ITP001 Axioms: ITP148+5.ax
%------------------------------------------------------------------------------
% File : ITP148+5 : TPTP v9.0.0. 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 : lebesgue+2.ax [Gau20]
% : HL4148+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 291 ( 2 unt; 0 def)
% Number of atoms : 1990 ( 295 equ)
% Maximal formula atoms : 42 ( 6 avg)
% Number of connectives : 1728 ( 29 ~; 1 |; 414 &)
% ( 8 <=>;1276 =>; 0 <=; 0 <~>)
% Maximal formula depth : 46 ( 11 avg)
% Maximal term depth : 14 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 239 ( 239 usr; 45 con; 0-8 aty)
% Number of variables : 1445 (1384 !; 61 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Elebesgue_2ERADON__F,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2ERADON__F(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(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),bool)))) ) ).
fof(mem_c_2Elebesgue_2ERADON__F__integrals,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2ERADON__F__integrals(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(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_2Eextreal_2Eextreal,bool)))) ) ).
fof(mem_c_2Elebesgue_2Efinite__space__integral,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Efinite__space__integral(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),ty_2Eextreal_2Eextreal))) ) ).
fof(mem_c_2Elebesgue_2Efn__seq,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Efn__seq(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(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal))))) ) ).
fof(mem_c_2Elebesgue_2Efn__seq__integral,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Efn__seq__integral(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(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))) ) ).
fof(mem_c_2Elebesgue_2Eintegrable,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Eintegrable(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),bool))) ) ).
fof(mem_c_2Elebesgue_2Eintegral,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Eintegral(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),ty_2Eextreal_2Eextreal))) ) ).
fof(mem_c_2Elebesgue_2Emax__fn__seq,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Emax__fn__seq(A_27a),arr(arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)),arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))) ) ).
fof(mem_c_2Elebesgue_2Emeasure__absolutely__continuous,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Emeasure__absolutely__continuous(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(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_2Elebesgue_2Enegative__set,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Enegative__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_2Elebesgue_2Epos__fn__integral,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Epos__fn__integral(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),ty_2Eextreal_2Eextreal))) ) ).
fof(mem_c_2Elebesgue_2Epos__simple__fn__integral,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Epos__simple__fn__integral(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(ty_2Enum_2Enum,bool),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Eextreal_2Eextreal))))) ) ).
fof(mem_c_2Elebesgue_2Eprod__measure,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Elebesgue_2Eprod__measure(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(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal)))) ) ) ).
fof(mem_c_2Elebesgue_2Eprod__measure3,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Elebesgue_2Eprod__measure3(A_27a,A_27b,A_27c),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(ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal))))) ) ) ) ).
fof(mem_c_2Elebesgue_2Eprod__measure__space,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Elebesgue_2Eprod__measure__space(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))),ty_2Epair_2Eprod(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal)))))) ) ) ).
fof(mem_c_2Elebesgue_2Eprod__measure__space3,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Elebesgue_2Eprod__measure__space3(A_27a,A_27b,A_27c),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(ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))),ty_2Epair_2Eprod(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal))))))) ) ) ) ).
fof(mem_c_2Elebesgue_2Eprod__sets3,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Elebesgue_2Eprod__sets3(A_27a,A_27b,A_27c),arr(arr(arr(A_27a,bool),bool),arr(arr(arr(A_27b,bool),bool),arr(arr(arr(A_27c,bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool))))) ) ) ) ).
fof(mem_c_2Elebesgue_2Epsfis,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Epsfis(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(ty_2Eextreal_2Eextreal,bool)))) ) ).
fof(mem_c_2Elebesgue_2Epsfs,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Epsfs(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(ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))),bool)))) ) ).
fof(mem_c_2Elebesgue_2Eseq__sup,axiom,
mem(c_2Elebesgue_2Eseq__sup,arr(arr(ty_2Eextreal_2Eextreal,bool),arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Elebesgue_2Esigned__measure__space,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Elebesgue_2Esigned__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(lameq_f3991,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))))
=> ! [V2a] :
( mem(V2a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4i] : ap(f3991(A_27a,V0m,V2a,V3x),V4i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V3x,V4i)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(V2a,V4i))) ) ) ) ).
fof(ax_thm_2Elebesgue_2Epos__simple__fn__integral__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(ty_2Enum_2Enum,bool))
=> ! [V2a] :
( mem(V2a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V1s),V2a),V3x) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Enum_2Enum),f3991(A_27a,V0m,V2a,V3x)),V1s)) ) ) ) ) ) ).
fof(lameq_f3992,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))))
=> ! [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] : ap(f3992(A_27a,V0m,V1f,V2s,V3a),V4x) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))),bool),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))),V2s),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V3a),V4x))),ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x)) ) ) ) ) ).
fof(lameq_f3993,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s] :
( mem(V2s,arr(ty_2Enum_2Enum,bool))
=> ! [V3a] : ap(f3993(A_27a,V0m,V1f,V2s),V3a) = f3992(A_27a,V0m,V1f,V2s,V3a) ) ) ) ).
fof(lameq_f3994,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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))))
=> ! [V2s] : ap(f3994(A_27a,V1f,V0m),V2s) = ap(c_2Epair_2EUNCURRY(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Epair_2Eprod(ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))),bool)),f3993(A_27a,V0m,V1f,V2s)) ) ) ).
fof(ax_thm_2Elebesgue_2Epsfs__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))
=> ap(ap(c_2Elebesgue_2Epsfs(A_27a),V0m),V1f) = ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))),ap(c_2Epair_2EUNCURRY(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),ty_2Epair_2Eprod(ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))),bool)),f3994(A_27a,V1f,V0m))) ) ) ) ).
fof(lameq_f3995,axiom,
! [A_27a,V3a] :
( mem(V3a,arr(ty_2Enum_2Enum,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))))
=> ! [V2s] :
( mem(V2s,arr(ty_2Enum_2Enum,bool))
=> ! [V4x] : ap(f3995(A_27a,V3a,V0m,V2s),V4x) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x) ) ) ) ).
fof(lameq_f3996,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))))
=> ! [V2s] :
( mem(V2s,arr(ty_2Enum_2Enum,bool))
=> ! [V3a] : ap(f3996(A_27a,V0m,V2s),V3a) = f3995(A_27a,V3a,V0m,V2s) ) ) ).
fof(lameq_f3997,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))))
=> ! [V2s] : ap(f3997(A_27a,V0m),V2s) = ap(c_2Epair_2EUNCURRY(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Eextreal_2Eextreal),f3996(A_27a,V0m,V2s)) ) ).
fof(ax_thm_2Elebesgue_2Epsfis__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))
=> ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))),ty_2Eextreal_2Eextreal),ap(c_2Epair_2EUNCURRY(arr(ty_2Enum_2Enum,bool),ty_2Epair_2Eprod(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),ty_2Eextreal_2Eextreal),f3997(A_27a,V0m))),ap(ap(c_2Elebesgue_2Epsfs(A_27a),V0m),V1f)) ) ) ) ).
fof(lameq_f3998,axiom,
! [A_27a,V3g] :
( mem(V3g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4x] : ap(f3998(A_27a,V3g,V1f),V4x) = ap(ap(c_2Eextreal_2Eextreal__le,ap(V3g,V4x)),ap(V1f,V4x)) ) ) ).
fof(lameq_f3999,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))))
=> ! [V2r] :
( mem(V2r,ty_2Eextreal_2Eextreal)
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3g] : ap(f3999(A_27a,V0m,V2r,V1f),V3g) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V2r),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V3g))),ap(c_2Ebool_2E_21(A_27a),f3998(A_27a,V3g,V1f))) ) ) ) ).
fof(lameq_f4000,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2r] : ap(f4000(A_27a,V0m,V1f),V2r) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V2r),ap(c_2Ebool_2E_3F(arr(A_27a,ty_2Eextreal_2Eextreal)),f3999(A_27a,V0m,V2r,V1f))) ) ) ).
fof(ax_thm_2Elebesgue_2Epos__fn__integral__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))
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) = ap(c_2Eextreal_2Eextreal__sup,ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f4000(A_27a,V0m,V1f))) ) ) ) ).
fof(ax_thm_2Elebesgue_2Eintegral__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))
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(c_2Emeasure_2Efn__plus(A_27a),V1f))),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(c_2Emeasure_2Efn__minus(A_27a),V1f))) ) ) ) ).
fof(ax_thm_2Elebesgue_2Eintegrable__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))
=> ( p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
<=> ( 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)))
& ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(c_2Emeasure_2Efn__plus(A_27a),V1f)) != c_2Eextreal_2EPosInf
& ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(c_2Emeasure_2Efn__minus(A_27a),V1f)) != c_2Eextreal_2EPosInf ) ) ) ) ) ).
fof(lameq_f4001,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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(f4001(A_27a,V1f,V0m),V2r) = ap(ap(c_2Eextreal_2Eextreal__mul,V2r),ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1f),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),V2r),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal)))),ap(c_2Emeasure_2Em__space(A_27a),V0m))))) ) ) ).
fof(ax_thm_2Elebesgue_2Efinite__space__integral__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))
=> ap(ap(c_2Elebesgue_2Efinite__space__integral(A_27a),V0m),V1f) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Eextreal_2Eextreal),f4001(A_27a,V1f,V0m)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) ) ).
fof(lameq_f4002,axiom,
! [A_27a,A_27b,V3s0] :
( mem(V3s0,A_27a)
=> ! [V4s1] : ap(f4002(A_27a,A_27b,V3s0),V4s1) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V3s0),V4s1) ) ).
fof(lameq_f4003,axiom,
! [A_27b,A_27a,V1m1] :
( mem(V1m1,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(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V3s0] : ap(f4003(A_27b,A_27a,V1m1,V2a),V3s0) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m1),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,ty_2Epair_2Eprod(A_27a,A_27b)),f4002(A_27a,A_27b,V3s0)),V2a))) ) ) ).
fof(lameq_f4004,axiom,
! [A_27a,A_27b,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_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2a] : ap(f4004(A_27a,A_27b,V0m0,V1m1),V2a) = ap(c_2Eextreal_2Ereal,ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m0),f4003(A_27b,A_27a,V1m1,V2a))) ) ) ).
fof(ax_thm_2Elebesgue_2Eprod__measure__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [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_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ap(ap(c_2Elebesgue_2Eprod__measure(A_27a,A_27b),V0m0),V1m1) = f4004(A_27a,A_27b,V0m0,V1m1) ) ) ) ) ).
fof(ax_thm_2Elebesgue_2Eprod__measure__space__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [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_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ap(ap(c_2Elebesgue_2Eprod__measure__space(A_27a,A_27b),V0m0),V1m1) = ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Em__space(A_27b),V1m1))),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Emeasure_2Esigma(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Em__space(A_27b),V1m1))),ap(ap(c_2Eutil__prob_2Eprod__sets(A_27a,A_27b),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1))))),ap(ap(c_2Elebesgue_2Eprod__measure(A_27a,A_27b),V0m0),V1m1))) ) ) ) ) ).
fof(lameq_f4005,axiom,
! [A_27a,A_27b,A_27c,V0a] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1b] :
( mem(V1b,arr(arr(A_27b,bool),bool))
=> ! [V2c] :
( mem(V2c,arr(arr(A_27c,bool),bool))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ! [V4t] :
( mem(V4t,arr(A_27b,bool))
=> ! [V5u] : ap(f4005(A_27a,A_27b,A_27c,V0a,V1b,V2c,V3s,V4t),V5u) = ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V3s),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V4t),V5u))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V0a)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4t),V1b)),ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V5u),V2c)))) ) ) ) ) ) ).
fof(lameq_f4006,axiom,
! [A_27c,A_27a,A_27b,V0a] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1b] :
( mem(V1b,arr(arr(A_27b,bool),bool))
=> ! [V2c] :
( mem(V2c,arr(arr(A_27c,bool),bool))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ! [V4t] : ap(f4006(A_27c,A_27a,A_27b,V0a,V1b,V2c,V3s),V4t) = f4005(A_27a,A_27b,A_27c,V0a,V1b,V2c,V3s,V4t) ) ) ) ) ).
