoms : 2187 ( 205 equ)
% Maximal formula atoms : 28 ( 6 avg)
% Number of connectives : 1863 ( 25 ~; 0 |; 416 &)
% ( 34 <=>;1388 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 197 ( 197 usr; 43 con; 0-5 aty)
% Number of variables : 1314 (1308 !; 6 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Emeasure_2EBorel,axiom,
mem(c_2Emeasure_2EBorel,ty_2Epair_2Eprod(arr(ty_2Eextreal_2Eextreal,bool),arr(arr(ty_2Eextreal_2Eextreal,bool),bool))) ).
fof(mem_c_2Emeasure_2Eadditive,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Eadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) ).
fof(mem_c_2Emeasure_2Ealgebra,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Ealgebra(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ) ).
fof(mem_c_2Emeasure_2Eclosed__cdi,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Eclosed__cdi(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ) ).
fof(mem_c_2Emeasure_2Ecountably__additive,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Ecountably__additive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) ).
fof(mem_c_2Emeasure_2Ecountably__subadditive,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Ecountably__subadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) ).
fof(mem_c_2Emeasure_2Efn__abs,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Efn__abs(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ) ).
fof(mem_c_2Emeasure_2Efn__minus,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Efn__minus(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ) ).
fof(mem_c_2Emeasure_2Efn__plus,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Efn__plus(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ) ).
fof(mem_c_2Emeasure_2Eincreasing,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Eincreasing(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) ).
fof(mem_c_2Emeasure_2Eindicator__fn,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Eindicator__fn(A_27a),arr(arr(A_27a,bool),arr(A_27a,ty_2Eextreal_2Eextreal))) ) ).
fof(mem_c_2Emeasure_2Einf__measure,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Einf__measure(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))) ) ).
fof(mem_c_2Emeasure_2Elambda__system,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Elambda__system(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(arr(arr(A_27a,bool),ty_2Erealax_2Ereal),arr(arr(A_27a,bool),bool)))) ) ).
fof(mem_c_2Emeasure_2Em__space,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Em__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(A_27a,bool))) ) ).
fof(mem_c_2Emeasure_2Emeasurable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Emeasure_2Emeasurable(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),arr(arr(A_27a,A_27b),bool)))) ) ) ).
fof(mem_c_2Emeasure_2Emeasurable__sets,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Emeasurable__sets(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Emeasure_2Emeasure,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Emeasure(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))) ) ).
fof(mem_c_2Emeasure_2Emeasure__preserving,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,A_27b),bool)))) ) ) ).
fof(mem_c_2Emeasure_2Emeasure__space,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Emeasure__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) ).
fof(mem_c_2Emeasure_2Enull__set,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Enull__set(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Emeasure_2Eouter__measure__space,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Eouter__measure__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) ).
fof(mem_c_2Emeasure_2Epos__simple__fn,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Epos__simple__fn(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(arr(ty_2Enum_2Enum,bool),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)))))) ) ).
fof(mem_c_2Emeasure_2Epositive,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Epositive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) ).
fof(mem_c_2Emeasure_2Esigma,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Esigma(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))))) ) ).
fof(mem_c_2Emeasure_2Esigma__algebra,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Esigma__algebra(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ) ).
fof(mem_c_2Emeasure_2Esmallest__closed__cdi,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Esmallest__closed__cdi(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))) ) ).
fof(mem_c_2Emeasure_2Espace,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Espace(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(A_27a,bool))) ) ).
fof(mem_c_2Emeasure_2Esubadditive,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Esubadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) ).
fof(mem_c_2Emeasure_2Esubset__class,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Esubset__class(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),bool))) ) ).
fof(mem_c_2Emeasure_2Esubsets,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emeasure_2Esubsets(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(arr(A_27a,bool),bool))) ) ).
fof(ax_thm_2Emeasure_2Espace__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y] :
( mem(V1y,arr(arr(A_27a,bool),bool))
=> ap(c_2Emeasure_2Espace(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0x),V1y)) = V0x ) ) ) ).
fof(ax_thm_2Emeasure_2Esubsets__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y] :
( mem(V1y,arr(arr(A_27a,bool),bool))
=> ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0x),V1y)) = V1y ) ) ) ).
fof(ax_thm_2Emeasure_2Esubset__class__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1sts))
<=> ! [V2x] :
( mem(V2x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1sts))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2x),V0sp)) ) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Ealgebra__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Esigma__algebra__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& ! [V1c] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
fof(lameq_f3417,axiom,
! [A_27a,V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1st] :
( mem(V1st,arr(arr(A_27a,bool),bool))
=> ! [V2s] : ap(f3417(A_27a,V0sp,V1st),V2s) = ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),bool),V2s),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1st),V2s)),ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2s)))) ) ) ).
fof(ax_thm_2Emeasure_2Esigma__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1st] :
( mem(V1st,arr(arr(A_27a,bool),bool))
=> ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1st) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EBIGINTER(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f3417(A_27a,V0sp,V1st)))) ) ) ) ).
fof(ax_thm_2Emeasure_2Em__space__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ! [V2mu] :
( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ap(c_2Emeasure_2Em__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V0sp ) ) ) ) ).
fof(ax_thm_2Emeasure_2Emeasurable__sets__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ! [V2mu] :
( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ap(c_2Emeasure_2Emeasurable__sets(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V1sts ) ) ) ) ).
fof(ax_thm_2Emeasure_2Emeasure__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ! [V2mu] :
( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ap(c_2Emeasure_2Emeasure(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V2mu ) ) ) ) ).
fof(ax_thm_2Emeasure_2Epositive__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
<=> ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Eadditive__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))
<=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) )
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)) ) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Ecountably__additive__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m))
<=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( V2m != V3n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,V3n))) ) ) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Esubadditive__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m))
<=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))) ) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Ecountably__subadditive__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m))
<=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Eincreasing__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
<=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Emeasure__space__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) ) ) ) ) ).
fof(lameq_f3418,axiom,
! [A_27a,V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,arr(A_27a,bool))
=> ! [V0gen] :
( mem(V0gen,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V3s] : ap(f3418(A_27a,V1lam,V2l,V0gen),V3s) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27a),V0gen))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V2l),V3s))),ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0gen)),V2l)),V3s)))),ap(V1lam,V3s))) ) ) ) ).
fof(lameq_f3419,axiom,
! [A_27a,V0gen] :
( mem(V0gen,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l] : ap(f3419(A_27a,V0gen,V1lam),V2l) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V2l),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l),ap(c_2Emeasure_2Esubsets(A_27a),V0gen))),ap(c_2Ebool_2E_21(arr(A_27a,bool)),f3418(A_27a,V1lam,V2l,V0gen)))) ) ) ).
fof(ax_thm_2Emeasure_2Elambda__system__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0gen] :
( mem(V0gen,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gen),V1lam) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3419(A_27a,V0gen,V1lam)) ) ) ) ).
fof(ax_thm_2Emeasure_2Eouter__measure__space__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),V0m))
<=> ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) ) ) ) ) ).
fof(lameq_f3420,axiom,
! [A_27a,V3f] :
( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4m] :
( mem(V4m,ty_2Enum_2Enum)
=> ! [V5n] : ap(f3420(A_27a,V3f,V4m),V5n) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V4m),V5n))),ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,V4m)),ap(V3f,V5n))) ) ) ).
fof(lameq_f3421,axiom,
! [A_27a,V3f] :
( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4m] : ap(f3421(A_27a,V3f),V4m) = ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3420(A_27a,V3f,V4m)) ) ).
