ITP001 Axioms: ITP146_5.ax
%------------------------------------------------------------------------------
% File : ITP146_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : measure_2.ax [Gau20]
% : HL4146_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 469 ( 64 unt; 120 typ; 0 def)
% Number of atoms : 13920 ( 205 equ)
% Maximal formula atoms : 21 ( 29 avg)
% Number of connectives : 1376 ( 25 ~; 0 |; 415 &)
% ( 34 <=>; 902 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 12220 (12220 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 281 ( 118 >; 163 *; 0 +; 0 <<)
% Number of predicates : 89 ( 88 usr; 45 prp; 0-3 aty)
% Number of functors : 120 ( 120 usr; 2 con; 0-5 aty)
% Number of variables : 1314 (1308 !; 6 ?;1314 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Emeasure_2EBorel,type,
c_2Emeasure_2EBorel: $i ).
tff(mem_c_2Emeasure_2EBorel,axiom,
mem(c_2Emeasure_2EBorel,ty_2Epair_2Eprod(arr(ty_2Eextreal_2Eextreal,bool),arr(arr(ty_2Eextreal_2Eextreal,bool),bool))) ).
tff(tp_c_2Emeasure_2Eadditive,type,
c_2Emeasure_2Eadditive: del > $i ).
tff(mem_c_2Emeasure_2Eadditive,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Eadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Emeasure_2Ealgebra,type,
c_2Emeasure_2Ealgebra: del > $i ).
tff(mem_c_2Emeasure_2Ealgebra,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Ealgebra(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ).
tff(tp_c_2Emeasure_2Eclosed__cdi,type,
c_2Emeasure_2Eclosed__cdi: del > $i ).
tff(mem_c_2Emeasure_2Eclosed__cdi,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Eclosed__cdi(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ).
tff(tp_c_2Emeasure_2Ecountably__additive,type,
c_2Emeasure_2Ecountably__additive: del > $i ).
tff(mem_c_2Emeasure_2Ecountably__additive,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Ecountably__additive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Emeasure_2Ecountably__subadditive,type,
c_2Emeasure_2Ecountably__subadditive: del > $i ).
tff(mem_c_2Emeasure_2Ecountably__subadditive,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Ecountably__subadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Emeasure_2Efn__abs,type,
c_2Emeasure_2Efn__abs: del > $i ).
tff(mem_c_2Emeasure_2Efn__abs,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Efn__abs(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ).
tff(tp_c_2Emeasure_2Efn__minus,type,
c_2Emeasure_2Efn__minus: del > $i ).
tff(mem_c_2Emeasure_2Efn__minus,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Efn__minus(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ).
tff(tp_c_2Emeasure_2Efn__plus,type,
c_2Emeasure_2Efn__plus: del > $i ).
tff(mem_c_2Emeasure_2Efn__plus,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Efn__plus(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ).
tff(tp_c_2Emeasure_2Eincreasing,type,
c_2Emeasure_2Eincreasing: del > $i ).
tff(mem_c_2Emeasure_2Eincreasing,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Eincreasing(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Emeasure_2Eindicator__fn,type,
c_2Emeasure_2Eindicator__fn: del > $i ).
tff(mem_c_2Emeasure_2Eindicator__fn,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Eindicator__fn(A_27a),arr(arr(A_27a,bool),arr(A_27a,ty_2Eextreal_2Eextreal))) ).
tff(tp_c_2Emeasure_2Einf__measure,type,
c_2Emeasure_2Einf__measure: del > $i ).
tff(mem_c_2Emeasure_2Einf__measure,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Einf__measure(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))) ).
tff(tp_c_2Emeasure_2Elambda__system,type,
c_2Emeasure_2Elambda__system: del > $i ).
tff(mem_c_2Emeasure_2Elambda__system,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Elambda__system(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(arr(arr(A_27a,bool),ty_2Erealax_2Ereal),arr(arr(A_27a,bool),bool)))) ).
tff(tp_c_2Emeasure_2Em__space,type,
c_2Emeasure_2Em__space: del > $i ).
tff(mem_c_2Emeasure_2Em__space,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Em__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(A_27a,bool))) ).
tff(tp_c_2Emeasure_2Emeasurable,type,
c_2Emeasure_2Emeasurable: ( del * del ) > $i ).
tff(mem_c_2Emeasure_2Emeasurable,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Emeasure_2Emeasurable(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),arr(arr(A_27a,A_27b),bool)))) ).
tff(tp_c_2Emeasure_2Emeasurable__sets,type,
c_2Emeasure_2Emeasurable__sets: del > $i ).
tff(mem_c_2Emeasure_2Emeasurable__sets,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Emeasurable__sets(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Emeasure_2Emeasure,type,
c_2Emeasure_2Emeasure: del > $i ).
tff(mem_c_2Emeasure_2Emeasure,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Emeasure(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))) ).
tff(tp_c_2Emeasure_2Emeasure__preserving,type,
c_2Emeasure_2Emeasure__preserving: ( del * del ) > $i ).