fof(lameq_f4007,axiom,
! [A_27b,A_27c,A_27a,V2c] :
( mem(V2c,arr(arr(A_27c,bool),bool))
=> ! [V1b] :
( mem(V1b,arr(arr(A_27b,bool),bool))
=> ! [V0a] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V3s] : ap(f4007(A_27b,A_27c,A_27a,V2c,V1b,V0a),V3s) = ap(c_2Epair_2EUNCURRY(arr(A_27b,bool),arr(A_27c,bool),ty_2Epair_2Eprod(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool)),f4006(A_27c,A_27a,A_27b,V0a,V1b,V2c,V3s)) ) ) ) ).
fof(ax_thm_2Elebesgue_2Eprod__sets3__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0a] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1b] :
( mem(V1b,arr(arr(A_27b,bool),bool))
=> ! [V2c] :
( mem(V2c,arr(arr(A_27c,bool),bool))
=> ap(ap(ap(c_2Elebesgue_2Eprod__sets3(A_27a,A_27b,A_27c),V0a),V1b),V2c) = ap(c_2Epred__set_2EGSPEC(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27b,bool),arr(A_27c,bool)))),ap(c_2Epair_2EUNCURRY(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27b,bool),arr(A_27c,bool)),ty_2Epair_2Eprod(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool)),f4007(A_27b,A_27c,A_27a,V2c,V1b,V0a))) ) ) ) ) ) ) ).
fof(ax_thm_2Elebesgue_2Eprod__measure3__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [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_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2m2] :
( mem(V2m2,ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))))
=> ap(ap(ap(c_2Elebesgue_2Eprod__measure3(A_27a,A_27b,A_27c),V0m0),V1m1),V2m2) = ap(ap(c_2Elebesgue_2Eprod__measure(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V0m0),ap(ap(c_2Elebesgue_2Eprod__measure__space(A_27b,A_27c),V1m1),V2m2)) ) ) ) ) ) ) ).
fof(ax_thm_2Elebesgue_2Eprod__measure__space3__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [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_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2m2] :
( mem(V2m2,ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))))
=> ap(ap(ap(c_2Elebesgue_2Eprod__measure__space3(A_27a,A_27b,A_27c),V0m0),V1m1),V2m2) = ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(c_2Emeasure_2Em__space(A_27b),V1m1)),ap(c_2Emeasure_2Em__space(A_27c),V2m2)))),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),ap(ap(c_2Emeasure_2Esigma(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(c_2Emeasure_2Em__space(A_27b),V1m1)),ap(c_2Emeasure_2Em__space(A_27c),V2m2)))),ap(ap(ap(c_2Elebesgue_2Eprod__sets3(A_27a,A_27b,A_27c),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27c),V2m2))))),ap(ap(ap(c_2Elebesgue_2Eprod__measure3(A_27a,A_27b,A_27c),V0m0),V1m1),V2m2))) ) ) ) ) ) ) ).
fof(lameq_f4008,axiom,
! [A_27a,V13t] :
( mem(V13t,A_27a)
=> ! [V11c] :
( mem(V11c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V9z] :
( mem(V9z,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V14i] : ap(f4008(A_27a,V13t,V11c,V9z),V14i) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,ap(V9z,V14i))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V11c,V14i)),V13t)) ) ) ) ).
fof(lameq_f4009,axiom,
! [A_27a,V15t] :
( mem(V15t,A_27a)
=> ! [V11c] :
( mem(V11c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V10z_27] :
( mem(V10z_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V16i] : ap(f4009(A_27a,V15t,V11c,V10z_27),V16i) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,ap(V10z_27,V16i))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V11c,V16i)),V15t)) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__present,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))
=> ! [V5g] :
( mem(V5g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V6s_27] :
( mem(V6s_27,arr(ty_2Enum_2Enum,bool))
=> ! [V7b] :
( mem(V7b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V8y] :
( mem(V8y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V5g),V6s_27),V7b),V8y)) )
=> ? [V9z] :
( mem(V9z,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ? [V10z_27] :
( mem(V10z_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ? [V11c] :
( mem(V11c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V12k] :
( mem(V12k,arr(ty_2Enum_2Enum,bool))
& ! [V13t] :
( mem(V13t,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V13t),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(V1f,V13t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f4008(A_27a,V13t,V11c,V9z)),V12k) ) )
& ! [V15t] :
( mem(V15t,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V15t),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(V5g,V15t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f4009(A_27a,V15t,V11c,V10z_27)),V12k) ) )
& ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V12k),V11c),V9z)
& ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V6s_27),V7b),V8y) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V12k),V11c),V10z_27)
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V12k))
& ! [V17i] :
( mem(V17i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V17i),V12k))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V9z,V17i))) ) )
& ! [V18i] :
( mem(V18i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V18i),V12k))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V10z_27,V18i))) ) )
& ! [V19i] :
( mem(V19i,ty_2Enum_2Enum)
=> ! [V20j] :
( mem(V20j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V19i),V12k))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V20j),V12k))
& V19i != V20j )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V11c,V19i)),ap(V11c,V20j))) ) ) )
& ! [V21i] :
( mem(V21i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V21i),V12k))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V11c,V21i)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
& ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V11c),V12k)) = ap(c_2Emeasure_2Em__space(A_27a),V0m) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4010,axiom,
! [A_27a,V9t] :
( mem(V9t,A_27a)
=> ! [V7c] :
( mem(V7c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5z] :
( mem(V5z,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V10i] : ap(f4010(A_27a,V9t,V7c,V5z),V10i) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,ap(V5z,V10i))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V7c,V10i)),V9t)) ) ) ) ).
fof(lameq_f4011,axiom,
! [A_27a,V11t] :
( mem(V11t,A_27a)
=> ! [V7c] :
( mem(V7c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V6z_27] :
( mem(V6z_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V12i] : ap(f4011(A_27a,V11t,V7c,V6z_27),V12i) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,ap(V6z_27,V12i))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V7c,V12i)),V11t)) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__present,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3a] :
( mem(V3a,ty_2Eextreal_2Eextreal)
=> ! [V4b] :
( mem(V4b,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V3a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V4b),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V2g))) )
=> ? [V5z] :
( mem(V5z,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ? [V6z_27] :
( mem(V6z_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ? [V7c] :
( mem(V7c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V8k] :
( mem(V8k,arr(ty_2Enum_2Enum,bool))
& ! [V9t] :
( mem(V9t,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V9t),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(V1f,V9t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f4010(A_27a,V9t,V7c,V5z)),V8k) ) )
& ! [V11t] :
( mem(V11t,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V11t),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(V2g,V11t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f4011(A_27a,V11t,V7c,V6z_27)),V8k) ) )
& V3a = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V8k),V7c),V5z)
& V4b = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V8k),V7c),V6z_27)
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V8k))
& ! [V13i] :
( mem(V13i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V13i),V8k))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V5z,V13i))) ) )
& ! [V14i] :
( mem(V14i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V14i),V8k))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V6z_27,V14i))) ) )
& ! [V15i] :
( mem(V15i,ty_2Enum_2Enum)
=> ! [V16j] :
( mem(V16j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V15i),V8k))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V16j),V8k))
& V15i != V16j )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V7c,V15i)),ap(V7c,V16j))) ) ) )
& ! [V17i] :
( mem(V17i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V17i),V8k))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V7c,V17i)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
& ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V7c),V8k)) = ap(c_2Emeasure_2Em__space(A_27a),V0m) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__thm1,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))
=> ! [V5i] :
( mem(V5i,ty_2Enum_2Enum)
=> ! [V6y] :
( mem(V6y,A_27a)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5i),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V6y),ap(V3a,V5i))) )
=> ap(V1f,V6y) = ap(c_2Eextreal_2ENormal,ap(V4x,V5i)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3s] :
( mem(V3s,arr(ty_2Enum_2Enum,bool))
=> ! [V4a] :
( mem(V4a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5x] :
( mem(V5x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6x_27] :
( mem(V6x_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7i] :
( mem(V7i,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V3s),V4a),V5x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V2g),V3s),V4a),V6x_27))
& ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V2g,V8x)),ap(V1f,V8x))) ) )
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V7i),V3s))
& ap(V4a,V7i) != c_2Epred__set_2EEMPTY(A_27a) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2ENormal,ap(V6x_27,V7i))),ap(c_2Eextreal_2ENormal,ap(V5x,V7i)))) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4012,axiom,
! [A_27a,V4f] :
( mem(V4f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ! [V9t] : ap(f4012(A_27a,V4f,V5z),V9t) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V5z)),ap(V4f,V9t)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__cmul,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ! [V1a] :
( mem(V1a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2x] :
( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3m] :
( mem(V3m,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] :
( mem(V4f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V3m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V3m),V4f),V0s),V1a),V2x))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5z)) )
=> ? [V6s_27] :
( mem(V6s_27,arr(ty_2Enum_2Enum,bool))
& ? [V7a_27] :
( mem(V7a_27,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V8x_27] :
( mem(V8x_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V3m),f4012(A_27a,V4f,V5z)),V6s_27),V7a_27),V8x_27)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4013,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ! [V6t] : ap(f4013(A_27a,V1f,V5z),V6t) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V5z)),ap(V1f,V6t)) ) ) ).
fof(lameq_f4014,axiom,
! [V4x] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ! [V7i] : ap(f4014(V4x,V5z),V7i) = ap(ap(c_2Erealax_2Ereal__mul,V5z),ap(V4x,V7i)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__cmul__alt,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))
=> ! [V5z] :
( mem(V5z,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)),V5z))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x)) )
=> p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),f4013(A_27a,V1f,V5z)),V2s),V3a),f4014(V4x,V5z))) ) ) ) ) ) ) ) ) ).
fof(lameq_f4015,axiom,
! [A_27a,V7f] :
( mem(V7f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V8g] :
( mem(V8g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V12t] : ap(f4015(A_27a,V7f,V8g),V12t) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V7f,V12t)),ap(V8g,V12t)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__add,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ! [V1a] :
( mem(V1a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2x] :
( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3s_27] :
( mem(V3s_27,arr(ty_2Enum_2Enum,bool))
=> ! [V4a_27] :
( mem(V4a_27,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5x_27] :
( mem(V5x_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6m] :
( mem(V6m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V7f] :
( mem(V7f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V8g] :
( mem(V8g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V6m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V6m),V7f),V0s),V1a),V2x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V6m),V8g),V3s_27),V4a_27),V5x_27)) )
=> ? [V9s_27_27] :
( mem(V9s_27_27,arr(ty_2Enum_2Enum,bool))
& ? [V10a_27_27] :
( mem(V10a_27_27,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V11x_27_27] :
( mem(V11x_27_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V6m),f4015(A_27a,V7f,V8g)),V9s_27_27),V10a_27_27),V11x_27_27)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4016,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V7t] : ap(f4016(A_27a,V1f,V2g),V7t) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V7t)),ap(V2g,V7t)) ) ) ).
fof(lameq_f4017,axiom,
! [V5x] :
( mem(V5x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6y] :
( mem(V6y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V8i] : ap(f4017(V5x,V6y),V8i) = ap(ap(c_2Erealax_2Ereal__add,ap(V5x,V8i)),ap(V6y,V8i)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__add__alt,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3s] :
( mem(V3s,arr(ty_2Enum_2Enum,bool))
=> ! [V4a] :
( mem(V4a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5x] :
( mem(V5x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6y] :
( mem(V6y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V3s),V4a),V5x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V2g),V3s),V4a),V6y)) )
=> p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),f4016(A_27a,V1f,V2g)),V3s),V4a),f4017(V5x,V6y))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__indicator,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))) )
=> ? [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),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1A)),V2s),V3a),V4x)) ) ) ) ) ) ) ) ).