fof(lameq_f3422,axiom,
! [A_27a,V2r] :
( mem(V2r,ty_2Erealax_2Ereal)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3f] : ap(f3422(A_27a,V2r,V1s,V0m),V3f) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3421(A_27a,V3f))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V3f)),V2r)))) ) ) ) ).
fof(lameq_f3423,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2r] : ap(f3423(A_27a,V1s,V0m),V2r) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V2r),ap(c_2Ebool_2E_3F(arr(ty_2Enum_2Enum,arr(A_27a,bool))),f3422(A_27a,V2r,V1s,V0m))) ) ) ).
fof(ax_thm_2Emeasure_2Einf__measure__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s) = ap(c_2Ereal_2Einf,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3423(A_27a,V1s,V0m))) ) ) ) ).
fof(ax_thm_2Emeasure_2Eclosed__cdi__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) )
& ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))
& ap(V2f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a)
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,V3n)),ap(V2f,ap(c_2Enum_2ESUC,V3n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) )
& ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))
& ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ( V5m != V6n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,V5m)),ap(V4f,V6n))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ) ) ).
fof(lameq_f3424,axiom,
! [A_27a,V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] : ap(f3424(A_27a,V0a),V1b) = ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),bool),V1b),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)),V1b)),ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1b)))) ) ).
fof(ax_thm_2Emeasure_2Esmallest__closed__cdi__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Epred__set_2EBIGINTER(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f3424(A_27a,V0a)))) ) ) ).
fof(lameq_f3425,axiom,
! [A_27a,A_27b,V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V3s] : ap(f3425(A_27a,A_27b,V0a,V2f,V1b),V3s) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))),ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Espace(A_27a),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ).
fof(lameq_f3426,axiom,
! [A_27b,A_27a,V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f] : ap(f3426(A_27b,A_27a,V0a,V1b),V2f) = ap(ap(c_2Epair_2E_2C(arr(A_27a,A_27b),bool),V2f),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))),ap(c_2Ebool_2E_21(arr(A_27b,bool)),f3425(A_27a,A_27b,V0a,V2f,V1b)))))) ) ) ).
fof(ax_thm_2Emeasure_2Emeasurable__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,A_27b),arr(A_27a,A_27b)),f3426(A_27b,A_27a,V0a,V1b)) ) ) ) ) ).
fof(lameq_f3427,axiom,
! [A_27a,A_27b,V0m1] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V1m2] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V3s] : ap(f3427(A_27a,A_27b,V0m1,V2f,V1m2),V3s) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m1),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)))),ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m2),V3s))) ) ) ) ).
fof(lameq_f3428,axiom,
! [A_27b,A_27a,V0m1] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2f] : ap(f3428(A_27b,A_27a,V0m1,V1m2),V2f) = ap(ap(c_2Epair_2E_2C(arr(A_27a,A_27b),bool),V2f),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2))))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2)),ap(c_2Ebool_2E_21(arr(A_27b,bool)),f3427(A_27a,A_27b,V0m1,V2f,V1m2)))))) ) ) ).
fof(ax_thm_2Emeasure_2Emeasure__preserving__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0m1] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,A_27b),arr(A_27a,A_27b)),f3428(A_27b,A_27a,V0m1,V1m2)) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Eindicator__fn__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(c_2Emeasure_2Eindicator__fn(A_27a),V0s) = f3355(A_27a,V0s) ) ) ).
fof(lameq_f3429,axiom,
! [A_27a,V6t] :
( mem(V6t,A_27a)
=> ! [V3a] :
( mem(V3a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4x] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7i] : ap(f3429(A_27a,V6t,V3a,V4x),V7i) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,ap(V4x,V7i))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V3a,V7i)),V6t)) ) ) ) ).
fof(ax_thm_2Emeasure_2Epos__simple__fn__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s] :
( mem(V2s,arr(ty_2Enum_2Enum,bool))
=> ! [V3a] :
( mem(V3a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4x] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
<=> ( ! [V5t] :
( mem(V5t,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V5t))) )
& ! [V6t] :
( mem(V6t,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6t),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(V1f,V6t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3429(A_27a,V6t,V3a,V4x)),V2s) ) )
& ! [V8i] :
( mem(V8i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V8i),V2s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V3a,V8i)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V2s))
& ! [V9i] :
( mem(V9i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V9i),V2s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V4x,V9i))) ) )
& ! [V10i] :
( mem(V10i,ty_2Enum_2Enum)
=> ! [V11j] :
( mem(V11j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V10i),V2s))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V11j),V2s))
& V10i != V11j )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3a,V10i)),ap(V3a,V11j))) ) ) )
& ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3a),V2s)) = ap(c_2Emeasure_2Em__space(A_27a),V0m) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Emeasure_2Enull__set__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Emeasure_2Enull__set(A_27a),V0m),V1s))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)) = V0a ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__ALT__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__SPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__COMPL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__FINITE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1c] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__INTER__SPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s) = V1s
& ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),ap(c_2Emeasure_2Espace(A_27a),V0a)) = V1s ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__COMPL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0g0)),V2l)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l1] :
( mem(V2l1,arr(A_27a,bool))
=> ! [V3l2] :
( mem(V3l2,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l1),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3l2),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2l1),V3l2)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ALGEBRA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) )
=> p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__ADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,bool))
=> ! [V3l1] :
( mem(V3l1,arr(A_27a,bool))
=> ! [V4l2] :
( mem(V4l2,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2g),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V3l1),V4l2))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3l1),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4l2),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) )
=> ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V3l1),V4l2)),V2g)) = ap(ap(c_2Erealax_2Ereal__add,ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V3l1),V2g))),ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V4l2),V2g))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l1] :
( mem(V2l1,A_27b)
=> ! [V3l2] :
( mem(V3l2,A_27c)
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) )
=> p(ap(c_2Emeasure_2Eadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE__SUBADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m)) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__MONO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
& ap(V1f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a)
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,V2n)),ap(V1f,ap(c_2Enum_2ESUC,V2n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__DISJOINT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( V2m != V3n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,V3n))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESUBADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t))
& V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) )
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))
& V2s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EADDITIVE__SUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( V3m != V4n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V3m)),ap(V1f,V4n))) ) ) ) )
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V2n)),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),ap(c_2Epred__set_2Ecount,V2n)))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EINCREASING__ADDITIVE__SUMMABLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( V2m != V3n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,V3n))) ) ) ) )
=> p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__POSITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
=> p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INCREASING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) ) ) ).
fof(lameq_f3430,axiom,
! [A_27a,V2g] :
( mem(V2g,arr(A_27a,bool))
=> ! [V7s] : ap(f3430(A_27a,V2g),V7s) = ap(ap(c_2Epred__set_2EINTER(A_27a),V7s),V2g) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__SUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g0] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,bool))
=> ! [V3f] :
( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2g),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))))
& ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ( V5m != V6n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,V5m)),ap(V3f,V6n))) ) ) ) )
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V4n)),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),V1lam),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,arr(A_27a,bool),arr(A_27a,bool)),f3430(A_27a,V2g)),V3f))) = ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),ap(c_2Epred__set_2Ecount,V4n)))),V2g)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALGEBRA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
=> p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) ) ) ) ).