tff(mem_c_2Emeasure_2Emeasure__preserving,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,A_27b),bool)))) ).
tff(tp_c_2Emeasure_2Emeasure__space,type,
c_2Emeasure_2Emeasure__space: del > $i ).
tff(mem_c_2Emeasure_2Emeasure__space,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Emeasure__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Emeasure_2Enull__set,type,
c_2Emeasure_2Enull__set: del > $i ).
tff(mem_c_2Emeasure_2Enull__set,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Enull__set(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Emeasure_2Eouter__measure__space,type,
c_2Emeasure_2Eouter__measure__space: del > $i ).
tff(mem_c_2Emeasure_2Eouter__measure__space,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Eouter__measure__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Emeasure_2Epos__simple__fn,type,
c_2Emeasure_2Epos__simple__fn: del > $i ).
tff(mem_c_2Emeasure_2Epos__simple__fn,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Epos__simple__fn(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(arr(ty_2Enum_2Enum,bool),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)))))) ).
tff(tp_c_2Emeasure_2Epositive,type,
c_2Emeasure_2Epositive: del > $i ).
tff(mem_c_2Emeasure_2Epositive,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Epositive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Emeasure_2Esigma,type,
c_2Emeasure_2Esigma: del > $i ).
tff(mem_c_2Emeasure_2Esigma,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Esigma(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))))) ).
tff(tp_c_2Emeasure_2Esigma__algebra,type,
c_2Emeasure_2Esigma__algebra: del > $i ).
tff(mem_c_2Emeasure_2Esigma__algebra,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Esigma__algebra(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ).
tff(tp_c_2Emeasure_2Esmallest__closed__cdi,type,
c_2Emeasure_2Esmallest__closed__cdi: del > $i ).
tff(mem_c_2Emeasure_2Esmallest__closed__cdi,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Esmallest__closed__cdi(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))) ).
tff(tp_c_2Emeasure_2Espace,type,
c_2Emeasure_2Espace: del > $i ).
tff(mem_c_2Emeasure_2Espace,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Espace(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(A_27a,bool))) ).
tff(tp_c_2Emeasure_2Esubadditive,type,
c_2Emeasure_2Esubadditive: del > $i ).
tff(mem_c_2Emeasure_2Esubadditive,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Esubadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ).
tff(tp_c_2Emeasure_2Esubset__class,type,
c_2Emeasure_2Esubset__class: del > $i ).
tff(mem_c_2Emeasure_2Esubset__class,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Esubset__class(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),bool))) ).
tff(tp_c_2Emeasure_2Esubsets,type,
c_2Emeasure_2Esubsets: del > $i ).
tff(mem_c_2Emeasure_2Esubsets,axiom,
! [A_27a: del] : mem(c_2Emeasure_2Esubsets(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(arr(A_27a,bool),bool))) ).
tff(ax_thm_2Emeasure_2Espace__def,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y: $i] :
( mem(V1y,arr(arr(A_27a,bool),bool))
=> ( ap(c_2Emeasure_2Espace(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0x),V1y)) = V0x ) ) ) ).
tff(ax_thm_2Emeasure_2Esubsets__def,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y: $i] :
( mem(V1y,arr(arr(A_27a,bool),bool))
=> ( ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0x),V1y)) = V1y ) ) ) ).
tff(ax_thm_2Emeasure_2Esubset__class__def,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts: $i] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1sts))
<=> ! [V2x: $i] :
( mem(V2x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1sts))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2x),V0sp)) ) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Ealgebra__def,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Esigma__algebra__def,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& ! [V1c: $i] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
tff(lamtp_f3417,type,
f3417: ( del * $i * $i ) > $i ).
tff(lameq_f3417,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1st: $i] :
( mem(V1st,arr(arr(A_27a,bool),bool))
=> ! [V2s: $i] : ( ap(f3417(A_27a,V0sp,V1st),V2s) = ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),bool),V2s),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1st),V2s)),ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2s)))) ) ) ) ).
tff(ax_thm_2Emeasure_2Esigma__def,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1st: $i] :
( mem(V1st,arr(arr(A_27a,bool),bool))
=> ( ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1st) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EBIGINTER(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f3417(A_27a,V0sp,V1st)))) ) ) ) ).
tff(ax_thm_2Emeasure_2Em__space__def,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts: $i] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ! [V2mu: $i] :
( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ap(c_2Emeasure_2Em__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V0sp ) ) ) ) ).
tff(ax_thm_2Emeasure_2Emeasurable__sets__def,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts: $i] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ! [V2mu: $i] :
( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ap(c_2Emeasure_2Emeasurable__sets(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V1sts ) ) ) ) ).
tff(ax_thm_2Emeasure_2Emeasure__def,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts: $i] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ! [V2mu: $i] :
( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ap(c_2Emeasure_2Emeasure(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V2mu ) ) ) ) ).