fof(lameq_f4018,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))))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2i] : ap(f4018(A_27a,V0m,V1s),V2i) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,bool)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V2i),c_2Enum_2E0)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)),V1s)),V1s) ) ) ).
fof(lameq_f4019,axiom,
! [V3i] : ap(f4019,V3i) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V3i),c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__indicator__alt,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(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1s)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),c_2Enum_2E0),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)))),f4018(A_27a,V0m,V1s)),f4019)) ) ) ) ) ).
fof(lameq_f4020,axiom,
! [A_27a,V3f] :
( mem(V3f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V7g] :
( mem(V7g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V13x] : ap(f4020(A_27a,V3f,V7g),V13x) = ap(ap(c_2Eextreal_2Eextreal__max,ap(V3f,V13x)),ap(V7g,V13x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__max,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s_27] :
( mem(V0s_27,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))))
=> ! [V3f] :
( mem(V3f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4s] :
( mem(V4s,arr(ty_2Enum_2Enum,bool))
=> ! [V5a] :
( mem(V5a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V6x] :
( mem(V6x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7g] :
( mem(V7g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V8s_27b] :
( mem(V8s_27b,A_27b)
=> ! [V9y] :
( mem(V9y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V2m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V2m),V3f),V4s),V5a),V6x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V2m),V7g),V0s_27),V1b),V9y)) )
=> ? [V10s_27_27] :
( mem(V10s_27_27,arr(ty_2Enum_2Enum,bool))
& ? [V11a_27_27] :
( mem(V11a_27_27,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V12x_27_27] :
( mem(V12x_27_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V2m),f4020(A_27a,V3f,V7g)),V10s_27_27),V11a_27_27),V12x_27_27)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__not__infty,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))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ( ap(V1f,V5x) != c_2Eextreal_2ENegInf
& ap(V1f,V5x) != c_2Eextreal_2EPosInf ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4021,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5g] :
( mem(V5g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V12x] : ap(f4021(A_27a,V1f,V5g),V12x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V12x)),ap(V5g,V12x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__add,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))
=> ! [V5g] :
( mem(V5g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V6s_27] :
( mem(V6s_27,arr(ty_2Enum_2Enum,bool))
=> ! [V7b] :
( mem(V7b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V8y] :
( mem(V8y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V5g),V6s_27),V7b),V8y)) )
=> ? [V9s_27_27] :
( mem(V9s_27_27,arr(ty_2Enum_2Enum,bool))
& ? [V10c] :
( mem(V10c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V11z] :
( mem(V11z,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),f4021(A_27a,V1f,V5g)),V9s_27_27),V10c),V11z))
& ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x)),ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V6s_27),V7b),V8y)) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V9s_27_27),V10c),V11z) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4022,axiom,
! [V4x] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6y] :
( mem(V6y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7i] : ap(f4022(V4x,V6y),V7i) = ap(ap(c_2Erealax_2Ereal__add,ap(V4x,V7i)),ap(V6y,V7i)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__add__alt,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))
=> ! [V5g] :
( mem(V5g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V6y] :
( mem(V6y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V5g),V2s),V3a),V6y)) )
=> ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x)),ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V6y)) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),f4022(V4x,V6y)) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4023,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5x] : ap(f4023(A_27a,V1f,V2g),V5x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V5x)),ap(V2g,V5x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__add,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3a] :
( mem(V3a,ty_2Eextreal_2Eextreal)
=> ! [V4b] :
( mem(V4b,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V3a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V4b),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V2g))) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__add,V3a),V4b)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),f4023(A_27a,V1f,V2g)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__mono,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))
=> ! [V5g] :
( mem(V5g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V6s_27] :
( mem(V6s_27,arr(ty_2Enum_2Enum,bool))
=> ! [V7b] :
( mem(V7b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V8y] :
( mem(V8y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V5g),V6s_27),V7b),V8y))
& ! [V9x] :
( mem(V9x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V9x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1f,V9x)),ap(V5g,V9x))) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x)),ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V6s_27),V7b),V8y))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__mono,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3a] :
( mem(V3a,ty_2Eextreal_2Eextreal)
=> ! [V4b] :
( mem(V4b,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V3a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V4b),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V2g)))
& ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1f,V5x)),ap(V2g,V5x))) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V3a),V4b)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__unique,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))
=> ! [V5s_27] :
( mem(V5s_27,arr(ty_2Enum_2Enum,bool))
=> ! [V6b] :
( mem(V6b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V7y] :
( mem(V7y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V5s_27),V6b),V7y)) )
=> ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V5s_27),V6b),V7y) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__unique,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))
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ! [V3b] :
( mem(V3b,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V2a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V3b),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f))) )
=> V2a = V3b ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__indicator,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))) )
=> ? [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),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1A)),V2s),V3a),V4x))
& ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1A)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__indicator,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))) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1A))),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1A)))) ) ) ) ) ).
fof(lameq_f4024,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ! [V6x] : ap(f4024(A_27a,V1f,V5z),V6x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V5z)),ap(V1f,V6x)) ) ) ).
fof(lameq_f4025,axiom,
! [V4x] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ! [V8i] : ap(f4025(V4x,V5z),V8i) = ap(ap(c_2Erealax_2Ereal__mul,V5z),ap(V4x,V8i)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__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))))
=> ! [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))
=> ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5z)) )
=> ( p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),f4024(A_27a,V1f,V5z)),V2s),V3a),f4014(V4x,V5z)))
& ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),f4025(V4x,V5z)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V5z)),ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x)) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4026,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3z] :
( mem(V3z,ty_2Erealax_2Ereal)
=> ! [V4x] : ap(f4026(A_27a,V1f,V3z),V4x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V3z)),ap(V1f,V4x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ! [V3z] :
( mem(V3z,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V2a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3z)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V3z)),V2a)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),f4026(A_27a,V1f,V3z)))) ) ) ) ) ) ) ).
fof(lameq_f4027,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5z] :
( mem(V5z,ty_2Erealax_2Ereal)
=> ! [V9t] : ap(f4027(A_27a,V1f,V5z),V9t) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V5z)),ap(V1f,V9t)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__cmul__alt,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))
=> ! [V5z] :
( mem(V5z,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)),V5z))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x)) )
=> ? [V6s_27] :
( mem(V6s_27,arr(ty_2Enum_2Enum,bool))
& ? [V7a_27] :
( mem(V7a_27,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V8x_27] :
( mem(V8x_27,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),f4027(A_27a,V1f,V5z)),V6s_27),V7a_27),V8x_27))
& ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V6s_27),V7a_27),V8x_27) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V5z)),ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2EIN__psfis,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))))
=> ! [V1r] :
( mem(V1r,ty_2Eextreal_2Eextreal)
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1r),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V2f)))
=> ? [V3s] :
( mem(V3s,arr(ty_2Enum_2Enum,bool))
& ? [V4a] :
( mem(V4a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V5x] :
( mem(V5x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V2f),V3s),V4a),V5x))
& V1r = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V3s),V4a),V5x) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2EIN__psfis__eq,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))))
=> ! [V1r] :
( mem(V1r,ty_2Eextreal_2Eextreal)
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1r),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V2f)))
<=> ? [V3s] :
( mem(V3s,arr(ty_2Enum_2Enum,bool))
& ? [V4a] :
( mem(V4a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V5x] :
( mem(V5x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V2f),V3s),V4a),V5x))
& V1r = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V3s),V4a),V5x) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V2a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f))) )
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__zero,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,bool))
=> ! [V2a] :
( mem(V2a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))),V1s),V2a),V3x)) )
=> ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V1s),V2a),V3x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__zero__alt,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,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))))
=> ! [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(c_2Emeasure_2Emeasure__space(A_27a),V1m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V1m),V0g),V2s),V3a),V4x))
& ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Em__space(A_27a),V1m)))
=> ap(V0g,V5x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) )
=> ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V1m),V2s),V3a),V4x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__zero,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,ty_2Eextreal_2Eextreal)
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))))
<=> V1a = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__not__infty,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,bool))
=> ! [V2a] :
( mem(V2a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V1s),V2a),V3x) != c_2Eextreal_2ENegInf
& ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V1s),V2a),V3x) != c_2Eextreal_2EPosInf ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__not__infty,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))
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V2a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f)))
=> ( V2a != c_2Eextreal_2ENegInf
& V2a != c_2Eextreal_2EPosInf ) ) ) ) ) ) ).
fof(lameq_f4028,axiom,
! [A_27a,A_27b,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V7t] :
( mem(V7t,A_27a)
=> ! [V8i] : ap(f4028(A_27a,A_27b,V1f,V7t),V8i) = ap(ap(V1f,V8i),V7t) ) ) ).
fof(lameq_f4029,axiom,
! [A_27b,A_27a,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V5P] :
( mem(V5P,arr(A_27b,bool))
=> ! [V7t] : ap(f4029(A_27b,A_27a,V1f,V5P),V7t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4028(A_27a,A_27b,V1f,V7t)),V5P) ) ) ).
fof(lameq_f4030,axiom,
! [A_27b,V4x] :
( mem(V4x,arr(A_27b,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V9i] :
( mem(V9i,ty_2Enum_2Enum)
=> ! [V10j] : ap(f4030(A_27b,V4x,V9i),V10j) = ap(ap(V4x,V10j),V9i) ) ) ).
fof(lameq_f4031,axiom,
! [A_27b,V4x] :
( mem(V4x,arr(A_27b,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V5P] :
( mem(V5P,arr(A_27b,bool))
=> ! [V9i] : ap(f4031(A_27b,V4x,V5P),V9i) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f4030(A_27b,V4x,V9i)),V5P) ) ) ).
fof(lameq_f4032,axiom,
! [A_27b,V4x] :
( mem(V4x,arr(A_27b,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V11j] :
( mem(V11j,ty_2Enum_2Enum)
=> ! [V12i] : ap(f4032(A_27b,V4x,V11j),V12i) = ap(ap(V4x,V12i),V11j) ) ) ).
fof(lameq_f4033,axiom,
! [A_27b,V4x] :
( mem(V4x,arr(A_27b,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V5P] :
( mem(V5P,arr(A_27b,bool))
=> ! [V11j] : ap(f4033(A_27b,V4x,V5P),V11j) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f4032(A_27b,V4x,V11j)),V5P) ) ) ).
fof(lameq_f4034,axiom,
! [A_27a,A_27b,V3a] :
( mem(V3a,arr(ty_2Enum_2Enum,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))))
=> ! [V2s] :
( mem(V2s,arr(ty_2Enum_2Enum,bool))
=> ! [V4x] :
( mem(V4x,arr(A_27b,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V13i] : ap(f4034(A_27a,A_27b,V3a,V0m,V2s,V4x),V13i) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),ap(V4x,V13i)) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__sum,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_27b,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(A_27b,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V5P] :
( mem(V5P,arr(A_27b,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V6i] :
( mem(V6i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6i),V5P))
=> p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),ap(V1f,V6i)),V2s),V3a),ap(V4x,V6i))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27b),V5P))
& V5P != c_2Epred__set_2EEMPTY(A_27b) )
=> ( p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),f4029(A_27b,A_27a,V1f,V5P)),V2s),V3a),f4031(A_27b,V4x,V5P)))
& ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),f4033(A_27b,V4x,V5P)) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4034(A_27a,A_27b,V3a,V0m,V2s,V4x)),V5P) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4035,axiom,
! [A_27a,A_27b,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V10t] :
( mem(V10t,A_27a)
=> ! [V11i] : ap(f4035(A_27a,A_27b,V1f,V10t),V11i) = ap(ap(V1f,V11i),V10t) ) ) ).
fof(lameq_f4036,axiom,
! [A_27b,A_27a,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V5P] :
( mem(V5P,arr(A_27b,bool))
=> ! [V10t] : ap(f4036(A_27b,A_27a,V1f,V5P),V10t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4035(A_27a,A_27b,V1f,V10t)),V5P) ) ) ).