fof(conj_thm_2Emeasure_2EOUTER__MEASURE__SPACE__POSITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),V0m))
=> p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) ) ) ).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__CARATHEODORY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0gsig] :
( mem(V0gsig,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0gsig))
& p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0gsig)),V1lam)))) )
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam))))
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( V3m != V4n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,V3m)),ap(V2f,V4n))) ) ) ) )
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam)))
& p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),V1lam),V2f)),ap(V1lam,ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ECARATHEODORY__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0gsig] :
( mem(V0gsig,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0gsig))
& p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0gsig)),V1lam)))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam)),V1lam)))) ) ) ) ) ).
fof(lameq_f3431,axiom,
! [A_27a,V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6n] : ap(f3431(A_27a,V4f,V5m),V6n) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V5m),V6n))),ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,V5m)),ap(V4f,V6n))) ) ) ).
fof(lameq_f3432,axiom,
! [A_27a,V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5m] : ap(f3432(A_27a,V4f),V5m) = ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3431(A_27a,V4f,V5m)) ) ).
fof(lameq_f3433,axiom,
! [A_27a,V3r] :
( mem(V3r,ty_2Erealax_2Ereal)
=> ! [V1g] :
( mem(V1g,arr(A_27a,bool))
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V4f] : ap(f3433(A_27a,V3r,V1g,V0m),V4f) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3432(A_27a,V4f))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V4f)),V3r)))) ) ) ) ).
fof(lameq_f3434,axiom,
! [A_27a,V1g] :
( mem(V1g,arr(A_27a,bool))
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3r] : ap(f3434(A_27a,V1g,V0m),V3r) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V3r),ap(c_2Ebool_2E_3F(arr(ty_2Enum_2Enum,arr(A_27a,bool))),f3433(A_27a,V3r,V1g,V0m))) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__NONEMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1g] :
( mem(V1g,arr(A_27a,bool))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),V2s)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3434(A_27a,V1g,V0m)))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__POS0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1g] :
( mem(V1g,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3434(A_27a,V1g,V0m)))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2x)) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__POS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1g] :
( mem(V1g,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),ap(c_2Emeasure_2Em__space(A_27a),V0m))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1g))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EINCREASING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EADDITIVE__INCREASING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) ) ) ) ).
fof(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE__ADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) ) ) ).
fof(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( V3m != V4n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,V3m)),ap(V2f,V4n))) ) ) )
& V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__AGREES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__DOWN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m0] :
( mem(V0m0,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m1] :
( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0))))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))
& ap(c_2Emeasure_2Emeasure(A_27a),V0m0) = ap(c_2Emeasure_2Emeasure(A_27a),V1m1)
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1)) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m0)) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__SIGMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1sts))
=> p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1sts))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EPOW__ALGEBRA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EPOW(A_27a),V0sp)))) ) ) ).
fof(conj_thm_2Emeasure_2EPOW__SIGMA__ALGEBRA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EPOW(A_27a),V0sp)))) ) ) ).
fof(conj_thm_2Emeasure_2EUNIV__SIGMA__ALGEBRA,axiom,
! [A_27a] :
( ne(A_27a)
=> p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),c_2Epred__set_2EUNIV(A_27a)),c_2Epred__set_2EUNIV(arr(A_27a,bool))))) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) )
=> ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__POSITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__INCREASING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) ) ) ).
fof(lameq_f3435,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V3r] :
( mem(V3r,ty_2Erealax_2Ereal)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V4f] : ap(f3435(A_27a,V1s,V3r,V0m),V4f) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V4f)),V3r))) ) ) ) ).
fof(lameq_f3436,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3r] : ap(f3436(A_27a,V1s,V0m),V3r) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V3r),ap(c_2Ebool_2E_3F(arr(ty_2Enum_2Enum,arr(A_27a,bool))),f3435(A_27a,V1s,V3r,V0m))) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3436(A_27a,V1s,V0m)))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)),V2x)) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__CLOSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2e] :
( mem(V2e,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Emeasure_2Em__space(A_27a),V0m))) )
=> ? [V3f] :
( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V4l] :
( mem(V4l,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))
& ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ( V5m != V6n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,V5m)),ap(V3f,V6n))) ) ) )
& p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V3f)),V4l))
& p(ap(ap(c_2Ereal_2Ereal__lte,V4l),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)),V2e))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__COUNTABLY__SUBADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) ) ) ).
fof(conj_thm_2Emeasure_2EINF__MEASURE__OUTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1b))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Esubsets(A_27a),V1b))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Espace(A_27a),V1b)),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V1b))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EALGEBRA__SUBSET__LAMBDA__SYSTEM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m)))) ) ) ) ).
fof(conj_thm_2Emeasure_2ECARATHEODORY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m0] :
( mem(V0m0,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m0))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m0)) )
=> ? [V1m] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
& ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)))
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V2s) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m0),V2s) ) )
& ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)) = ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__SUBSET__SUBSETS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)))) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__SIGMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2x] :
( mem(V2x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1a))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0p))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) )
& ! [V2c] :
( mem(V2c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V2c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2c),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V2c)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__COUNTABLE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1c] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ENUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__COMPL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)),V1s)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,arr(arr(A_27a,bool),bool))
=> ! [V2a] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V1p))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p))
& ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) )
& ! [V4c] :
( mem(V4c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V4c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V4c),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V4c)),V1p)) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__SUBSET__MEASURABLE__SETS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1m] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V1m)),V0a))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__DISJOINT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
& ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
& ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ( V5m != V6n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,V5m)),ap(V4f,V6n))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__ALT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,arr(arr(A_27a,bool),bool))
=> ! [V2a] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V1p))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p))
& ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) )
& ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESPACE__SMALLEST__CLOSED__CDI,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ap(c_2Emeasure_2Espace(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)) = ap(c_2Emeasure_2Espace(A_27a),V0a) ) ) ).
fof(conj_thm_2Emeasure_2ESMALLEST__CLOSED__CDI,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))))
& p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))
& p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ECLOSED__CDI__DUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ECLOSED__CDI__COMPL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ECLOSED__CDI__DISJOINT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( V2m != V3n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,V3n))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ECLOSED__CDI__INCREASING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))
& ap(V1f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a)
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,V2n)),ap(V1f,ap(c_2Enum_2ESUC,V2n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2p] :
( mem(V2p,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2p))
& p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2p))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))),V2p)) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__WEAK,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,arr(arr(A_27a,bool),bool))
=> ! [V2a] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2a)))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p))
& p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p))
& ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V1p))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) )
& ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1p)))
& ap(V4f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a)
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V4f,V5n)),ap(V4f,ap(c_2Enum_2ESUC,V5n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) )
& ! [V6f] :
( mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V6f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1p)))
& ! [V7m] :
( mem(V7m,ty_2Enum_2Enum)
=> ! [V8n] :
( mem(V8n,ty_2Enum_2Enum)
=> ( V7m != V8n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V6f,V7m)),ap(V6f,V8n))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V6f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,arr(arr(A_27a,bool),bool))
=> ! [V2a] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2a)))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p))
& ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) )
& ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))))
& ap(V4f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a)
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V4f,V5n)),ap(V4f,ap(c_2Enum_2ESUC,V5n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) )
& ! [V6f] :
( mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V6f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))))
& ! [V7m] :
( mem(V7m,ty_2Enum_2Enum)
=> ! [V8n] :
( mem(V8n,ty_2Enum_2Enum)
=> ( V7m != V8n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V6f,V7m)),ap(V6f,V8n))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V6f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__COUNTABLY__ADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( V3m != V4n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,V3m)),ap(V2f,V4n))) ) ) )
& V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) )
=> p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__ADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__ADDITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t))
& V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) )
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__COUNTABLE__INCREASING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ap(V2f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a)
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,V3n)),ap(V2f,ap(c_2Enum_2ESUC,V3n)))) )
& V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__REDUCE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),ap(c_2Emeasure_2Emeasure(A_27a),V0m))) = V0m ) ) ).