tff(ax_thm_2Emeasure_2Epositive__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
<=> ( ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Eadditive__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))
<=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Ecountably__additive__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m))
<=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n)))) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Esubadditive__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m))
<=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))) ) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Ecountably__subadditive__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m))
<=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Eincreasing__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
<=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Emeasure__space__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) ) ) ) ).
tff(lamtp_f3418,type,
f3418: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3418,axiom,
! [A_27a: del,V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l: $i] :
( mem(V2l,arr(A_27a,bool))
=> ! [V0gen: $i] :
( mem(V0gen,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V3s: $i] : ( ap(f3418(A_27a,V1lam,V2l,V0gen),V3s) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27a),V0gen))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V2l),V3s))),ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0gen)),V2l)),V3s)))),ap(V1lam,V3s))) ) ) ) ) ).
tff(lamtp_f3419,type,
f3419: ( del * $i * $i ) > $i ).
tff(lameq_f3419,axiom,
! [A_27a: del,V0gen: $i] :
( mem(V0gen,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l: $i] : ( ap(f3419(A_27a,V0gen,V1lam),V2l) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V2l),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l),ap(c_2Emeasure_2Esubsets(A_27a),V0gen))),ap(c_2Ebool_2E_21(arr(A_27a,bool)),f3418(A_27a,V1lam,V2l,V0gen)))) ) ) ) ).
tff(ax_thm_2Emeasure_2Elambda__system__def,axiom,
! [A_27a: del,V0gen: $i] :
( mem(V0gen,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gen),V1lam) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3419(A_27a,V0gen,V1lam)) ) ) ) ).
tff(ax_thm_2Emeasure_2Eouter__measure__space__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),V0m))
<=> ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) ) ) ) ).
tff(lamtp_f3420,type,
f3420: ( del * $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3420,axiom,
! [A_27a: del,V3f: $i] :
( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4m: tp__ty_2Enum_2Enum,V5n: tp__ty_2Enum_2Enum] : ( ap(f3420(A_27a,V3f,V4m),inj__ty_2Enum_2Enum(V5n)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4m)),inj__ty_2Enum_2Enum(V5n)))),ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,inj__ty_2Enum_2Enum(V4m))),ap(V3f,inj__ty_2Enum_2Enum(V5n)))) ) ) ).
tff(lamtp_f3421,type,
f3421: ( del * $i ) > $i ).
tff(lameq_f3421,axiom,
! [A_27a: del,V3f: $i] :
( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4m: tp__ty_2Enum_2Enum] : ( ap(f3421(A_27a,V3f),inj__ty_2Enum_2Enum(V4m)) = ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3420(A_27a,V3f,V4m)) ) ) ).
tff(lamtp_f3422,type,
f3422: ( del * tp__ty_2Erealax_2Ereal * $i * $i ) > $i ).
tff(lameq_f3422,axiom,
! [A_27a: del,V2r: tp__ty_2Erealax_2Ereal,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3f: $i] : ( ap(f3422(A_27a,V2r,V1s,V0m),V3f) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3421(A_27a,V3f))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V3f)),inj__ty_2Erealax_2Ereal(V2r))))) ) ) ) ).
tff(lamtp_f3423,type,
f3423: ( del * $i * $i ) > $i ).
tff(lameq_f3423,axiom,
! [A_27a: del,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2r: tp__ty_2Erealax_2Ereal] : ( ap(f3423(A_27a,V1s,V0m),inj__ty_2Erealax_2Ereal(V2r)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V2r)),ap(c_2Ebool_2E_3F(arr(ty_2Enum_2Enum,arr(A_27a,bool))),f3422(A_27a,V2r,V1s,V0m))) ) ) ) ).
tff(ax_thm_2Emeasure_2Einf__measure__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Einf,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3423(A_27a,V1s,V0m)))) ) ) ) ).
tff(ax_thm_2Emeasure_2Eclosed__cdi__def,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) )
& ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))
& ( ap(V2f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3n))),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) )
& ! [V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5m))),ap(V4f,inj__ty_2Enum_2Enum(V6n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ) ).
tff(lamtp_f3424,type,
f3424: ( del * $i ) > $i ).
tff(lameq_f3424,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] : ( ap(f3424(A_27a,V0a),V1b) = ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),bool),V1b),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)),V1b)),ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1b)))) ) ) ).
tff(ax_thm_2Emeasure_2Esmallest__closed__cdi__def,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Epred__set_2EBIGINTER(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f3424(A_27a,V0a)))) ) ) ).
tff(lamtp_f3425,type,
f3425: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3425,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V3s: $i] : ( ap(f3425(A_27a,A_27b,V0a,V2f,V1b),V3s) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))),ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Espace(A_27a),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
tff(lamtp_f3426,type,
f3426: ( del * del * $i * $i ) > $i ).
tff(lameq_f3426,axiom,
! [A_27b: del,A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f: $i] : ( ap(f3426(A_27b,A_27a,V0a,V1b),V2f) = ap(ap(c_2Epair_2E_2C(arr(A_27a,A_27b),bool),V2f),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))),ap(c_2Ebool_2E_21(arr(A_27b,bool)),f3425(A_27a,A_27b,V0a,V2f,V1b)))))) ) ) ) ).