fof(lameq_f4037,axiom,
! [A_27a,A_27b,V3a] :
( mem(V3a,arr(A_27b,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ! [V2s] :
( mem(V2s,arr(A_27b,arr(ty_2Enum_2Enum,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))))
=> ! [V4x] :
( mem(V4x,arr(A_27b,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V12i] : ap(f4037(A_27a,A_27b,V3a,V2s,V0m,V4x),V12i) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),ap(V2s,V12i)),ap(V3a,V12i)),ap(V4x,V12i)) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__sum__alt,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_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V2s] :
( mem(V2s,arr(A_27b,arr(ty_2Enum_2Enum,bool)))
=> ! [V3a] :
( mem(V3a,arr(A_27b,arr(ty_2Enum_2Enum,arr(A_27a,bool))))
=> ! [V4x] :
( mem(V4x,arr(A_27b,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V5P] :
( mem(V5P,arr(A_27b,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V6i] :
( mem(V6i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6i),V5P))
=> p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),ap(V1f,V6i)),ap(V2s,V6i)),ap(V3a,V6i)),ap(V4x,V6i))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27b),V5P))
& V5P != c_2Epred__set_2EEMPTY(A_27b) )
=> ? [V7c] :
( mem(V7c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V8k] :
( mem(V8k,arr(ty_2Enum_2Enum,bool))
& ? [V9z] :
( mem(V9z,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),f4036(A_27b,A_27a,V1f,V5P)),V8k),V7c),V9z))
& ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V8k),V7c),V9z) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4037(A_27a,A_27b,V3a,V2s,V0m,V4x)),V5P) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4038,axiom,
! [A_27a,A_27b,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V5t] :
( mem(V5t,A_27a)
=> ! [V6i] : ap(f4038(A_27a,A_27b,V1f,V5t),V6i) = ap(ap(V1f,V6i),V5t) ) ) ).
fof(lameq_f4039,axiom,
! [A_27b,A_27a,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V3P] :
( mem(V3P,arr(A_27b,bool))
=> ! [V5t] : ap(f4039(A_27b,A_27a,V1f,V3P),V5t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4038(A_27a,A_27b,V1f,V5t)),V3P) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__sum,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_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V2a] :
( mem(V2a,arr(A_27b,ty_2Eextreal_2Eextreal))
=> ! [V3P] :
( mem(V3P,arr(A_27b,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V4i] :
( mem(V4i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4i),V3P))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(V2a,V4i)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),ap(V1f,V4i)))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3P)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),V2a),V3P)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),f4039(A_27b,A_27a,V1f,V3P)))) ) ) ) ) ) ) ) ).
fof(lameq_f4040,axiom,
! [A_27a,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))))
=> ! [V1a] :
( mem(V1a,arr(A_27b,arr(A_27a,bool)))
=> ! [V2x] :
( mem(V2x,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V6i] : ap(f4040(A_27a,A_27b,V0m,V1a,V2x),V6i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V2x,V6i)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(V1a,V6i))) ) ) ) ).
fof(lameq_f4041,axiom,
! [A_27a,A_27b,V7t] :
( mem(V7t,A_27a)
=> ! [V1a] :
( mem(V1a,arr(A_27b,arr(A_27a,bool)))
=> ! [V2x] :
( mem(V2x,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V8i] : ap(f4041(A_27a,A_27b,V7t,V1a,V2x),V8i) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,ap(V2x,V8i))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V1a,V8i)),V7t)) ) ) ) ).
fof(lameq_f4042,axiom,
! [A_27b,A_27a,V1a] :
( mem(V1a,arr(A_27b,arr(A_27a,bool)))
=> ! [V2x] :
( mem(V2x,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V3P] :
( mem(V3P,arr(A_27b,bool))
=> ! [V7t] : ap(f4042(A_27b,A_27a,V1a,V2x,V3P),V7t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4041(A_27a,A_27b,V7t,V1a,V2x)),V3P) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__intro,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))))
=> ! [V1a] :
( mem(V1a,arr(A_27b,arr(A_27a,bool)))
=> ! [V2x] :
( mem(V2x,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V3P] :
( mem(V3P,arr(A_27b,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V4i] :
( mem(V4i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4i),V3P))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1a,V4i)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
& ! [V5i] :
( mem(V5i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5i),V3P))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V2x,V5i))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3P)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f4040(A_27a,A_27b,V0m,V1a,V2x)),V3P))),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),f4042(A_27b,A_27a,V1a,V2x,V3P)))) ) ) ) ) ) ) ) ).
fof(lameq_f4043,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5g] :
( mem(V5g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V14x] : ap(f4043(A_27a,V1f,V5g),V14x) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V1f,V14x)),ap(V5g,V14x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__simple__fn__integral__sub,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))
=> ! [V5g] :
( mem(V5g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V6s_27] :
( mem(V6s_27,arr(ty_2Enum_2Enum,bool))
=> ! [V7b] :
( mem(V7b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V8y] :
( mem(V8y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V9x] :
( mem(V9x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V5g,V9x)),ap(V1f,V9x))) )
& ! [V10x] :
( mem(V10x,A_27a)
=> ap(V5g,V10x) != c_2Eextreal_2EPosInf )
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V5g),V6s_27),V7b),V8y)) )
=> ? [V11s_27_27] :
( mem(V11s_27_27,arr(ty_2Enum_2Enum,bool))
& ? [V12c] :
( mem(V12c,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V13z] :
( mem(V13z,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),f4043(A_27a,V1f,V5g)),V11s_27_27),V12c),V13z))
& ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x)),ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V6s_27),V7b),V8y)) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V11s_27_27),V12c),V13z) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4044,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V7x] : ap(f4044(A_27a,V1f,V2g),V7x) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V1f,V7x)),ap(V2g,V7x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__sub,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3a] :
( mem(V3a,ty_2Eextreal_2Eextreal)
=> ! [V4b] :
( mem(V4b,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V5x] :
( mem(V5x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V2g,V5x)),ap(V1f,V5x))) )
& ! [V6x] :
( mem(V6x,A_27a)
=> ap(V2g,V6x) != c_2Eextreal_2EPosInf )
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V3a),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V1f)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V4b),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V2g))) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__sub,V3a),V4b)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),f4044(A_27a,V1f,V2g)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__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))
=> ! [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(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x)) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) = ap(ap(ap(ap(c_2Elebesgue_2Epos__simple__fn__integral(A_27a),V0m),V2s),V3a),V4x) ) ) ) ) ) ) ) ).
fof(lameq_f4045,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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(f4045(A_27a,V1f,V0m),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V3x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)),V3x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4045(A_27a,V1f,V0m)) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__zero,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_2Elebesgue_2Epos__fn__integral(A_27a),V0m),k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__mono,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1f,V3x)),ap(V2g,V3x))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f)),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V2g))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__mono__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V2g,V3x)),ap(V1f,V3x))) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V4x))) )
& ! [V5x] :
( mem(V5x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V2g,V5x))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V2g)),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f))) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x))) ) )
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2c)) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f3354(A_27a,V1f,V2c)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f)) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__indicator,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_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1s)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) ) ) ) ) ).
fof(lameq_f4046,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f4046(A_27a,V1s,V2c),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V1s),V3x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__cmul__indicator,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))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ( 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_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2c)) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4046(A_27a,V1s,V2c)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Erealax_2Ereal__mul,V2c),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ).
fof(lameq_f4047,axiom,
! [A_27a,V6t] :
( mem(V6t,A_27a)
=> ! [V2a] :
( mem(V2a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7i] : ap(f4047(A_27a,V6t,V2a,V3x),V7i) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,ap(V3x,V7i))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V2a,V7i)),V6t)) ) ) ) ).
fof(lameq_f4048,axiom,
! [A_27a,V2a] :
( mem(V2a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ! [V6t] : ap(f4048(A_27a,V2a,V3x,V1s),V6t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f4047(A_27a,V6t,V2a,V3x)),V1s) ) ) ) ).
fof(lameq_f4049,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))))
=> ! [V2a] :
( mem(V2a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V8i] : ap(f4049(A_27a,V0m,V2a,V3x),V8i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V3x,V8i)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(V2a,V8i))) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__sum__cmul__indicator,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,bool))
=> ! [V2a] :
( mem(V2a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s))
& ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4i),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V3x,V4i))) ) )
& ! [V5i] :
( mem(V5i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5i),V1s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V2a,V5i)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4048(A_27a,V2a,V3x,V1s)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Enum_2Enum),f4049(A_27a,V0m,V2a,V3x)),V1s)) ) ) ) ) ) ) ).
fof(lameq_f4050,axiom,
! [A_27a,V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7i] : ap(f4050(A_27a,V2fi,V6x),V7i) = ap(ap(V2fi,V7i),V6x) ) ) ).
fof(lameq_f4051,axiom,
! [A_27a,V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V8x] :
( mem(V8x,A_27a)
=> ! [V9i] : ap(f4051(A_27a,V2fi,V8x),V9i) = ap(ap(V2fi,V9i),V8x) ) ) ).
fof(lameq_f4052,axiom,
! [A_27a,V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [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))))
=> ! [V13i] : ap(f4052(A_27a,V2fi,V0m),V13i) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(V2fi,V13i)) ) ) ).
fof(conj_thm_2Elebesgue_2Elebesgue__monotone__convergence__lemma,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))
=> ! [V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V3g] :
( mem(V3g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4r_27] :
( mem(V4r_27,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V5i] :
( mem(V5i,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V2fi,V5i)),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))) )
& ! [V6x] :
( mem(V6x,A_27a)
=> p(ap(c_2Eextreal_2Eext__mono__increasing,f4050(A_27a,V2fi,V6x))) )
& ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4051(A_27a,V2fi,V8x)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(V1f,V8x) ) )
& p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V4r_27),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),V3g)))
& ! [V10x] :
( mem(V10x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V3g,V10x)),ap(V1f,V10x))) )
& ! [V11i] :
( mem(V11i,ty_2Enum_2Enum)
=> ! [V12x] :
( mem(V12x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(V2fi,V11i),V12x))) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V4r_27),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4052(A_27a,V2fi,V0m)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) ) ) ) ) ) ) ) ).
fof(lameq_f4053,axiom,
! [A_27a,V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V7x] :
( mem(V7x,A_27a)
=> ! [V8i] : ap(f4053(A_27a,V2fi,V7x),V8i) = ap(ap(V2fi,V8i),V7x) ) ) ).
fof(lameq_f4054,axiom,
! [A_27a,V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V9x] :
( mem(V9x,A_27a)
=> ! [V10i] : ap(f4054(A_27a,V2fi,V9x),V10i) = ap(ap(V2fi,V10i),V9x) ) ) ).
fof(lameq_f4055,axiom,
! [A_27a,V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [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))))
=> ! [V11i] : ap(f4055(A_27a,V2fi,V0m),V11i) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(V2fi,V11i)) ) ) ).
fof(conj_thm_2Elebesgue_2Elebesgue__monotone__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V3i] :
( mem(V3i,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V2fi,V3i)),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))) )
& ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> ! [V5x] :
( mem(V5x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(V2fi,V4i),V5x))) ) )
& ! [V6x] :
( mem(V6x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V6x))) )
& ! [V7x] :
( mem(V7x,A_27a)
=> p(ap(c_2Eextreal_2Eext__mono__increasing,f4053(A_27a,V2fi,V7x))) )
& ! [V9x] :
( mem(V9x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V9x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4054(A_27a,V2fi,V9x)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(V1f,V9x) ) ) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4055(A_27a,V2fi,V0m)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) ) ).
fof(lameq_f4056,axiom,
! [A_27a,V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V10x] :
( mem(V10x,A_27a)
=> ! [V11i] : ap(f4056(A_27a,V2fi,V10x),V11i) = ap(ap(V2fi,V11i),V10x) ) ) ).
fof(lameq_f4057,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3A] :
( mem(V3A,arr(A_27a,bool))
=> ! [V12x] : ap(f4057(A_27a,V1f,V3A),V12x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V12x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V3A),V12x)) ) ) ).