fof(conj_thm_2Emeasure_2ESPACE__SIGMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ap(c_2Emeasure_2Espace(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)) = V0sp ) ) ) ).
fof(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,V3n)),ap(V2f,ap(c_2Enum_2ESUC,V3n)))) )
& V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__REDUCE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))) = ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__SETS__SUBSET__SPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__DIFF__PROPERTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
=> ! [V4s] :
( mem(V4s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),ap(ap(c_2Epred__set_2EDIFF(A_27b),ap(c_2Emeasure_2Espace(A_27b),V1b)),V4s)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V4s)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__BIGUNION__PROPERTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
=> ! [V4c] :
( mem(V4c,arr(arr(A_27b,bool),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27b,bool)),V4c),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),ap(c_2Epred__set_2EBIGUNION(A_27b),V4c)) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),V4c)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASUBABLE__BIGUNION__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
=> ! [V4c] :
( mem(V4c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V4c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V4c),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V4c)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__SIGMA__PREIMAGES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
=> p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b))))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Espace(A_27a),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__SIGMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2b] :
( mem(V2b,arr(arr(A_27b,bool),bool))
=> ! [V3sp] :
( mem(V3sp,arr(A_27b,bool))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Emeasure_2Esubset__class(A_27b),V3sp),V2b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V1a)),V3sp)))
& ! [V4s] :
( mem(V4s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4s),V2b))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V4s)),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),ap(ap(c_2Emeasure_2Esigma(A_27b),V3sp),V2b)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Espace(A_27b),V1b)),ap(c_2Emeasure_2Esubsets(A_27b),V1b))))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__LIFT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2b] :
( mem(V2b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),V2b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Espace(A_27b),V2b)),ap(c_2Emeasure_2Esubsets(A_27b),V2b))))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURE__PRESERVING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0m1] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2)))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2)))))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2))
& ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2)))
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m1),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Em__space(A_27a),V0m1))) = ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m2),V3s) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__LIFT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0m1] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2a] :
( mem(V2a,arr(arr(A_27b,bool),bool))
=> ! [V3f] :
( mem(V3f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2))
& ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2) = ap(c_2Emeasure_2Esubsets(A_27b),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),V2a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27b),V1m2)))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0m1] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2a] :
( mem(V2a,arr(arr(A_27b,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2))
& ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2) = ap(c_2Emeasure_2Esubsets(A_27b),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),V2a)) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27b),V1m2))))),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__I,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27a)),c_2Ecombin_2EI(A_27a)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27a),V0a),V0a))) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__COMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2a] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V3b] :
( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4c] :
( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27b,A_27c),V3b),V4c))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2a] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V3b] :
( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4c] :
( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27c),V4c))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Epred__set_2EFUNSET(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V3b)),ap(c_2Emeasure_2Espace(A_27c),V4c))))
& ! [V5x] :
( mem(V5x,arr(A_27c,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V5x),ap(c_2Emeasure_2Esubsets(A_27c),V4c)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V5x)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Emeasure_2Espace(A_27a),V2a)))),ap(c_2Emeasure_2Esubsets(A_27b),V3b))) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONGER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2a] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V3b] :
( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4c] :
( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ! [V5t] :
( mem(V5t,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27c),V4c))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Epred__set_2EFUNSET(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V3b)),ap(c_2Emeasure_2Espace(A_27c),V4c))))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Emeasure_2Espace(A_27a),V2a))),V5t))
& ! [V6s] :
( mem(V6s,arr(A_27c,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V6s),ap(c_2Emeasure_2Esubsets(A_27c),V4c)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V6s)),V5t)),ap(c_2Emeasure_2Esubsets(A_27b),V3b))) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__UP__LIFT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2b] :
( mem(V2b,arr(arr(A_27a,bool),bool))
=> ! [V3c] :
( mem(V3c,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4f] :
( mem(V4f,arr(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V4f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)),V3c)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2b)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V4f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)),V3c))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__UP__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2b] :
( mem(V2b,arr(arr(A_27a,bool),bool))
=> ! [V3c] :
( mem(V3c,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2b))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)),V3c)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)),V3c))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__UP__SIGMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))),V1b))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__LIFT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1m1] :
( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2m2] :
( mem(V2m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V3f] :
( mem(V3f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V0a),ap(c_2Emeasure_2Emeasure(A_27a),V1m1)))),V2m2)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V1m1),V2m2))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1m1] :
( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2m2] :
( mem(V2m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V0a),ap(c_2Emeasure_2Emeasure(A_27a),V1m1)))),V2m2)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V1m1),V2m2))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SIGMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0m1] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2a] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1))
& ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1) = ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),V2a)) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27a),V0m1)))),V1m2)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__PROD__SIGMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1a1] :
( mem(V1a1,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2a2] :
( mem(V2a2,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ! [V3f] :
( mem(V3f,arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,ty_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2EFST(A_27b,A_27c)),V3f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1a1)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,ty_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2ESND(A_27b,A_27c)),V3f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V0a),V2a2))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),V3f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V0a),ap(ap(c_2Emeasure_2Esigma(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V1a1)),ap(c_2Emeasure_2Espace(A_27c),V2a2))),ap(ap(c_2Eutil__prob_2Eprod__sets(A_27b,A_27c),ap(c_2Emeasure_2Esubsets(A_27b),V1a1)),ap(c_2Emeasure_2Esubsets(A_27c),V2a2)))))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__RANGE__REDUCE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),V2s),ap(c_2Epred__set_2EPOW(A_27b),V2s)))))
& V2s != c_2Epred__set_2EEMPTY(A_27b) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))))))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),c_2Epred__set_2EUNIV(A_27b)),ap(c_2Epred__set_2EPOW(A_27b),c_2Epred__set_2EUNIV(A_27b)))))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))))))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1s_27] :
( mem(V1s_27,arr(A_27a,bool))
=> ! [V2m] :
( mem(V2m,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s_27),V0s))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0s),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V0s)),V2m)))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s_27),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),V2m)))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESTRONG__MEASURE__SPACE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s_27] :
( mem(V1s_27,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s_27),ap(c_2Emeasure_2Em__space(A_27a),V0s)))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0s))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0s))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s_27),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),ap(c_2Emeasure_2Emeasure(A_27a),V0s))))) ) ) ) ) ).
fof(lameq_f3437,axiom,
! [A_27a,V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3x] : ap(f3437(A_27a,V0m),V3x) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EINSERT(A_27a),V3x),c_2Epred__set_2EEMPTY(A_27a))) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__REAL__SUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) )
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3437(A_27a,V0m)),V1s) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Emeasure_2Esigma(A_27a),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s)) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s)) ) ) ).
fof(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1m] :
( mem(V1m,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0s)))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m))))
& p(ap(c_2Emeasure_2Eadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) ) ) ) ).