tff(ax_thm_2Emeasure_2Emeasurable__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ( ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,A_27b),arr(A_27a,A_27b)),f3426(A_27b,A_27a,V0a,V1b)) ) ) ) ).
tff(lamtp_f3427,type,
f3427: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3427,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V3s: $i] : ( ap(f3427(A_27a,A_27b,V0m1,V2f,V1m2),V3s) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2))),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m1),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)))),ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m2),V3s))) ) ) ) ) ).
tff(lamtp_f3428,type,
f3428: ( del * del * $i * $i ) > $i ).
tff(lameq_f3428,axiom,
! [A_27b: del,A_27a: del,V0m1: $i] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2f: $i] : ( ap(f3428(A_27b,A_27a,V0m1,V1m2),V2f) = ap(ap(c_2Epair_2E_2C(arr(A_27a,A_27b),bool),V2f),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2))))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2)),ap(c_2Ebool_2E_21(arr(A_27b,bool)),f3427(A_27a,A_27b,V0m1,V2f,V1m2)))))) ) ) ) ).
tff(ax_thm_2Emeasure_2Emeasure__preserving__def,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ( ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,A_27b),arr(A_27a,A_27b)),f3428(A_27b,A_27a,V0m1,V1m2)) ) ) ) ).
tff(ax_thm_2Emeasure_2Eindicator__fn__def,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( ap(c_2Emeasure_2Eindicator__fn(A_27a),V0s) = f3355(A_27a,V0s) ) ) ).
tff(lamtp_f3429,type,
f3429: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3429,axiom,
! [A_27a: del,V6t: $i] :
( mem(V6t,A_27a)
=> ! [V3a: $i] :
( mem(V3a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4x: $i] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7i: tp__ty_2Enum_2Enum] : ( ap(f3429(A_27a,V6t,V3a,V4x),inj__ty_2Enum_2Enum(V7i)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,ap(V4x,inj__ty_2Enum_2Enum(V7i)))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V3a,inj__ty_2Enum_2Enum(V7i))),V6t)) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Epos__simple__fn__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s: $i] :
( mem(V2s,arr(ty_2Enum_2Enum,bool))
=> ! [V3a: $i] :
( mem(V3a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4x: $i] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x))
<=> ( ! [V5t: $i] :
( mem(V5t,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V1f,V5t))) )
& ! [V6t: $i] :
( mem(V6t,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6t),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V1f,V6t)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3429(A_27a,V6t,V3a,V4x)),V2s)) ) ) )
& ! [V8i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V8i)),V2s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V3a,inj__ty_2Enum_2Enum(V8i))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V2s))
& ! [V9i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V9i)),V2s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V4x,inj__ty_2Enum_2Enum(V9i)))) )
& ! [V10i: tp__ty_2Enum_2Enum,V11j: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V10i)),V2s))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V11j)),V2s))
& ( V10i != V11j ) )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3a,inj__ty_2Enum_2Enum(V10i))),ap(V3a,inj__ty_2Enum_2Enum(V11j)))) )
& ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3a),V2s)) = ap(c_2Emeasure_2Em__space(A_27a),V0m) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Emeasure_2Enull__set__def,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Emeasure_2Enull__set(A_27a),V0m),V1s))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESPACE,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)) = V0a ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__ALT__INTER,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__EMPTY,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__SPACE,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__COMPL,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__UNION,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__INTER,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__DIFF,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__FINITE__UNION,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1c: $i] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__INTER__SPACE,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s) = V1s )
& ( ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),ap(c_2Emeasure_2Espace(A_27a),V0a)) = V1s ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__EMPTY,axiom,
! [A_27a: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__COMPL,axiom,
! [A_27a: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l: $i] :
( mem(V2l,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0g0)),V2l)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INTER,axiom,
! [A_27a: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l1: $i] :
( mem(V2l1,arr(A_27a,bool))
=> ! [V3l2: $i] :
( mem(V3l2,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l1),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3l2),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2l1),V3l2)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ALGEBRA,axiom,
! [A_27a: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) )
=> p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))) ) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__ADDITIVE,axiom,
! [A_27a: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,bool))
=> ! [V3l1: $i] :
( mem(V3l1,arr(A_27a,bool))
=> ! [V4l2: $i] :
( mem(V4l2,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2g),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V3l1),V4l2))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3l1),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4l2),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) )
=> ( surj__ty_2Erealax_2Ereal(ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V3l1),V4l2)),V2g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V3l1),V2g))),ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V4l2),V2g)))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ADDITIVE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2l1: $i] :
( mem(V2l1,A_27b)
=> ! [V3l2: $i] :
( mem(V3l2,A_27c)
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) )
=> p(ap(c_2Emeasure_2Eadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE__SUBADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m)) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__MONO,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
& ( ap(V1f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a) )
& ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2n))),ap(V1f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__DISJOINT,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
& ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESUBADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ( V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t))
& ( V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))
& ( V2s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EADDITIVE__SUM,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V3m))),ap(V1f,inj__ty_2Enum_2Enum(V4n)))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2n))),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V2n)))))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EINCREASING__ADDITIVE__SUMMABLE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n)))) ) )
=> p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) ) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__POSITIVE,axiom,
! [A_27a: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
=> p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INCREASING,axiom,
! [A_27a: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) ) ).