fof(lameq_f4058,axiom,
! [A_27a,V13i] :
( mem(V13i,ty_2Enum_2Enum)
=> ! [V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V3A] :
( mem(V3A,arr(A_27a,bool))
=> ! [V14x] : ap(f4058(A_27a,V13i,V2fi,V3A),V14x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(V2fi,V13i),V14x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V3A),V14x)) ) ) ) ).
fof(lameq_f4059,axiom,
! [A_27a,V3A] :
( mem(V3A,arr(A_27a,bool))
=> ! [V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [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))))
=> ! [V13i] : ap(f4059(A_27a,V3A,V2fi,V0m),V13i) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4058(A_27a,V13i,V2fi,V3A)) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elebesgue__monotone__convergence__subset,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))
=> ! [V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V3A] :
( mem(V3A,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V2fi,V4i)),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))) )
& ! [V5i] :
( mem(V5i,ty_2Enum_2Enum)
=> ! [V6x] :
( mem(V6x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(V2fi,V5i),V6x))) ) )
& ! [V7x] :
( mem(V7x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V7x))) )
& ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4051(A_27a,V2fi,V8x)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(V1f,V8x) ) )
& ! [V10x] :
( mem(V10x,A_27a)
=> p(ap(c_2Eextreal_2Eext__mono__increasing,f4056(A_27a,V2fi,V10x))) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4057(A_27a,V1f,V3A)) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4059(A_27a,V3A,V2fi,V0m)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) ) ) ).
fof(lameq_f4060,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4k] :
( mem(V4k,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V5x] : ap(f4060(A_27a,V0m,V1f,V4k,V2n),V5x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V5x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V4k)),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n))),ap(V1f,V5x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V1f,V5x)),ap(ap(c_2Eextreal_2Eextreal__div,ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eextreal__of__num,V4k)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)))))) ) ) ) ) ).
fof(lameq_f4061,axiom,
! [A_27a,V3x] :
( mem(V3x,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))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V4k] : ap(f4061(A_27a,V3x,V0m,V1f,V2n),V4k) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V4k)),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f4060(A_27a,V0m,V1f,V4k,V2n))),V3x)) ) ) ) ) ).
fof(lameq_f4062,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V6x] : ap(f4062(A_27a,V0m,V1f,V2n),V6x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V6x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V6x),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)),ap(V1f,V6x)))) ) ) ) ).
fof(lameq_f4063,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3x] : ap(f4063(A_27a,V0m,V1f,V2n),V3x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f4061(A_27a,V3x,V0m,V1f,V2n)),ap(c_2Epred__set_2Ecount,ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n)))),ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f4062(A_27a,V0m,V1f,V2n))),V3x))) ) ) ) ).
fof(lameq_f4064,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2n] : ap(f4064(A_27a,V0m,V1f),V2n) = f4063(A_27a,V0m,V1f,V2n) ) ) ).
fof(ax_thm_2Elebesgue_2Efn__seq__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))
=> ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f) = f4064(A_27a,V0m,V1f) ) ) ) ).
fof(lameq_f4065,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3k] :
( mem(V3k,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V4x] : ap(f4065(A_27a,V0m,V1f,V3k,V2n),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V3k)),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n))),ap(V1f,V4x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V1f,V4x)),ap(ap(c_2Eextreal_2Eextreal__div,ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eextreal__of__num,V3k)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)))))) ) ) ) ) ).
fof(lameq_f4066,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3k] : ap(f4066(A_27a,V0m,V1f,V2n),V3k) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,V3k)),ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n))),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f4065(A_27a,V0m,V1f,V3k,V2n)))) ) ) ) ).
fof(lameq_f4067,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V5x] : ap(f4067(A_27a,V0m,V1f,V2n),V5x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V5x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)),ap(V1f,V5x)))) ) ) ) ).
fof(lameq_f4068,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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))))
=> ! [V2n] : ap(f4068(A_27a,V1f,V0m),V2n) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Enum_2Enum),f4066(A_27a,V0m,V1f,V2n)),ap(c_2Epred__set_2Ecount,ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n)))),ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f4067(A_27a,V0m,V1f,V2n)))))) ) ) ).
fof(ax_thm_2Elebesgue_2Efn__seq__integral__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))
=> ap(ap(c_2Elebesgue_2Efn__seq__integral(A_27a),V0m),V1f) = f4068(A_27a,V1f,V0m) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__1,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))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4k] :
( mem(V4k,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4k),ap(c_2Epred__set_2Ecount,ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V4k)),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n))),ap(V1f,V3x)))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(V1f,V3x)),ap(ap(c_2Eextreal_2Eextreal__div,ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eextreal__of__num,V4k)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)))) )
=> ap(ap(ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f),V2n),V3x) = ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V4k)),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__2,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))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)),ap(V1f,V3x))) )
=> ap(ap(ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f),V2n),V3x) = ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__3,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))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x))) )
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)),ap(V1f,V3x)))
| ? [V4k] :
( mem(V4k,ty_2Enum_2Enum)
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4k),ap(c_2Epred__set_2Ecount,ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,ap(c_2Eextreal_2Eextreal__of__num,V4k)),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n))),ap(V1f,V3x)))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(V1f,V3x)),ap(ap(c_2Eextreal_2Eextreal__div,ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eextreal__of__num,V4k)),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eextreal_2Eextreal__pow,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V2n)))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__4,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))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(ap(ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f),V2n),V3x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__5,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))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f),V2n),V3x))) ) ) ) ) ) ) ).
fof(lameq_f4069,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3n] : ap(f4069(A_27a,V0m,V1f,V2x),V3n) = ap(ap(ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f),V3n),V2x) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__mono__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x)))
=> p(ap(c_2Eextreal_2Eext__mono__increasing,f4069(A_27a,V0m,V1f,V2x))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__upper__bounded,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))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f),V2n),V3x)),ap(V1f,V3x))) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__sup,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))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) )
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4069(A_27a,V0m,V1f,V2x)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(V1f,V2x) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__fn__in__psfis,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))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x))) )
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& 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))) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(ap(c_2Elebesgue_2Efn__seq__integral(A_27a),V0m),V1f),V2n)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),ap(ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f),V2n)))) ) ) ) ) ) ).
fof(lameq_f4070,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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))))
=> ! [V3i] : ap(f4070(A_27a,V1f,V0m),V3i) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(ap(ap(c_2Elebesgue_2Efn__seq(A_27a),V0m),V1f),V3i)) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__sequence,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))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) )
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& 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))) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4070(A_27a,V1f,V0m)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) ).
fof(lameq_f4071,axiom,
! [A_27a,V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5i] : ap(f4071(A_27a,V2fi,V4x),V5i) = ap(ap(V2fi,V5i),V4x) ) ) ).
fof(lameq_f4072,axiom,
! [A_27a,V14gi] :
( mem(V14gi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V16x] :
( mem(V16x,A_27a)
=> ! [V17i] : ap(f4072(A_27a,V14gi,V16x),V17i) = ap(ap(V14gi,V17i),V16x) ) ) ).
fof(lameq_f4073,axiom,
! [A_27a,V14gi] :
( mem(V14gi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V18x] :
( mem(V18x,A_27a)
=> ! [V19i] : ap(f4073(A_27a,V14gi,V18x),V19i) = ap(ap(V14gi,V19i),V18x) ) ) ).
fof(lameq_f4074,axiom,
! [A_27a,V14gi] :
( mem(V14gi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [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))))
=> ! [V25i] : ap(f4074(A_27a,V14gi,V0m),V25i) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(V14gi,V25i)) ) ) ).
fof(conj_thm_2Elebesgue_2Emeasurable__sequence,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))
& 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))) )
=> ( ? [V2fi] :
( mem(V2fi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
& ? [V3ri] :
( mem(V3ri,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(c_2Eextreal_2Eext__mono__increasing,f4071(A_27a,V2fi,V4x))) )
& ! [V6x] :
( mem(V6x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4050(A_27a,V2fi,V6x)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(ap(c_2Emeasure_2Efn__plus(A_27a),V1f),V6x) ) )
& ! [V8i] :
( mem(V8i,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(V3ri,V8i)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),ap(V2fi,V8i)))) )
& ! [V9i] :
( mem(V9i,ty_2Enum_2Enum)
=> ! [V10x] :
( mem(V10x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(V2fi,V9i),V10x)),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V1f),V10x))) ) )
& ! [V11i] :
( mem(V11i,ty_2Enum_2Enum)
=> ! [V12x] :
( mem(V12x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(V2fi,V11i),V12x))) ) )
& ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(c_2Emeasure_2Efn__plus(A_27a),V1f)) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4052(A_27a,V2fi,V0m)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )
& ? [V14gi] :
( mem(V14gi,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
& ? [V15vi] :
( mem(V15vi,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
& ! [V16x] :
( mem(V16x,A_27a)
=> p(ap(c_2Eextreal_2Eext__mono__increasing,f4072(A_27a,V14gi,V16x))) )
& ! [V18x] :
( mem(V18x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V18x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4073(A_27a,V14gi,V18x)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(ap(c_2Emeasure_2Efn__minus(A_27a),V1f),V18x) ) )
& ! [V20i] :
( mem(V20i,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(V15vi,V20i)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),ap(V14gi,V20i)))) )
& ! [V21i] :
( mem(V21i,ty_2Enum_2Enum)
=> ! [V22x] :
( mem(V22x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(V14gi,V21i),V22x)),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V1f),V22x))) ) )
& ! [V23i] :
( mem(V23i,ty_2Enum_2Enum)
=> ! [V24x] :
( mem(V24x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(V14gi,V23i),V24x))) ) )
& ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(c_2Emeasure_2Efn__minus(A_27a),V1f)) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4074(A_27a,V14gi,V0m)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) ) ) ) ).
fof(lameq_f4075,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4x] : ap(f4075(A_27a,V1f,V2g),V4x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V4x)),ap(V2g,V4x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__add,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V2g,V3x))) ) )
& 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)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),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))) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4075(A_27a,V1f,V2g)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f)),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V2g)) ) ) ) ) ) ).
fof(lameq_f4076,axiom,
! [A_27a,A_27b,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7i] : ap(f4076(A_27a,A_27b,V1f,V6x),V7i) = ap(ap(V1f,V7i),V6x) ) ) ).
fof(lameq_f4077,axiom,
! [A_27b,A_27a,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ! [V6x] : ap(f4077(A_27b,A_27a,V1f,V2s),V6x) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4076(A_27a,A_27b,V1f,V6x)),V2s) ) ) ).
fof(lameq_f4078,axiom,
! [A_27a,A_27b,V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [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))))
=> ! [V8i] : ap(f4078(A_27a,A_27b,V1f,V0m),V8i) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(V1f,V8i)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__sum,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_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V3i] :
( mem(V3i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3i),V2s))
=> ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(V1f,V3i),V4x))) ) ) )
& ! [V5i] :
( mem(V5i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5i),V2s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V1f,V5i)),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))) ) ) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4077(A_27b,A_27a,V1f,V2s)) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4078(A_27a,A_27b,V1f,V0m)),V2s) ) ) ) ) ) ) ).
fof(lameq_f4079,axiom,
! [A_27a,V3t] :
( mem(V3t,arr(A_27a,bool))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5x] : ap(f4079(A_27a,V3t,V2s,V1f),V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V5x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),V5x)) ) ) ) ).
fof(lameq_f4080,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V6x] : ap(f4080(A_27a,V1f,V2s),V6x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V6x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V2s),V6x)) ) ) ).
fof(lameq_f4081,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ! [V7x] : ap(f4081(A_27a,V1f,V3t),V7x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V7x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V3t),V7x)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__disjoint__sets,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(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V2s),V3t))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& 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)))
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V4x))) ) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4079(A_27a,V3t,V2s,V1f)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4080(A_27a,V1f,V2s))),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4081(A_27a,V1f,V3t))) ) ) ) ) ) ) ).