fof(conj_thm_2Emeasure_2EIMAGE__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))) = ap(ap(c_2Epred__set_2EINSERT(A_27b),ap(V0f,V1x)),c_2Epred__set_2EEMPTY(A_27b)) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESUBSET__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0X] :
( mem(V0X,arr(A_27a,bool))
=> ! [V1P] :
( mem(V1P,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),ap(c_2Epred__set_2EBIGINTER(A_27a),V1P)))
<=> ! [V2Y] :
( mem(V2Y,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2Y),V1P))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),V2Y)) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__INCREASING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__POSITIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__COMPL__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,arr(A_27a,bool))
=> ! [V1m] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0t),V2s)) )
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),ap(ap(c_2Epred__set_2EDIFF(A_27a),V2s),V0t)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V2s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V0t)) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1s,V2n)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__IN__MSPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1A] :
( mem(V1A,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1A))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET__MSPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1m] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0A),ap(c_2Emeasure_2Em__space(A_27a),V1m))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__EMPTY__MEASURABLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__MSPACE__MEASURABLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1s,V2n)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,V2n)),ap(V1f,ap(c_2Enum_2ESUC,V2n)))) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,ap(c_2Enum_2ESUC,V3n))),ap(V2f,V3n))) )
& V1s = ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,ap(c_2Enum_2ESUC,V2n))),ap(V1f,V2n))) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ).
fof(lameq_f3438,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] : ap(f3438(A_27a,V1s),V2t) = ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__RESTRICTED,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),f3438(A_27a,V1s)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(c_2Emeasure_2Emeasure(A_27a),V0m))))) ) ) ) ) ).
fof(lameq_f3439,axiom,
! [A_27a,V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2a] : ap(f3439(A_27a,V0m,V1c),V2a) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2a)) ) ) ).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__CMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1c)) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),f3439(A_27a,V0m,V1c))))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EBIGUNION__IMAGE__Q,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(ty_2Eextreal_2Eextreal,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Eextreal_2Eextreal,arr(A_27a,bool)),c_2Eextreal_2EQ__set),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,arr(A_27a,bool)),V1f),c_2Eextreal_2EQ__set))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
fof(lameq_f3440,axiom,
! [V0a] :
( mem(V0a,ty_2Eextreal_2Eextreal)
=> ! [V1x] : ap(f3440(V0a),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V1x),ap(ap(c_2Eextreal_2Eextreal__lt,V1x),V0a)) ) ).
fof(lameq_f3441,axiom,
! [V0a] : ap(f3441,V0a) = ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3440(V0a)) ).
fof(ax_thm_2Emeasure_2EBorel__def,axiom,
c_2Emeasure_2EBorel = ap(ap(c_2Emeasure_2Esigma(ty_2Eextreal_2Eextreal),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)),f3441),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))) ).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__BOREL,axiom,
p(ap(c_2Emeasure_2Esigma__algebra(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)) ).
fof(lameq_f3442,axiom,
! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3x] : ap(f3442(V2c),V3x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__lt,V3x),V2c)) ) ).
fof(conj_thm_2Emeasure_2EMEASURABLE__BOREL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,ty_2Eextreal_2Eextreal),V0f),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3442(V2c)))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ).
fof(lameq_f3443,axiom,
! [A_27a,V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3443(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V3x)),V2c)) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3443(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ).
fof(lameq_f3444,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x] : ap(f3444(A_27a,V0f),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V2x)),c_2Eextreal_2ENegInf)) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__NEGINF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3444(A_27a,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
fof(lameq_f3445,axiom,
! [A_27a,V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3445(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__le,V2c),ap(V0f,V3x))) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3445(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ).
fof(lameq_f3446,axiom,
! [A_27a,V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3446(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V3x)),V2c)) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3446(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ).
fof(lameq_f3447,axiom,
! [A_27a,V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3447(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__lt,V2c),ap(V0f,V3x))) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT3,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3447(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ).
fof(lameq_f3448,axiom,
! [A_27a,V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V4x] : ap(f3448(A_27a,V3d,V0f,V2c),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,V2c),ap(V0f,V4x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V4x)),V3d))) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT4,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3448(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ) ).
fof(lameq_f3449,axiom,
! [A_27a,V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V4x] : ap(f3449(A_27a,V3d,V0f,V2c),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V2c),ap(V0f,V4x))),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V4x)),V3d))) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT5,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3449(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ) ).
fof(lameq_f3450,axiom,
! [A_27a,V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V4x] : ap(f3450(A_27a,V3d,V0f,V2c),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,V2c),ap(V0f,V4x))),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V4x)),V3d))) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT6,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3450(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ) ).
fof(lameq_f3451,axiom,
! [A_27a,V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V4x] : ap(f3451(A_27a,V3d,V0f,V2c),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V2c),ap(V0f,V4x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V4x)),V3d))) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT7,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3d] :
( mem(V3d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3451(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ) ).
fof(lameq_f3452,axiom,
! [A_27a,V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3452(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V3x)),V2c)) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT8,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3452(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ).
fof(lameq_f3453,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3x] : ap(f3453(A_27a,V0f,V2c),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V3x)),V2c))) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT9,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3453(A_27a,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ).
fof(lameq_f3454,axiom,
! [A_27a,V4c] :
( mem(V4c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5x] : ap(f3454(A_27a,V4c,V0f),V5x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V5x),ap(ap(c_2Eextreal_2Eextreal__le,V4c),ap(V0f,V5x))) ) ) ).
fof(lameq_f3455,axiom,
! [A_27a,V6c] :
( mem(V6c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V7x] : ap(f3455(A_27a,V6c,V0f),V7x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V7x),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V7x)),V6c)) ) ) ).
fof(lameq_f3456,axiom,
! [A_27a,V8c] :
( mem(V8c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V9x] : ap(f3456(A_27a,V8c,V0f),V9x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V9x),ap(ap(c_2Eextreal_2Eextreal__lt,V8c),ap(V0f,V9x))) ) ) ).
fof(lameq_f3457,axiom,
! [A_27a,V11d] :
( mem(V11d,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V10c] :
( mem(V10c,ty_2Eextreal_2Eextreal)
=> ! [V12x] : ap(f3457(A_27a,V11d,V0f,V10c),V12x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V12x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V10c),ap(V0f,V12x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V12x)),V11d))) ) ) ) ).
fof(lameq_f3458,axiom,
! [A_27a,V14d] :
( mem(V14d,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V13c] :
( mem(V13c,ty_2Eextreal_2Eextreal)
=> ! [V15x] : ap(f3458(A_27a,V14d,V0f,V13c),V15x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V15x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,V13c),ap(V0f,V15x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V15x)),V14d))) ) ) ) ).
fof(lameq_f3459,axiom,
! [A_27a,V17d] :
( mem(V17d,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V16c] :
( mem(V16c,ty_2Eextreal_2Eextreal)
=> ! [V18x] : ap(f3459(A_27a,V17d,V0f,V16c),V18x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V18x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V16c),ap(V0f,V18x))),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V18x)),V17d))) ) ) ) ).
fof(lameq_f3460,axiom,
! [A_27a,V20d] :
( mem(V20d,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V19c] :
( mem(V19c,ty_2Eextreal_2Eextreal)
=> ! [V21x] : ap(f3460(A_27a,V20d,V0f,V19c),V21x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V21x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,V19c),ap(V0f,V21x))),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V21x)),V20d))) ) ) ) ).
fof(lameq_f3461,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V22c] :
( mem(V22c,ty_2Eextreal_2Eextreal)
=> ! [V23x] : ap(f3461(A_27a,V0f,V22c),V23x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V23x),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V23x)),V22c))) ) ) ).