tff(lamtp_f3430,type,
f3430: ( del * $i ) > $i ).
tff(lameq_f3430,axiom,
! [A_27a: del,V2g: $i] :
( mem(V2g,arr(A_27a,bool))
=> ! [V7s: $i] : ( ap(f3430(A_27a,V2g),V7s) = ap(ap(c_2Epred__set_2EINTER(A_27a),V7s),V2g) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__SUM,axiom,
! [A_27a: del,V0g0: $i] :
( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,bool))
=> ! [V3f: $i] :
( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V4n: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2g),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))))
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,inj__ty_2Enum_2Enum(V5m))),ap(V3f,inj__ty_2Enum_2Enum(V6n)))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V4n))),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),V1lam),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,arr(A_27a,bool),arr(A_27a,bool)),f3430(A_27a,V2g)),V3f)))) = surj__ty_2Erealax_2Ereal(ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V4n))))),V2g))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALGEBRA,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
=> p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) ) ) ).
tff(conj_thm_2Emeasure_2EOUTER__MEASURE__SPACE__POSITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),V0m))
=> p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) ) ).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__CARATHEODORY,axiom,
! [A_27a: del,V0gsig: $i] :
( mem(V0gsig,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0gsig))
& p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0gsig)),V1lam)))) )
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam))))
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3m))),ap(V2f,inj__ty_2Enum_2Enum(V4n)))) ) )
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam)))
& p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),V1lam),V2f)),ap(V1lam,ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ECARATHEODORY__LEMMA,axiom,
! [A_27a: del,V0gsig: $i] :
( mem(V0gsig,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1lam: $i] :
( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0gsig))
& p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0gsig)),V1lam)))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam)),V1lam)))) ) ) ) ).
tff(lamtp_f3431,type,
f3431: ( del * $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3431,axiom,
! [A_27a: del,V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] : ( ap(f3431(A_27a,V4f,V5m),inj__ty_2Enum_2Enum(V6n)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V6n)))),ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5m))),ap(V4f,inj__ty_2Enum_2Enum(V6n)))) ) ) ).
tff(lamtp_f3432,type,
f3432: ( del * $i ) > $i ).
tff(lameq_f3432,axiom,
! [A_27a: del,V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5m: tp__ty_2Enum_2Enum] : ( ap(f3432(A_27a,V4f),inj__ty_2Enum_2Enum(V5m)) = ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3431(A_27a,V4f,V5m)) ) ) ).
tff(lamtp_f3433,type,
f3433: ( del * tp__ty_2Erealax_2Ereal * $i * $i ) > $i ).
tff(lameq_f3433,axiom,
! [A_27a: del,V3r: tp__ty_2Erealax_2Ereal,V1g: $i] :
( mem(V1g,arr(A_27a,bool))
=> ! [V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V4f: $i] : ( ap(f3433(A_27a,V3r,V1g,V0m),V4f) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3432(A_27a,V4f))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V4f)),inj__ty_2Erealax_2Ereal(V3r))))) ) ) ) ).
tff(lamtp_f3434,type,
f3434: ( del * $i * $i ) > $i ).
tff(lameq_f3434,axiom,
! [A_27a: del,V1g: $i] :
( mem(V1g,arr(A_27a,bool))
=> ! [V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3r: tp__ty_2Erealax_2Ereal] : ( ap(f3434(A_27a,V1g,V0m),inj__ty_2Erealax_2Ereal(V3r)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V3r)),ap(c_2Ebool_2E_3F(arr(ty_2Enum_2Enum,arr(A_27a,bool))),f3433(A_27a,V3r,V1g,V0m))) ) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__NONEMPTY,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,bool))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),V2s)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3434(A_27a,V1g,V0m)))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__POS0,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,bool))
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3434(A_27a,V1g,V0m)))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__POS,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),ap(c_2Emeasure_2Em__space(A_27a),V0m))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1g))) ) ) ) ).
tff(conj_thm_2Emeasure_2EINCREASING,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EADDITIVE__INCREASING,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) ) ) ).
tff(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) ) ).
tff(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3m))),ap(V2f,inj__ty_2Enum_2Enum(V4n)))) )
& ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__AGREES,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__DOWN,axiom,
! [A_27a: del,V0m0: $i] :
( mem(V0m0,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m1: $i] :
( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0))))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))
& ( ap(c_2Emeasure_2Emeasure(A_27a),V0m0) = ap(c_2Emeasure_2Emeasure(A_27a),V1m1) )
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1)) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m0)) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__SIGMA,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1sts: $i] :
( mem(V1sts,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1sts))
=> p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1sts))) ) ) ) ).