fof(lameq_f4082,axiom,
! [A_27b,A_27a,V3a] :
( mem(V3a,arr(A_27b,arr(A_27a,bool)))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V8x] : ap(f4082(A_27b,A_27a,V3a,V2s,V1f),V8x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V8x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(A_27b,arr(A_27a,bool)),V3a),V2s))),V8x)) ) ) ) ).
fof(lameq_f4083,axiom,
! [A_27b,A_27a,V9i] :
( mem(V9i,A_27b)
=> ! [V3a] :
( mem(V3a,arr(A_27b,arr(A_27a,bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V10x] : ap(f4083(A_27b,A_27a,V9i,V3a,V1f),V10x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V10x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V3a,V9i)),V10x)) ) ) ) ).
fof(lameq_f4084,axiom,
! [A_27a,A_27b,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3a] :
( mem(V3a,arr(A_27b,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))))
=> ! [V9i] : ap(f4084(A_27a,A_27b,V1f,V3a,V0m),V9i) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4083(A_27b,A_27a,V9i,V3a,V1f)) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__disjoint__sets__sum,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,ty_2Eextreal_2Eextreal))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ! [V3a] :
( mem(V3a,arr(A_27b,arr(A_27a,bool)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V4i] :
( mem(V4i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4i),V2s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V3a,V4i)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
& ! [V5x] :
( mem(V5x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V5x))) )
& ! [V6i] :
( mem(V6i,A_27b)
=> ! [V7j] :
( mem(V7j,A_27b)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6i),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27b),V7j),V2s))
& V6i != V7j )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3a,V6i)),ap(V3a,V7j))) ) ) )
& 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))) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4082(A_27b,A_27a,V3a,V2s,V1f)) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f4084(A_27a,A_27b,V1f,V3a,V0m)),V2s) ) ) ) ) ) ) ) ).
fof(lameq_f4085,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V4x] : ap(f4085(A_27a,V1f,V2s),V4x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V4x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V2s),V4x)) ) ) ).
fof(lameq_f4086,axiom,
! [A_27a,V2s] :
( mem(V2s,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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5x] : ap(f4086(A_27a,V2s,V0m,V1f),V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V5x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)),V2s)),V5x)) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__split,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(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x))) )
& 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))) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4085(A_27a,V1f,V2s))),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4086(A_27a,V2s,V0m,V1f))) ) ) ) ) ) ).
fof(lameq_f4087,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] : ap(f4087(A_27a,V1s),V2x) = ap(ap(c_2Eextreal_2Eextreal__mul,c_2Eextreal_2EPosInf),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V1s),V2x)) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__cmul__infty,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_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4087(A_27a,V1s)) = ap(ap(c_2Eextreal_2Eextreal__mul,c_2Eextreal_2EPosInf),ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ).
fof(lameq_f4088,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ! [V6i] : ap(f4088(A_27a,V1f,V5x),V6i) = ap(ap(V1f,V6i),V5x) ) ) ).
fof(lameq_f4089,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V5x] : ap(f4089(A_27a,V1f),V5x) = ap(c_2Eextreal_2Eext__suminf,f4088(A_27a,V1f,V5x)) ) ).
fof(lameq_f4090,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [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))))
=> ! [V7i] : ap(f4090(A_27a,V1f,V0m),V7i) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),ap(V1f,V7i)) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__suminf,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,ty_2Eextreal_2Eextreal)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(ap(V1f,V2i),V3x))) ) )
& ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> 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),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))) ) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4089(A_27a,V1f)) = ap(c_2Eextreal_2Eext__suminf,f4090(A_27a,V1f,V0m)) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__pos__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))
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> ( p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
<=> ( 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)))
& ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) != c_2Eextreal_2EPosInf ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__infty,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(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ap(V1f,V3x) = c_2Eextreal_2EPosInf ) ) )
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ).
fof(lameq_f4091,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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))))
=> ! [V2x] : ap(f4091(A_27a,V1f,V0m),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V2x)),c_2Eextreal_2EPosInf))) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__infty__null,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))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f)) )
=> p(ap(ap(c_2Emeasure_2Enull__set(A_27a),V0m),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f4091(A_27a,V1f,V0m)))) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__bounded,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),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)))
& ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__abs,ap(V2g,V3x))),ap(V1f,V3x))) ) )
=> p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V2g)) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__fn__plus,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))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f)) )
=> p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),ap(c_2Emeasure_2Efn__plus(A_27a),V1f))) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__fn__minus,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))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f)) )
=> p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),ap(c_2Emeasure_2Efn__minus(A_27a),V1f))) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__const,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_2Elebesgue_2Eintegrable(A_27a),V0m),k(A_27a,ap(c_2Eextreal_2ENormal,V1c)))) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__zero,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))))
=> ! [V1c] :
( mem(V1c,A_27b)
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__plus__minus,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))
=> ( p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
<=> ( 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)))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),ap(c_2Emeasure_2Efn__plus(A_27a),V1f)))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),ap(c_2Emeasure_2Efn__minus(A_27a),V1f))) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__add__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V2g))
& ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x))) )
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V2g,V4x))) ) )
=> p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4023(A_27a,V1f,V2g))) ) ) ) ) ) ).
fof(lameq_f4092,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f4092(A_27a,V1f,V2g),V3x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Emeasure_2Efn__plus(A_27a),V1f),V3x)),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V2g),V3x)) ) ) ).
fof(lameq_f4093,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4x] : ap(f4093(A_27a,V1f,V2g),V4x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Emeasure_2Efn__minus(A_27a),V1f),V4x)),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V2g),V4x)) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__add__lemma,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V2g)) )
=> ( p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4092(A_27a,V1f,V2g)))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4093(A_27a,V1f,V2g))) ) ) ) ) ) ) ).
fof(lameq_f4094,axiom,
! [A_27a,V1f1] :
( mem(V1f1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f2] :
( mem(V2f2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f4094(A_27a,V1f1,V2f2),V3x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f1,V3x)),ap(V2f2,V3x)) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__add,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))))
=> ! [V1f1] :
( mem(V1f1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f2] :
( mem(V2f2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f1))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V2f2)) )
=> p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4094(A_27a,V1f1,V2f2))) ) ) ) ) ) ).
fof(lameq_f4095,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f4095(A_27a,V1f,V2c),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),ap(V1f,V3x)) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f)) )
=> p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4095(A_27a,V1f,V2c))) ) ) ) ) ) ).
fof(lameq_f4096,axiom,
! [A_27a,V1f1] :
( mem(V1f1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f2] :
( mem(V2f2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f4096(A_27a,V1f1,V2f2),V3x) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V1f1,V3x)),ap(V2f2,V3x)) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__sub,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))))
=> ! [V1f1] :
( mem(V1f1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f2] :
( mem(V2f2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f1))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V2f2)) )
=> p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4096(A_27a,V1f1,V2f2))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__indicator,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_2Elebesgue_2Eintegrable(A_27a),V0m),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1s))) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__not__infty,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))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> ? [V3g] :
( mem(V3g,arr(A_27a,ty_2Eextreal_2Eextreal))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V3g))
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V3g,V4x))) )
& ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ap(V3g,V5x) != c_2Eextreal_2EPosInf ) )
& ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V3g) ) ) ) ) ) ).
fof(lameq_f4097,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f4097(A_27a,V1f),V3x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V3x)),c_2Eextreal_2EPosInf)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x)) ) ).
fof(lameq_f4098,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4x] : ap(f4098(A_27a,V1f),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V4x)),c_2Eextreal_2EPosInf)),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V4x)) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__not__infty__alt,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))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> ( p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4097(A_27a,V1f)))
& ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),f4098(A_27a,V1f)) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__not__infty__alt2,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))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> ( p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4097(A_27a,V1f)))
& ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4098(A_27a,V1f)) ) ) ) ) ) ).
fof(lameq_f4099,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x] : ap(f4099(A_27a,V1f),V2x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V2x)),c_2Eextreal_2ENegInf)),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V2x)),c_2Eextreal_2EPosInf))),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V2x)) ) ).
fof(lameq_f4100,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f4100(A_27a,V1f),V3x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V3x)),c_2Eextreal_2ENegInf)),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V3x)),c_2Eextreal_2EPosInf))),ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V3x)) ) ).
fof(conj_thm_2Elebesgue_2Eintegrable__not__infty__alt3,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))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f)) )
=> ( p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),f4099(A_27a,V1f)))
& ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),f4100(A_27a,V1f)) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__indicator,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_2Elebesgue_2Eintegral(A_27a),V0m),ap(c_2Emeasure_2Eindicator__fn(A_27a),V1s)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) ) ) ) ) ).
fof(lameq_f4101,axiom,
! [A_27a,V2f1] :
( mem(V2f1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3f2] :
( mem(V3f2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4x] : ap(f4101(A_27a,V2f1,V3f2),V4x) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V2f1,V4x)),ap(V3f2,V4x)) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__add__lemma,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))
=> ! [V2f1] :
( mem(V2f1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3f2] :
( mem(V3f2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V2f1))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V3f2))
& V1f = f4101(A_27a,V2f1,V3f2)
& ! [V5x] :
( mem(V5x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V2f1,V5x))) )
& ! [V6x] :
( mem(V6x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V3f2,V6x))) ) )
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V2f1)),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V3f2)) ) ) ) ) ) ) ).
fof(lameq_f4102,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f4102(A_27a,V1f,V2g),V3x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V3x)),ap(V2g,V3x)) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__add,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))
=> ! [V2g] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V2g)) )
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),f4102(A_27a,V1f,V2g)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f)),ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V2g)) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Elebesgue_2Eintegrable(A_27a),V0m),V1f)) )
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),f4095(A_27a,V1f,V2c)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f)) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__cmul__indicator,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))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ( 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_2Elebesgue_2Eintegral(A_27a),V0m),f4046(A_27a,V1s,V2c)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Erealax_2Ereal__mul,V2c),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__zero,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_2Elebesgue_2Eintegral(A_27a),V0m),k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ).
fof(lameq_f4103,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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))))
=> ! [V2x] : ap(f4103(A_27a,V1f,V0m),V2x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V2x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)),V2x)) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),f4103(A_27a,V1f,V0m)) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Eintegral__mono,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))))
=> ! [V1f1] :
( mem(V1f1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f2] :
( mem(V2f2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V3t] :
( mem(V3t,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3t),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1f1,V3t)),ap(V2f2,V3t))) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f1)),ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V2f2))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Efinite__space__integral__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& 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)))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ( ap(V1f,V2x) != c_2Eextreal_2ENegInf
& ap(V1f,V2x) != c_2Eextreal_2EPosInf ) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))) )
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Efinite__space__integral(A_27a),V0m),V1f) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Efinite__support__integral__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& 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)))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ( ap(V1f,V2x) != c_2Eextreal_2ENegInf
& ap(V1f,V2x) != c_2Eextreal_2EPosInf ) ) )
& p(ap(c_2Epred__set_2EFINITE(ty_2Eextreal_2Eextreal),ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))) )
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Elebesgue_2Efinite__space__integral(A_27a),V0m),V1f) ) ) ) ) ).