fof(lameq_f3462,axiom,
! [A_27a,V24c] :
( mem(V24c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V25x] : ap(f3462(A_27a,V24c,V0f),V25x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V25x),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V25x)),V24c)) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> ( ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3443(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) )
& ! [V4c] :
( mem(V4c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3454(A_27a,V4c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) )
& ! [V6c] :
( mem(V6c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3455(A_27a,V6c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) )
& ! [V8c] :
( mem(V8c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3456(A_27a,V8c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) )
& ! [V10c] :
( mem(V10c,ty_2Eextreal_2Eextreal)
=> ! [V11d] :
( mem(V11d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3457(A_27a,V11d,V0f,V10c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) )
& ! [V13c] :
( mem(V13c,ty_2Eextreal_2Eextreal)
=> ! [V14d] :
( mem(V14d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3458(A_27a,V14d,V0f,V13c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) )
& ! [V16c] :
( mem(V16c,ty_2Eextreal_2Eextreal)
=> ! [V17d] :
( mem(V17d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3459(A_27a,V17d,V0f,V16c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) )
& ! [V19c] :
( mem(V19c,ty_2Eextreal_2Eextreal)
=> ! [V20d] :
( mem(V20d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3460(A_27a,V20d,V0f,V19c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) )
& ! [V22c] :
( mem(V22c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3461(A_27a,V0f,V22c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) )
& ! [V24c] :
( mem(V24c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3462(A_27a,V24c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ).
fof(lameq_f3463,axiom,
! [A_27a,V22c] :
( mem(V22c,ty_2Eextreal_2Eextreal)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V23x] : ap(f3463(A_27a,V22c,V0f),V23x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V23x),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V23x)),V22c)) ) ) ).
fof(lameq_f3464,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V24c] :
( mem(V24c,ty_2Eextreal_2Eextreal)
=> ! [V25x] : ap(f3464(A_27a,V0f,V24c),V25x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V25x),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V25x)),V24c))) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL__MEASURE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1m] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))),c_2Emeasure_2EBorel)))
=> ( ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3443(A_27a,V2c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
& ! [V4c] :
( mem(V4c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3454(A_27a,V4c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
& ! [V6c] :
( mem(V6c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3455(A_27a,V6c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
& ! [V8c] :
( mem(V8c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3456(A_27a,V8c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
& ! [V10c] :
( mem(V10c,ty_2Eextreal_2Eextreal)
=> ! [V11d] :
( mem(V11d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3457(A_27a,V11d,V0f,V10c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) )
& ! [V13c] :
( mem(V13c,ty_2Eextreal_2Eextreal)
=> ! [V14d] :
( mem(V14d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3458(A_27a,V14d,V0f,V13c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) )
& ! [V16c] :
( mem(V16c,ty_2Eextreal_2Eextreal)
=> ! [V17d] :
( mem(V17d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3459(A_27a,V17d,V0f,V16c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) )
& ! [V19c] :
( mem(V19c,ty_2Eextreal_2Eextreal)
=> ! [V20d] :
( mem(V20d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3460(A_27a,V20d,V0f,V19c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) )
& ! [V22c] :
( mem(V22c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3463(A_27a,V22c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
& ! [V24c] :
( mem(V24c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3464(A_27a,V0f,V24c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) ) ) ) ).
fof(lameq_f3465,axiom,
! [A_27a,V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3465(A_27a,V1g,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V3x)),ap(V1g,V3x))) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3465(A_27a,V1g,V0f))),ap(c_2Emeasure_2Espace(A_27a),V2a))),ap(c_2Emeasure_2Esubsets(A_27a),V2a))) ) ) ) ) ) ).
fof(lameq_f3466,axiom,
! [A_27a,V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3466(A_27a,V1g,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V3x)),ap(V1g,V3x))) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3466(A_27a,V1g,V0f))),ap(c_2Emeasure_2Espace(A_27a),V2a))),ap(c_2Emeasure_2Esubsets(A_27a),V2a))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2ESPACE__BOREL,axiom,
ap(c_2Emeasure_2Espace(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel) = c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal) ).
fof(lameq_f3467,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1x] : ap(f3467(V0c),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V1x),ap(ap(c_2Eextreal_2Eextreal__lt,V1x),V0c)) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS1,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3467(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ).
fof(lameq_f3468,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1x] : ap(f3468(V0c),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V1x),ap(ap(c_2Eextreal_2Eextreal__le,V0c),V1x)) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS2,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3468(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ).
fof(lameq_f3469,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1x] : ap(f3469(V0c),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V1x),ap(ap(c_2Eextreal_2Eextreal__le,V1x),V0c)) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS3,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3469(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ).
fof(lameq_f3470,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1x] : ap(f3470(V0c),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V1x),ap(ap(c_2Eextreal_2Eextreal__lt,V0c),V1x)) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS4,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3470(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ).
fof(lameq_f3471,axiom,
! [V1d] :
( mem(V1d,ty_2Eextreal_2Eextreal)
=> ! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V2x] : ap(f3471(V1d,V0c),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,V0c),V2x)),ap(ap(c_2Eextreal_2Eextreal__lt,V2x),V1d))) ) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS5,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1d] :
( mem(V1d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3471(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ).
fof(lameq_f3472,axiom,
! [V1d] :
( mem(V1d,ty_2Eextreal_2Eextreal)
=> ! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V2x] : ap(f3472(V1d,V0c),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V0c),V2x)),ap(ap(c_2Eextreal_2Eextreal__le,V2x),V1d))) ) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS6,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1d] :
( mem(V1d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3472(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ).
fof(lameq_f3473,axiom,
! [V1d] :
( mem(V1d,ty_2Eextreal_2Eextreal)
=> ! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V2x] : ap(f3473(V1d,V0c),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,V0c),V2x)),ap(ap(c_2Eextreal_2Eextreal__le,V2x),V1d))) ) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS7,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1d] :
( mem(V1d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3473(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ).
fof(lameq_f3474,axiom,
! [V1d] :
( mem(V1d,ty_2Eextreal_2Eextreal)
=> ! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V2x] : ap(f3474(V1d,V0c),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V0c),V2x)),ap(ap(c_2Eextreal_2Eextreal__lt,V2x),V1d))) ) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS8,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1d] :
( mem(V1d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3474(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS9,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),V0c),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ).
fof(lameq_f3475,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> ! [V1x] : ap(f3475(V0c),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V1x),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V1x),V0c))) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS10,axiom,
! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3475(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ).
fof(lameq_f3476,axiom,
! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> ! [V3x] : ap(f3476(V2c),V3x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__le,V2c),V3x)) ) ).
fof(lameq_f3477,axiom,
! [V4c] :
( mem(V4c,ty_2Eextreal_2Eextreal)
=> ! [V5x] : ap(f3477(V4c),V5x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V5x),ap(ap(c_2Eextreal_2Eextreal__lt,V4c),V5x)) ) ).
fof(lameq_f3478,axiom,
! [V6c] :
( mem(V6c,ty_2Eextreal_2Eextreal)
=> ! [V7x] : ap(f3478(V6c),V7x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V7x),ap(ap(c_2Eextreal_2Eextreal__le,V7x),V6c)) ) ).
fof(lameq_f3479,axiom,
! [V9d] :
( mem(V9d,ty_2Eextreal_2Eextreal)
=> ! [V8c] :
( mem(V8c,ty_2Eextreal_2Eextreal)
=> ! [V10x] : ap(f3479(V9d,V8c),V10x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V10x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V8c),V10x)),ap(ap(c_2Eextreal_2Eextreal__lt,V10x),V9d))) ) ) ).