tff(conj_thm_2Emeasure_2EPOW__ALGEBRA,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EPOW(A_27a),V0sp)))) ) ).
tff(conj_thm_2Emeasure_2EPOW__SIGMA__ALGEBRA,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EPOW(A_27a),V0sp)))) ) ).
tff(conj_thm_2Emeasure_2EUNIV__SIGMA__ALGEBRA,axiom,
! [A_27a: del] : p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),c_2Epred__set_2EUNIV(A_27a)),c_2Epred__set_2EUNIV(arr(A_27a,bool))))) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__EMPTY,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__POSITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__INCREASING,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) ) ).
tff(lamtp_f3435,type,
f3435: ( del * $i * tp__ty_2Erealax_2Ereal * $i ) > $i ).
tff(lameq_f3435,axiom,
! [A_27a: del,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V3r: tp__ty_2Erealax_2Ereal,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V4f: $i] : ( ap(f3435(A_27a,V1s,V3r,V0m),V4f) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V4f)),inj__ty_2Erealax_2Ereal(V3r)))) ) ) ) ).
tff(lamtp_f3436,type,
f3436: ( del * $i * $i ) > $i ).
tff(lameq_f3436,axiom,
! [A_27a: del,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3r: tp__ty_2Erealax_2Ereal] : ( ap(f3436(A_27a,V1s,V0m),inj__ty_2Erealax_2Ereal(V3r)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V3r)),ap(c_2Ebool_2E_3F(arr(ty_2Enum_2Enum,arr(A_27a,bool))),f3435(A_27a,V1s,V3r,V0m))) ) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__LE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3436(A_27a,V1s,V0m)))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)),inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__CLOSE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Emeasure_2Em__space(A_27a),V0m))) )
=> ? [V3f: $i] :
( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V4l: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,inj__ty_2Enum_2Enum(V5m))),ap(V3f,inj__ty_2Enum_2Enum(V6n)))) )
& p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V3f)),inj__ty_2Erealax_2Ereal(V4l)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4l)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)),inj__ty_2Erealax_2Ereal(V2e)))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__COUNTABLY__SUBADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) ) ).
tff(conj_thm_2Emeasure_2EINF__MEASURE__OUTER,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__SUBSET,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1b))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Esubsets(A_27a),V1b))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Espace(A_27a),V1b)),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V1b))) ) ) ) ).
tff(conj_thm_2Emeasure_2EALGEBRA__SUBSET__LAMBDA__SYSTEM,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m)))) ) ) ).
tff(conj_thm_2Emeasure_2ECARATHEODORY,axiom,
! [A_27a: del,V0m0: $i] :
( mem(V0m0,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0))))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m0))
& p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m0)) )
=> ? [V1m: $i] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
& ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V2s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m0),V2s)) ) ) )
& ( ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)) = ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)) )
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__SUBSET__SUBSETS,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__SIGMA,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2x: $i] :
( mem(V2x,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1a))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0p))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) )
& ! [V2c: $i] :
( mem(V2c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V2c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2c),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V2c)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__COUNTABLE__UNION,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1c: $i] :
( mem(V1c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ENUM,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__COMPL,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)),V1s))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1p: $i] :
( mem(V1p,arr(arr(A_27a,bool),bool))
=> ! [V2a: $i] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V1p))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p))
& ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) )
& ! [V4c: $i] :
( mem(V4c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V4c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V4c),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V4c)),V1p)) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__EMPTY,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__SUBSET__MEASURABLE__SETS,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1m: $i] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V1m)),V0a))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__DISJOINT,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
<=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
& ! [V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( V5m != V6n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5m))),ap(V4f,inj__ty_2Enum_2Enum(V6n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__ALT,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1p: $i] :
( mem(V1p,arr(arr(A_27a,bool),bool))
=> ! [V2a: $i] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V1p))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p))
& ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) )
& ! [V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESPACE__SMALLEST__CLOSED__CDI,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ap(c_2Emeasure_2Espace(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)) = ap(c_2Emeasure_2Espace(A_27a),V0a) ) ) ).
tff(conj_thm_2Emeasure_2ESMALLEST__CLOSED__CDI,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))))
& p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))
& p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) ).
tff(conj_thm_2Emeasure_2ECLOSED__CDI__DUNION,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ECLOSED__CDI__COMPL,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ).
tff(conj_thm_2Emeasure_2ECLOSED__CDI__DISJOINT,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ).