fof(lameq_f4104,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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(f4104(A_27a,V1f,V0m),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V3x)),ap(c_2Eextreal_2ENormal,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_2Elebesgue_2Efinite__space__POW__integral__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))))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ap(c_2Epred__set_2EPOW(A_27a),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)))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ( ap(V1f,V2x) != c_2Eextreal_2ENegInf
& ap(V1f,V2x) != c_2Eextreal_2EPosInf ) ) ) )
=> ap(ap(c_2Elebesgue_2Eintegral(A_27a),V0m),V1f) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f4104(A_27a,V1f,V0m)),ap(c_2Emeasure_2Em__space(A_27a),V0m)) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Efinite__POW__prod__measure__reduce,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [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_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),V0m0))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m1))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)))
& p(ap(c_2Epred__set_2EFINITE(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)))
& ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)
& ap(c_2Epred__set_2EPOW(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)) = ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1) )
=> ! [V2a0] :
( mem(V2a0,arr(A_27a,bool))
=> ! [V3a1] :
( mem(V3a1,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2a0),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3a1),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1))) )
=> ap(ap(ap(c_2Elebesgue_2Eprod__measure(A_27a,A_27b),V0m0),V1m1),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V2a0),V3a1)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m0),V2a0)),ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m1),V3a1)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [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_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),V0m0))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m1))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)))
& p(ap(c_2Epred__set_2EFINITE(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)))
& ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)
& ap(c_2Epred__set_2EPOW(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)) = ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1) )
=> p(ap(c_2Emeasure_2Emeasure__space(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Elebesgue_2Eprod__measure__space(A_27a,A_27b),V0m0),V1m1))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [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_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),V0m0))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m1))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)))
& p(ap(c_2Epred__set_2EFINITE(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)))
& ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)
& ap(c_2Epred__set_2EPOW(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)) = ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1) )
=> p(ap(c_2Emeasure_2Emeasure__space(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Em__space(A_27b),V1m1))),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Em__space(A_27b),V1m1)))),ap(ap(c_2Elebesgue_2Eprod__measure(A_27a,A_27b),V0m0),V1m1))))) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Efinite__prod__measure__space__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27b,bool))
=> ! [V2u] :
( mem(V2u,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V3v] :
( mem(V3v,arr(arr(A_27b,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1s2)) )
=> ap(ap(c_2Elebesgue_2Eprod__measure__space(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))),V0s1),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),V0s1)),V2u))),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))),V1s2),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27b),V1s2)),V3v))) = ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s1),V1s2)),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s1),V1s2))),ap(ap(c_2Elebesgue_2Eprod__measure(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))),V0s1),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),V0s1)),V2u))),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))),V1s2),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27b),V1s2)),V3v))))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Efinite__POW__prod__measure__reduce3,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [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_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2m2] :
( mem(V2m2,ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m0))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m1))
& p(ap(c_2Emeasure_2Emeasure__space(A_27c),V2m2))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)))
& p(ap(c_2Epred__set_2EFINITE(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)))
& p(ap(c_2Epred__set_2EFINITE(A_27c),ap(c_2Emeasure_2Em__space(A_27c),V2m2)))
& ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)
& ap(c_2Epred__set_2EPOW(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)) = ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1)
& ap(c_2Epred__set_2EPOW(A_27c),ap(c_2Emeasure_2Em__space(A_27c),V2m2)) = ap(c_2Emeasure_2Emeasurable__sets(A_27c),V2m2) )
=> ! [V3a0] :
( mem(V3a0,arr(A_27a,bool))
=> ! [V4a1] :
( mem(V4a1,arr(A_27b,bool))
=> ! [V5a2] :
( mem(V5a2,arr(A_27c,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3a0),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4a1),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V5a2),ap(c_2Emeasure_2Emeasurable__sets(A_27c),V2m2))) )
=> ap(ap(ap(ap(c_2Elebesgue_2Eprod__measure3(A_27a,A_27b,A_27c),V0m0),V1m1),V2m2),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V3a0),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V4a1),V5a2))) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m0),V3a0)),ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m1),V4a1))),ap(ap(c_2Emeasure_2Emeasure(A_27c),V2m2),V5a2)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Emeasure__space__finite__prod__measure__POW3,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [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_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2m2] :
( mem(V2m2,ty_2Epair_2Eprod(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m0))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m1))
& p(ap(c_2Emeasure_2Emeasure__space(A_27c),V2m2))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)))
& p(ap(c_2Epred__set_2EFINITE(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)))
& p(ap(c_2Epred__set_2EFINITE(A_27c),ap(c_2Emeasure_2Em__space(A_27c),V2m2)))
& ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)
& ap(c_2Epred__set_2EPOW(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m1)) = ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m1)
& ap(c_2Epred__set_2EPOW(A_27c),ap(c_2Emeasure_2Em__space(A_27c),V2m2)) = ap(c_2Emeasure_2Emeasurable__sets(A_27c),V2m2) )
=> p(ap(c_2Emeasure_2Emeasure__space(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(c_2Emeasure_2Em__space(A_27b),V1m1)),ap(c_2Emeasure_2Em__space(A_27c),V2m2)))),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(c_2Emeasure_2Em__space(A_27b),V1m1)),ap(c_2Emeasure_2Em__space(A_27c),V2m2))))),ap(ap(ap(c_2Elebesgue_2Eprod__measure3(A_27a,A_27b,A_27c),V0m0),V1m1),V2m2))))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Efinite__prod__measure__space__POW3,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27b,bool))
=> ! [V2s3] :
( mem(V2s3,arr(A_27c,bool))
=> ! [V3u] :
( mem(V3u,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V4v] :
( mem(V4v,arr(arr(A_27b,bool),ty_2Erealax_2Ereal))
=> ! [V5w] :
( mem(V5w,arr(arr(A_27c,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1s2))
& p(ap(c_2Epred__set_2EFINITE(A_27c),V2s3)) )
=> ap(ap(ap(c_2Elebesgue_2Eprod__measure__space3(A_27a,A_27b,A_27c),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))),V0s1),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),V0s1)),V3u))),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))),V1s2),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27b),V1s2)),V4v))),ap(ap(c_2Epair_2E_2C(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))),V2s3),ap(ap(c_2Epair_2E_2C(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27c),V2s3)),V5w))) = ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Epair_2Eprod(arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V0s1),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V1s2),V2s3))),ap(ap(c_2Epair_2E_2C(arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),ap(ap(c_2Epred__set_2ECROSS(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V0s1),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),V1s2),V2s3)))),ap(ap(ap(c_2Elebesgue_2Eprod__measure3(A_27a,A_27b,A_27c),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))),V0s1),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),V0s1)),V3u))),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))),V1s2),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27b),V1s2)),V4v))),ap(ap(c_2Epair_2E_2C(arr(A_27c,bool),ty_2Epair_2Eprod(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal))),V2s3),ap(ap(c_2Epair_2E_2C(arr(arr(A_27c,bool),bool),arr(arr(A_27c,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27c),V2s3)),V5w))))) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f4105,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1r] : ap(f4105(V0P),V1r) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V1r),V0P)),ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__sup,V0P)),ap(ap(c_2Eextreal_2Eextreal__add,V1r),ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) ) ).
fof(lameq_f4106,axiom,
! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V2P] :
( mem(V2P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V4r] : ap(f4106(V3n,V2P),V4r) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V4r),V2P)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__sup,V2P)),ap(ap(c_2Eextreal_2Eextreal__add,V4r),ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(c_2Enum_2ESUC,V3n)))))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Elebesgue_2Eseq__sup,V2P),V3n)),V4r)),ap(ap(c_2Eextreal_2Eextreal__lt,V4r),ap(c_2Eextreal_2Eextreal__sup,V2P))))) ) ) ).
fof(ax_thm_2Elebesgue_2Eseq__sup__def,axiom,
( ! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ap(ap(c_2Elebesgue_2Eseq__sup,V0P),c_2Enum_2E0) = ap(c_2Emin_2E_40(ty_2Eextreal_2Eextreal),f4105(V0P)) )
& ! [V2P] :
( mem(V2P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Elebesgue_2Eseq__sup,V2P),ap(c_2Enum_2ESUC,V3n)) = ap(c_2Emin_2E_40(ty_2Eextreal_2Eextreal),f4106(V3n,V2P)) ) ) ) ).
fof(lameq_f4107,axiom,
! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V2P] :
( mem(V2P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V4r] : ap(f4107(V3n,V2P),V4r) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V4r),V2P)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__sup,V2P)),ap(ap(c_2Eextreal_2Eextreal__add,V4r),ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))))))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Elebesgue_2Eseq__sup,V2P),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V4r)),ap(ap(c_2Eextreal_2Eextreal__lt,V4r),ap(c_2Eextreal_2Eextreal__sup,V2P))))) ) ) ).
fof(lameq_f4108,axiom,
! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ! [V5P] :
( mem(V5P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V7r] : ap(f4108(V6n,V5P),V7r) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),V7r),V5P)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__sup,V5P)),ap(ap(c_2Eextreal_2Eextreal__add,V7r),ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V6n))))))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,ap(ap(c_2Elebesgue_2Eseq__sup,V5P),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V6n)))),V7r)),ap(ap(c_2Eextreal_2Eextreal__lt,V7r),ap(c_2Eextreal_2Eextreal__sup,V5P))))) ) ) ).
fof(conj_thm_2Elebesgue_2Eseq__sup__def__compute,axiom,
( ! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ap(ap(c_2Elebesgue_2Eseq__sup,V0P),c_2Enum_2E0) = ap(c_2Emin_2E_40(ty_2Eextreal_2Eextreal),f4105(V0P)) )
& ! [V2P] :
( mem(V2P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Elebesgue_2Eseq__sup,V2P),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))) = ap(c_2Emin_2E_40(ty_2Eextreal_2Eextreal),f4107(V3n,V2P)) ) )
& ! [V5P] :
( mem(V5P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ap(ap(c_2Elebesgue_2Eseq__sup,V5P),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V6n))) = ap(c_2Emin_2E_40(ty_2Eextreal_2Eextreal),f4108(V6n,V5P)) ) ) ) ).
fof(conj_thm_2Elebesgue_2EEXTREAL__SUP__SEQ,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ( ( ? [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
& p(ap(V0P,V1x)) )
& ? [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
& V2z != c_2Eextreal_2EPosInf
& ! [V3x] :
( mem(V3x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0P,V3x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V3x),V2z)) ) ) ) )
=> ? [V4x] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(V4x,V5n)),V0P)) )
& ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V4x,V6n)),ap(V4x,ap(c_2Enum_2ESUC,V6n)))) )
& ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V4x),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Eextreal_2Eextreal__sup,V0P) ) ) ) ).
fof(lameq_f4109,axiom,
! [A_27a,V6g] :
( mem(V6g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V2f] :
( mem(V2f,arr(arr(A_27a,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal))
=> ! [V8n] : ap(f4109(A_27a,V6g,V2f),V8n) = ap(V2f,ap(V6g,V8n)) ) ) ).
fof(conj_thm_2Elebesgue_2EEXTREAL__SUP__FUN__SEQ__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1P_27] :
( mem(V1P_27,arr(arr(A_27a,ty_2Eextreal_2Eextreal),bool))
=> ! [V2f] :
( mem(V2f,arr(arr(A_27a,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal))
=> ( ( ? [V3x] :
( mem(V3x,ty_2Eextreal_2Eextreal)
& p(ap(V0P,V3x)) )
& ? [V4z] :
( mem(V4z,ty_2Eextreal_2Eextreal)
& V4z != c_2Eextreal_2EPosInf
& ! [V5x] :
( mem(V5x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0P,V5x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V5x),V4z)) ) ) )
& V0P = ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal),V2f),V1P_27) )
=> ? [V6g] :
( mem(V6g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
& ! [V7n] :
( mem(V7n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V6g,V7n)),V1P_27)) )
& ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4109(A_27a,V6g,V2f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Eextreal_2Eextreal__sup,V0P) ) ) ) ) ) ) ).
fof(ax_thm_2Elebesgue_2Emax__fn__seq__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0g] :
( mem(V0g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V0g),c_2Enum_2E0),V1x) = ap(ap(V0g,c_2Enum_2E0),V1x) ) )
& ! [V2g] :
( mem(V2g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V4x] :
( mem(V4x,A_27a)
=> ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V2g),ap(c_2Enum_2ESUC,V3n)),V4x) = ap(ap(c_2Eextreal_2Eextreal__max,ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V2g),V3n),V4x)),ap(ap(V2g,ap(c_2Enum_2ESUC,V3n)),V4x)) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Emax__fn__seq__def__compute,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0g] :
( mem(V0g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V0g),c_2Enum_2E0),V1x) = ap(ap(V0g,c_2Enum_2E0),V1x) ) )
& ! [V2g] :
( mem(V2g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V4x] :
( mem(V4x,A_27a)
=> ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V2g),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))),V4x) = ap(ap(c_2Eextreal_2Eextreal__max,ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V2g),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V4x)),ap(ap(V2g,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V3n))),V4x)) ) ) )
& ! [V5g] :
( mem(V5g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ! [V7x] :
( mem(V7x,A_27a)
=> ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V5g),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V6n))),V7x) = ap(ap(c_2Eextreal_2Eextreal__max,ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V5g),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V6n))),V7x)),ap(ap(V5g,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V6n))),V7x)) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Emax__fn__seq__mono,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V0g),V1n),V2x)),ap(ap(ap(c_2Elebesgue_2Emax__fn__seq(A_27a),V0g),ap(c_2Enum_2ESUC,V1n)),V2x))) ) ) ) ) ).