fof(lameq_f3480,axiom,
! [V12d] :
( mem(V12d,ty_2Eextreal_2Eextreal)
=> ! [V11c] :
( mem(V11c,ty_2Eextreal_2Eextreal)
=> ! [V13x] : ap(f3480(V12d,V11c),V13x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V13x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,V11c),V13x)),ap(ap(c_2Eextreal_2Eextreal__lt,V13x),V12d))) ) ) ).
fof(lameq_f3481,axiom,
! [V15d] :
( mem(V15d,ty_2Eextreal_2Eextreal)
=> ! [V14c] :
( mem(V14c,ty_2Eextreal_2Eextreal)
=> ! [V16x] : ap(f3481(V15d,V14c),V16x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V16x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,V14c),V16x)),ap(ap(c_2Eextreal_2Eextreal__le,V16x),V15d))) ) ) ).
fof(lameq_f3482,axiom,
! [V18d] :
( mem(V18d,ty_2Eextreal_2Eextreal)
=> ! [V17c] :
( mem(V17c,ty_2Eextreal_2Eextreal)
=> ! [V19x] : ap(f3482(V18d,V17c),V19x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V19x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,V17c),V19x)),ap(ap(c_2Eextreal_2Eextreal__le,V19x),V18d))) ) ) ).
fof(lameq_f3483,axiom,
! [V21c] :
( mem(V21c,ty_2Eextreal_2Eextreal)
=> ! [V22x] : ap(f3483(V21c),V22x) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V22x),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V22x),V21c))) ) ).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS,axiom,
( ! [V0c] :
( mem(V0c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3467(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) )
& ! [V2c] :
( mem(V2c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3476(V2c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) )
& ! [V4c] :
( mem(V4c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3477(V4c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) )
& ! [V6c] :
( mem(V6c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3478(V6c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) )
& ! [V8c] :
( mem(V8c,ty_2Eextreal_2Eextreal)
=> ! [V9d] :
( mem(V9d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3479(V9d,V8c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) )
& ! [V11c] :
( mem(V11c,ty_2Eextreal_2Eextreal)
=> ! [V12d] :
( mem(V12d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3480(V12d,V11c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) )
& ! [V14c] :
( mem(V14c,ty_2Eextreal_2Eextreal)
=> ! [V15d] :
( mem(V15d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3481(V15d,V14c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) )
& ! [V17c] :
( mem(V17c,ty_2Eextreal_2Eextreal)
=> ! [V18d] :
( mem(V18d,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3482(V18d,V17c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) )
& ! [V20c] :
( mem(V20c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),V20c),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) )
& ! [V21c] :
( mem(V21c,ty_2Eextreal_2Eextreal)
=> p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3483(V21c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CONST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1k] :
( mem(V1k,ty_2Eextreal_2Eextreal)
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V2f,V3x) = V1k ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__INDICATOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1A] :
( mem(V1A,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1A),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V2f,V3x) = ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V1A),V3x) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3z] :
( mem(V3z,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V2g,V4x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V3z)),ap(V1f,V4x)) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ABS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V2g,V3x) = ap(c_2Eextreal_2Eextreal__abs,ap(V1f,V3x)) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SQR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V2g,V3x) = ap(ap(c_2Eextreal_2Eextreal__pow,ap(V1f,V3x)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3h] :
( mem(V3h,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V3h,V4x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V4x)),ap(V2g,V4x)) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V3h),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SUB,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3h] :
( mem(V3h,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V3h,V4x) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V1f,V4x)),ap(V2g,V4x)) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V3h),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3h] :
( mem(V3h,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( ap(V1f,V4x) != c_2Eextreal_2ENegInf
& ap(V1f,V4x) != c_2Eextreal_2EPosInf
& ap(V2g,V4x) != c_2Eextreal_2ENegInf
& ap(V2g,V4x) != c_2Eextreal_2EPosInf ) ) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V3h,V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V5x)),ap(V2g,V5x)) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V3h),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ) ).
fof(lameq_f3484,axiom,
! [A_27a,A_27b,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ! [V6i] : ap(f3484(A_27a,A_27b,V1f,V5x),V6i) = ap(ap(V1f,V6i),V5x) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& ! [V4i] :
( mem(V4i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4i),V3s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V1f,V4i)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) )
& ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ap(V2g,V5x) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f3484(A_27a,A_27b,V1f,V5x)),V3s) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ) ) ).
fof(lameq_f3485,axiom,
! [A_27a,V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1z] :
( mem(V1z,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f3485(A_27a,V2s,V1z),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1z)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V2s),V3x)) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CMUL__INDICATOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1z] :
( mem(V1z,ty_2Erealax_2Ereal)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3485(A_27a,V2s,V1z)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(lameq_f3486,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3x] : ap(f3486(A_27a,V1f,V2s),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V3x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V2s),V3x)) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL__INDICATOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3486(A_27a,V1f,V2s)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(lameq_f3487,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2x] : ap(f3487(A_27a,V1f,V0a),V2x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V2x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V2x)) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL__INDICATOR__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
<=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3487(A_27a,V1f,V0a)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__POS__SIMPLE__FN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ? [V2s] :
( mem(V2s,arr(ty_2Enum_2Enum,bool))
& ? [V3a] :
( mem(V3a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V4x] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x)) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),c_2Emeasure_2EBorel))) ) ) ) ) ).
fof(lameq_f3488,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V4x] : ap(f3488(A_27a,V2f,V0n),V4x) = ap(ap(c_2Eextreal_2Eextreal__pow,ap(V2f,V4x)),V0n) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1a] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V1a)))
=> ( ap(V2f,V3x) != c_2Eextreal_2ENegInf
& ap(V2f,V3x) != c_2Eextreal_2EPosInf ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3488(A_27a,V2f,V0n)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(lameq_f3489,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3489(A_27a,V1f,V2g),V3x) = ap(ap(c_2Eextreal_2Eextreal__max,ap(V1f,V3x)),ap(V2g,V3x)) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MAX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3489(A_27a,V1f,V2g)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(lameq_f3490,axiom,
! [A_27a,V0fn] :
( mem(V0fn,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7n] : ap(f3490(A_27a,V0fn,V6x),V7n) = ap(ap(V0fn,V7n),V6x) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MONO__SUP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0fn] :
( mem(V0fn,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V2a))
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V0fn,V3n)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) )
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Espace(A_27a),V2a)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(V0fn,V4n),V5x)),ap(ap(V0fn,ap(c_2Enum_2ESUC,V4n)),V5x))) ) ) )
& ! [V6x] :
( mem(V6x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),ap(c_2Emeasure_2Espace(A_27a),V2a)))
=> ap(V1f,V6x) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3490(A_27a,V0fn,V6x)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(lameq_f3491,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x] : ap(f3491(A_27a,V0f),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0f,V1x))),ap(V0f,V1x)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) ) ).
fof(ax_thm_2Emeasure_2Efn__plus__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ap(c_2Emeasure_2Efn__plus(A_27a),V0f) = f3491(A_27a,V0f) ) ) ).
fof(lameq_f3492,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x] : ap(f3492(A_27a,V0f),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V1x)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))),ap(c_2Eextreal_2Eextreal__ainv,ap(V0f,V1x))),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) ) ).
fof(ax_thm_2Emeasure_2Efn__minus__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ap(c_2Emeasure_2Efn__minus(A_27a),V0f) = f3492(A_27a,V0f) ) ) ).
fof(lameq_f3493,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x] : ap(f3493(A_27a,V0f),V1x) = ap(c_2Eextreal_2Eextreal__abs,ap(V0f,V1x)) ) ).
fof(ax_thm_2Emeasure_2Efn__abs__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ap(c_2Emeasure_2Efn__abs(A_27a),V0f) = f3493(A_27a,V0f) ) ) ).
fof(conj_thm_2Emeasure_2EFN__PLUS__POS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V0g),V1x))) ) ) ) ).
fof(conj_thm_2Emeasure_2EFN__MINUS__POS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V0g),V1x))) ) ) ) ).
fof(conj_thm_2Emeasure_2EFN__PLUS__POS__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0g,V1x))) )
=> ap(c_2Emeasure_2Efn__plus(A_27a),V0g) = V0g ) ) ) ).
fof(conj_thm_2Emeasure_2EFN__MINUS__POS__ZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V0g,V1x))) )
=> ap(c_2Emeasure_2Efn__minus(A_27a),V0g) = k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)) ) ) ) ).
fof(lameq_f3494,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2x] : ap(f3494(A_27a,V0f,V1c),V2x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1c)),ap(V0f,V2x)) ) ) ).
fof(lameq_f3495,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f3495(A_27a,V0f,V1c),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1c)),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V0f),V3x)) ) ) ).
fof(lameq_f3496,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V4x] : ap(f3496(A_27a,V0f,V1c),V4x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1c)),ap(V0f,V4x)) ) ) ).
fof(lameq_f3497,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V5x] : ap(f3497(A_27a,V0f,V1c),V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2ENormal,V1c))),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V0f),V5x)) ) ) ).
fof(conj_thm_2Emeasure_2EFN__PLUS__CMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1c))
=> ap(c_2Emeasure_2Efn__plus(A_27a),f3494(A_27a,V0f,V1c)) = f3495(A_27a,V0f,V1c) )
& ( p(ap(ap(c_2Ereal_2Ereal__lte,V1c),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
=> ap(c_2Emeasure_2Efn__plus(A_27a),f3496(A_27a,V0f,V1c)) = f3497(A_27a,V0f,V1c) ) ) ) ) ) ).
fof(lameq_f3498,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f3498(A_27a,V0f,V1c),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V1c)),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V0f),V3x)) ) ) ).
fof(lameq_f3499,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V5x] : ap(f3499(A_27a,V0f,V1c),V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2ENormal,V1c))),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V0f),V5x)) ) ) ).
fof(conj_thm_2Emeasure_2EFN__MINUS__CMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1c))
=> ap(c_2Emeasure_2Efn__minus(A_27a),f3494(A_27a,V0f,V1c)) = f3498(A_27a,V0f,V1c) )
& ( p(ap(ap(c_2Ereal_2Ereal__lte,V1c),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
=> ap(c_2Emeasure_2Efn__minus(A_27a),f3496(A_27a,V0f,V1c)) = f3499(A_27a,V0f,V1c) ) ) ) ) ) ).
fof(lameq_f3500,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f3500(A_27a,V0f,V1g),V3x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Emeasure_2EFN__PLUS__ADD__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Emeasure_2Efn__plus(A_27a),f3500(A_27a,V0f,V1g)),V2x)),ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Emeasure_2Efn__plus(A_27a),V0f),V2x)),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V1g),V2x)))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EFN__MINUS__ADD__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Emeasure_2Efn__minus(A_27a),f3500(A_27a,V0f,V1g)),V2x)),ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Emeasure_2Efn__minus(A_27a),V0f),V2x)),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V1g),V2x)))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__FN__PLUS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(c_2Emeasure_2Efn__plus(A_27a),V1f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__FN__MINUS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(c_2Emeasure_2Efn__minus(A_27a),V1f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__PLUS__MINUS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(c_2Emeasure_2Efn__plus(A_27a),V1f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(c_2Emeasure_2Efn__minus(A_27a),V1f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
fof(lameq_f3501,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V2t] : ap(f3501(A_27a,V0A,V1B),V2t) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V0A),V2t)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V1B),V2t)) ) ) ).
fof(lameq_f3502,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V3t] : ap(f3502(A_27a,V0A,V1B),V3t) = ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0A),V1B)),V3t) ) ) ).
fof(conj_thm_2Emeasure_2EINDICATOR__FN__MUL__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( mem(V1B,arr(A_27a,bool))
=> f3501(A_27a,V0A,V1B) = f3502(A_27a,V0A,V1B) ) ) ) ).
fof(lameq_f3503,axiom,
! [A_27a,V2b] :
( mem(V2b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5a] :
( mem(V5a,arr(A_27a,bool))
=> ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7i] : ap(f3503(A_27a,V2b,V5a,V6x),V7i) = ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V5a),ap(V2b,V7i))),V6x) ) ) ) ).
fof(lameq_f3504,axiom,
! [A_27a,V2b] :
( mem(V2b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5a] :
( mem(V5a,arr(A_27a,bool))
=> ! [V0r] :
( mem(V0r,arr(ty_2Enum_2Enum,bool))
=> ! [V6x] : ap(f3504(A_27a,V2b,V5a,V0r),V6x) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3503(A_27a,V2b,V5a,V6x)),V0r) ) ) ) ).
fof(conj_thm_2Emeasure_2Eindicator__fn__split,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Enum_2Enum,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2b] :
( mem(V2b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0r))
& ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2b),V0r)) = V1s
& ! [V3i] :
( mem(V3i,ty_2Enum_2Enum)
=> ! [V4j] :
( mem(V4j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3i),V0r))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4j),V0r))
& V3i != V4j )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2b,V3i)),ap(V2b,V4j))) ) ) ) )
=> ! [V5a] :
( mem(V5a,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V5a),V1s))
=> ap(c_2Emeasure_2Eindicator__fn(A_27a),V5a) = f3504(A_27a,V2b,V5a,V0r) ) ) ) ) ) ) ) ).
fof(lameq_f3505,axiom,
! [A_27a,V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V4i] : ap(f3505(A_27a,V0a,V1x),V4i) = ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V0a,V4i)),V1x) ) ) ).
fof(conj_thm_2Emeasure_2Eindicator__fn__suminf,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( V2m != V3n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0a,V2m)),ap(V0a,V3n))) ) ) )
=> ap(c_2Eextreal_2Eext__suminf,f3505(A_27a,V0a,V1x)) = ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V0a),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1x) ) ) ) ) ).
fof(lameq_f3506,axiom,
! [A_27a,V6a] :
( mem(V6a,arr(A_27a,bool))
=> ! [V1b] :
( mem(V1b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2m] :
( mem(V2m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V7i] : ap(f3506(A_27a,V6a,V1b,V2m),V7i) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V2m),ap(ap(c_2Epred__set_2EINTER(A_27a),V6a),ap(V1b,V7i))) ) ) ) ).
fof(conj_thm_2Emeasure_2Emeasure__split,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(ty_2Enum_2Enum,bool))
=> ! [V1b] :
( mem(V1b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2m] :
( mem(V2m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V2m))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0r))
& ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1b),V0r)) = ap(c_2Emeasure_2Em__space(A_27a),V2m)
& ! [V3i] :
( mem(V3i,ty_2Enum_2Enum)
=> ! [V4j] :
( mem(V4j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3i),V0r))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4j),V0r))
& V3i != V4j )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1b,V3i)),ap(V1b,V4j))) ) ) )
& ! [V5i] :
( mem(V5i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5i),V0r))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1b,V5i)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V2m))) ) ) )
=> ! [V6a] :
( mem(V6a,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V6a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V2m)))
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V2m),V6a) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Enum_2Enum),f3506(A_27a,V6a,V1b,V2m)),V0r) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------