fof(lameq_f4110,axiom,
! [A_27a,V9g1] :
( mem(V9g1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V10g2] :
( mem(V10g2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V11x] : ap(f4110(A_27a,V9g1,V10g2),V11x) = ap(ap(c_2Eextreal_2Eextreal__max,ap(V9g1,V11x)),ap(V10g2,V11x)) ) ) ).
fof(lameq_f4111,axiom,
! [A_27a,V12g] :
( mem(V12g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal))
=> ! [V16n] : ap(f4111(A_27a,V12g,V0f),V16n) = ap(V0f,ap(V12g,V16n)) ) ) ).
fof(conj_thm_2Elebesgue_2EEXTREAL__SUP__FUN__SEQ__MONO__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal))
=> ! [V1P] :
( mem(V1P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V2P_27] :
( mem(V2P_27,arr(arr(A_27a,ty_2Eextreal_2Eextreal),bool))
=> ( ( ? [V3x] :
( mem(V3x,ty_2Eextreal_2Eextreal)
& p(ap(V1P,V3x)) )
& ? [V4z] :
( mem(V4z,ty_2Eextreal_2Eextreal)
& V4z != c_2Eextreal_2EPosInf
& ! [V5x] :
( mem(V5x,ty_2Eextreal_2Eextreal)
=> ( p(ap(V1P,V5x))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V5x),V4z)) ) ) )
& V1P = ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal),V0f),V2P_27)
& ! [V6g1] :
( mem(V6g1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V7g2] :
( mem(V7g2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V6g1),V2P_27))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V7g2),V2P_27))
& ! [V8x] :
( mem(V8x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V6g1,V8x)),ap(V7g2,V8x))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V6g1)),ap(V0f,V7g2))) ) ) )
& ! [V9g1] :
( mem(V9g1,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V10g2] :
( mem(V10g2,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V9g1),V2P_27))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V10g2),V2P_27)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f4110(A_27a,V9g1,V10g2)),V2P_27)) ) ) ) )
=> ? [V12g] :
( mem(V12g,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
& ! [V13n] :
( mem(V13n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V12g,V13n)),V2P_27)) )
& ! [V14x] :
( mem(V14x,A_27a)
=> ! [V15n] :
( mem(V15n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(V12g,V15n),V14x)),ap(ap(V12g,ap(c_2Enum_2ESUC,V15n)),V14x))) ) )
& ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4111(A_27a,V12g,V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Eextreal_2Eextreal__sup,V1P) ) ) ) ) ) ) ).
fof(ax_thm_2Elebesgue_2Esigned__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_2Elebesgue_2Esigned__measure__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_2Ecountably__additive(A_27a),V0m)) ) ) ) ) ).
fof(ax_thm_2Elebesgue_2Enegative__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))))
=> ! [V1A] :
( mem(V1A,arr(A_27a,bool))
=> ( p(ap(ap(c_2Elebesgue_2Enegative__set(A_27a),V0m),V1A))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ! [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),V0m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2s),V1A)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) ).
fof(lameq_f4112,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x] : ap(f4112(A_27a,V2f),V3x) = ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V2f,V3x)) ) ).
fof(lameq_f4113,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4A] :
( mem(V4A,arr(A_27a,bool))
=> ! [V5x] : ap(f4113(A_27a,V2f,V4A),V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V2f,V5x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V4A),V5x)) ) ) ).
fof(lameq_f4114,axiom,
! [A_27a,V1v] :
( mem(V1v,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,ty_2Eextreal_2Eextreal))
=> ! [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))))
=> ! [V4A] : ap(f4114(A_27a,V1v,V2f,V0m),V4A) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4113(A_27a,V2f,V4A))),ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V1v),V4A)))) ) ) ) ).
fof(lameq_f4115,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))))
=> ! [V1v] :
( mem(V1v,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] : ap(f4115(A_27a,V0m,V1v),V2f) = ap(ap(c_2Epair_2E_2C(arr(A_27a,ty_2Eextreal_2Eextreal),bool),V2f),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),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))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(A_27a),f4112(A_27a,V2f))),ap(c_2Ebool_2E_21(arr(A_27a,bool)),f4114(A_27a,V1v,V2f,V0m))))) ) ) ).
fof(ax_thm_2Elebesgue_2ERADON__F__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))))
=> ! [V1v] :
( mem(V1v,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_2Elebesgue_2ERADON__F(A_27a),V0m),V1v) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal)),f4115(A_27a,V0m,V1v)) ) ) ) ).
fof(lameq_f4116,axiom,
! [A_27a,V2r] :
( mem(V2r,ty_2Eextreal_2Eextreal)
=> ! [V1v] :
( mem(V1v,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),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))))
=> ! [V3f] : ap(f4116(A_27a,V2r,V1v,V0m),V3f) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),V2r),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V3f))),ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V3f),ap(ap(c_2Elebesgue_2ERADON__F(A_27a),V0m),V1v))) ) ) ) ).
fof(lameq_f4117,axiom,
! [A_27a,V1v] :
( mem(V1v,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),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))))
=> ! [V2r] : ap(f4117(A_27a,V1v,V0m),V2r) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),V2r),ap(c_2Ebool_2E_3F(arr(A_27a,ty_2Eextreal_2Eextreal)),f4116(A_27a,V2r,V1v,V0m))) ) ) ).
fof(ax_thm_2Elebesgue_2ERADON__F__integrals__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))))
=> ! [V1v] :
( mem(V1v,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_2Elebesgue_2ERADON__F__integrals(A_27a),V0m),V1v) = ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f4117(A_27a,V1v,V0m)) ) ) ) ).
fof(ax_thm_2Elebesgue_2Emeasure__absolutely__continuous__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))))
=> ! [V1v] :
( mem(V1v,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_2Elebesgue_2Emeasure__absolutely__continuous(A_27a),V0m),V1v))
<=> ! [V2A] :
( mem(V2A,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ap(ap(c_2Emeasure_2Emeasure(A_27a),V1v),V2A) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2A) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ).
fof(lameq_f4118,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V4x] : ap(f4118(A_27a,V0f,V1g),V4x) = ap(ap(c_2Eextreal_2Eextreal__max,ap(V0f,V4x)),ap(V1g,V4x)) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__radon__max__in__F,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [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))))
=> ! [V3v] :
( mem(V3v,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_2Emeasure_2Emeasure__space(A_27a),V3v))
& ap(c_2Emeasure_2Em__space(A_27a),V3v) = ap(c_2Emeasure_2Em__space(A_27a),V2m)
& ap(c_2Emeasure_2Emeasurable__sets(A_27a),V3v) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V2m)
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Elebesgue_2ERADON__F(A_27a),V2m),V3v)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Elebesgue_2ERADON__F(A_27a),V2m),V3v))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f4118(A_27a,V0f,V1g)),ap(ap(c_2Elebesgue_2ERADON__F(A_27a),V2m),V3v))) ) ) ) ) ) ) ).
fof(lameq_f4119,axiom,
! [A_27b,V3f__n] :
( mem(V3f__n,arr(ty_2Enum_2Enum,arr(A_27b,ty_2Eextreal_2Eextreal)))
=> ! [V1m] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V7n] : ap(f4119(A_27b,V3f__n,V1m),V7n) = ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27b),V1m),ap(V3f__n,V7n)) ) ) ).
fof(conj_thm_2Elebesgue_2Elemma__radon__seq__conv__sup,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,A_27a)
=> ! [V1m] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2v] :
( mem(V2v,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_27b),V1m))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V2v))
& ap(c_2Emeasure_2Em__space(A_27b),V2v) = ap(c_2Emeasure_2Em__space(A_27b),V1m)
& ap(c_2Emeasure_2Emeasurable__sets(A_27b),V2v) = ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m) )
=> ? [V3f__n] :
( mem(V3f__n,arr(ty_2Enum_2Enum,arr(A_27b,ty_2Eextreal_2Eextreal)))
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27b,ty_2Eextreal_2Eextreal)),ap(V3f__n,V4n)),ap(ap(c_2Elebesgue_2ERADON__F(A_27b),V1m),V2v))) )
& ! [V5x] :
( mem(V5x,A_27b)
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(V3f__n,V6n),V5x)),ap(ap(V3f__n,ap(c_2Enum_2ESUC,V6n)),V5x))) ) )
& ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f4119(A_27b,V3f__n,V1m)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Elebesgue_2ERADON__F__integrals(A_27b),V1m),V2v)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2ERN__lemma1,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))))
=> ! [V1v] :
( mem(V1v,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2e] :
( mem(V2e,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1v))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e))
& ap(c_2Emeasure_2Em__space(A_27a),V1v) = ap(c_2Emeasure_2Em__space(A_27a),V0m)
& ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1v) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) )
=> ? [V3A] :
( mem(V3A,arr(A_27a,bool))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ereal_2Ereal__lte,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),V1v),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3A)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V1v),V3A))))
& ! [V4B] :
( mem(V4B,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4B),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V4B),V3A)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Erealax_2Ereal__neg,V2e)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V4B)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V1v),V4B)))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2ERN__lemma2,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))))
=> ! [V1v] :
( mem(V1v,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_2Emeasure__space(A_27a),V1v))
& ap(c_2Emeasure_2Em__space(A_27a),V1v) = ap(c_2Emeasure_2Em__space(A_27a),V0m)
& ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1v) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) )
=> ? [V2A] :
( mem(V2A,arr(A_27a,bool))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ereal_2Ereal__lte,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),V1v),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2A)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V1v),V2A))))
& ! [V3B] :
( mem(V3B,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3B),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3B),V2A)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3B)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V1v),V3B)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2ERadon__Nikodym,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))))
=> ! [V1v] :
( mem(V1v,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_2Emeasure__space(A_27a),V1v))
& ap(c_2Emeasure_2Em__space(A_27a),V1v) = ap(c_2Emeasure_2Em__space(A_27a),V0m)
& ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1v) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)
& p(ap(ap(c_2Elebesgue_2Emeasure__absolutely__continuous(A_27a),V1v),V0m)) )
=> ? [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),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)))
& ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V2f,V3x))) )
& ! [V4A] :
( mem(V4A,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4113(A_27a,V2f,V4A)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V1v),V4A)) ) ) ) ) ) ) ) ).
fof(lameq_f4120,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5A] :
( mem(V5A,arr(A_27a,bool))
=> ! [V6x] : ap(f4120(A_27a,V2f,V5A),V6x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V2f,V6x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V5A),V6x)) ) ) ).
fof(conj_thm_2Elebesgue_2ERadon__Nikodym2,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))))
=> ! [V1v] :
( mem(V1v,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_2Emeasure__space(A_27a),V1v))
& ap(c_2Emeasure_2Em__space(A_27a),V1v) = ap(c_2Emeasure_2Em__space(A_27a),V0m)
& ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1v) = ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)
& p(ap(ap(c_2Elebesgue_2Emeasure__absolutely__continuous(A_27a),V1v),V0m)) )
=> ? [V2f] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),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)))
& ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V2f,V3x))) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ap(V2f,V4x) != c_2Eextreal_2EPosInf )
& ! [V5A] :
( mem(V5A,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V5A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f4120(A_27a,V2f,V5A)) = ap(c_2Eextreal_2ENormal,ap(ap(c_2Emeasure_2Emeasure(A_27a),V1v),V5A)) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------