tff(conj_thm_2Emeasure_2ECLOSED__CDI__INCREASING,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))
& ( ap(V1f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a) )
& ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2n))),ap(V1f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA1,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA2,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2p: $i] :
( mem(V2p,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2p))
& p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2p))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))),V2p)) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__WEAK,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1p: $i] :
( mem(V1p,arr(arr(A_27a,bool),bool))
=> ! [V2a: $i] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2a)))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p))
& p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p))
& ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V1p))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) )
& ! [V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1p)))
& ( ap(V4f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a) )
& ! [V5n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5n))),ap(V4f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V5n))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) )
& ! [V6f: $i] :
( mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V6f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1p)))
& ! [V7m: tp__ty_2Enum_2Enum,V8n: tp__ty_2Enum_2Enum] :
( ( V7m != V8n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V6f,inj__ty_2Enum_2Enum(V7m))),ap(V6f,inj__ty_2Enum_2Enum(V8n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V6f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1p: $i] :
( mem(V1p,arr(arr(A_27a,bool),bool))
=> ! [V2a: $i] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2a)))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p))
& ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) )
& ! [V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))))
& ( ap(V4f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a) )
& ! [V5n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5n))),ap(V4f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V5n))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) )
& ! [V6f: $i] :
( mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V6f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))))
& ! [V7m: tp__ty_2Enum_2Enum,V8n: tp__ty_2Enum_2Enum] :
( ( V7m != V8n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V6f,inj__ty_2Enum_2Enum(V7m))),ap(V6f,inj__ty_2Enum_2Enum(V8n)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V6f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__COUNTABLY__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3m))),ap(V2f,inj__ty_2Enum_2Enum(V4n)))) )
& ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )
=> p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__ADDITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t))
& ( V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__COUNTABLE__INCREASING,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ( ap(V2f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3n))),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))))
& ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__REDUCE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),ap(c_2Emeasure_2Emeasure(A_27a),V0m))) = V0m ) ) ).
tff(conj_thm_2Emeasure_2ESPACE__SIGMA,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ( ap(c_2Emeasure_2Espace(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)) = V0sp ) ) ) ).
tff(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3n))),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))))
& ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__REDUCE,axiom,
! [A_27a: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ( ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))) = ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__SETS__SUBSET__SPACE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__DIFF__PROPERTY,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
=> ! [V4s: $i] :
( mem(V4s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),ap(ap(c_2Epred__set_2EDIFF(A_27b),ap(c_2Emeasure_2Espace(A_27b),V1b)),V4s)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V4s)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__BIGUNION__PROPERTY,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
=> ! [V4c: $i] :
( mem(V4c,arr(arr(A_27b,bool),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27b,bool)),V4c),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),ap(c_2Epred__set_2EBIGUNION(A_27b),V4c)) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),V4c)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASUBABLE__BIGUNION__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
=> ! [V4c: $i] :
( mem(V4c,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V4c))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V4c),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V4c)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__SIGMA__PREIMAGES,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )
=> p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b))))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b))))
& ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Espace(A_27a),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__SIGMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2b: $i] :
( mem(V2b,arr(arr(A_27b,bool),bool))
=> ! [V3sp: $i] :
( mem(V3sp,arr(A_27b,bool))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Emeasure_2Esubset__class(A_27b),V3sp),V2b))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V1a)),V3sp)))
& ! [V4s: $i] :
( mem(V4s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4s),V2b))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V4s)),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),ap(ap(c_2Emeasure_2Esigma(A_27b),V3sp),V2b)))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Espace(A_27b),V1b)),ap(c_2Emeasure_2Esubsets(A_27b),V1b))))) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__LIFT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2b: $i] :
( mem(V2b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),V2b)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Espace(A_27b),V2b)),ap(c_2Emeasure_2Esubsets(A_27b),V2b))))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURE__PRESERVING,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2)))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2)))))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2))
& ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m1),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m2),V3s)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__LIFT,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2a: $i] :
( mem(V2a,arr(arr(A_27b,bool),bool))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2))
& ( ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2) = ap(c_2Emeasure_2Esubsets(A_27b),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),V2a)) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27b),V1m2)))))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2a: $i] :
( mem(V2a,arr(arr(A_27b,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1))
& p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2))
& ( ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2) = ap(c_2Emeasure_2Esubsets(A_27b),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),V2a)) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27b),V1m2))))),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__I,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27a)),c_2Ecombin_2EI(A_27a)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27a),V0a),V0a))) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__COMP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2a: $i] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V3b: $i] :
( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4c: $i] :
( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27b,A_27c),V3b),V4c))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2a: $i] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V3b: $i] :
( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4c: $i] :
( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27c),V4c))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Epred__set_2EFUNSET(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V3b)),ap(c_2Emeasure_2Espace(A_27c),V4c))))
& ! [V5x: $i] :
( mem(V5x,arr(A_27c,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V5x),ap(c_2Emeasure_2Esubsets(A_27c),V4c)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V5x)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Emeasure_2Espace(A_27a),V2a)))),ap(c_2Emeasure_2Esubsets(A_27b),V3b))) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONGER,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2a: $i] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V3b: $i] :
( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4c: $i] :
( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ! [V5t: $i] :
( mem(V5t,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27c),V4c))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Epred__set_2EFUNSET(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V3b)),ap(c_2Emeasure_2Espace(A_27c),V4c))))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Emeasure_2Espace(A_27a),V2a))),V5t))
& ! [V6s: $i] :
( mem(V6s,arr(A_27c,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V6s),ap(c_2Emeasure_2Esubsets(A_27c),V4c)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V6s)),V5t)),ap(c_2Emeasure_2Esubsets(A_27b),V3b))) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__UP__LIFT,axiom,
! [A_27a: del,A_27b: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2b: $i] :
( mem(V2b,arr(arr(A_27a,bool),bool))
=> ! [V3c: $i] :
( mem(V3c,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V4f: $i] :
( mem(V4f,arr(A_27a,A_27b))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V4f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)),V3c)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2b)) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V4f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)),V3c))) ) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__UP__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0sp: $i] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2b: $i] :
( mem(V2b,arr(arr(A_27a,bool),bool))
=> ! [V3c: $i] :
( mem(V3c,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2b))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)),V3c)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)),V3c))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__UP__SIGMA,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1b: $i] :
( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))),V1b))) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__LIFT,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1m1: $i] :
( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2m2: $i] :
( mem(V2m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V0a),ap(c_2Emeasure_2Emeasure(A_27a),V1m1)))),V2m2)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V1m1),V2m2))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1m1: $i] :
( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2m2: $i] :
( mem(V2m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V0a),ap(c_2Emeasure_2Emeasure(A_27a),V1m1)))),V2m2)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V1m1),V2m2))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SIGMA,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1m2: $i] :
( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))))
=> ! [V2a: $i] :
( mem(V2a,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1))
& ( ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1) = ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),V2a)) ) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27a),V0m1)))),V1m2)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__PROD__SIGMA,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1a1: $i] :
( mem(V1a1,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)))
=> ! [V2a2: $i] :
( mem(V2a2,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool)))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,ty_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2EFST(A_27b,A_27c)),V3f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1a1)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,ty_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2ESND(A_27b,A_27c)),V3f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V0a),V2a2))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),V3f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V0a),ap(ap(c_2Emeasure_2Esigma(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V1a1)),ap(c_2Emeasure_2Espace(A_27c),V2a2))),ap(ap(c_2Eutil__prob_2Eprod__sets(A_27b,A_27c),ap(c_2Emeasure_2Esubsets(A_27b),V1a1)),ap(c_2Emeasure_2Esubsets(A_27c),V2a2)))))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__RANGE__REDUCE,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27b,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),V2s),ap(c_2Epred__set_2EPOW(A_27b),V2s)))))
& ( V2s != c_2Epred__set_2EEMPTY(A_27b) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))))))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ( ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),c_2Epred__set_2EUNIV(A_27b)),ap(c_2Epred__set_2EPOW(A_27b),c_2Epred__set_2EUNIV(A_27b)))))) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ( ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))))))) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1s_27: $i] :
( mem(V1s_27,arr(A_27a,bool))
=> ! [V2m: $i] :
( mem(V2m,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s_27),V0s))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0s),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V0s)),V2m)))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s_27),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),V2m)))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESTRONG__MEASURE__SPACE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s_27: $i] :
( mem(V1s_27,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s_27),ap(c_2Emeasure_2Em__space(A_27a),V0s)))
& p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0s))
& p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0s))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s_27),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),ap(c_2Emeasure_2Emeasure(A_27a),V0s))))) ) ) ) ).
tff(lamtp_f3437,type,
f3437: ( del * $i ) > $i ).
tff(lameq_f3437,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V3x: $i] : ( ap(f3437(A_27a,V0m),V3x) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EINSERT(A_27a),V3x),c_2Epred__set_2EEMPTY(A_27a))) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__REAL__SUM__IMAGE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3437(A_27a,V0m)),V1s)) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__POW,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( ap(ap(c_2Emeasure_2Esigma(A_27a),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s)) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s)) ) ) ).
tff(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1m: $i] :
( mem(V1m,arr(arr(A_27a,bool),ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0s)))
& p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m))))
& p(ap(c_2Emeasure_2Eadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))) ) ) ) ).
tff(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces2,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)))
& p(ap(c_2Emeasure_2Epositive(A_27a),V0m))
& p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) ) ) ).
tff(conj_thm_2Emeasure_2EIMAGE__SING,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))) = ap(ap(c_2Epred__set_2EINSERT(A_27b),ap(V0f,V1x)),c_2Epred__set_2EEMPTY(A_27b)) ) ) ) ).
tff(conj_thm_2Emeasure_2ESUBSET__BIGINTER,axiom,
! [A_27a: del,V0X: $i] :
( mem(V0X,arr(A_27a,bool))
=> ! [V1P: $i] :
( mem(V1P,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),ap(c_2Epred__set_2EBIGINTER(A_27a),V1P)))
<=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2Y),V1P))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),V2Y)) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__INCREASING,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__POSITIVE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__INTER,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__UNION,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__DIFF,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__COMPL__SUBSET,axiom,
! [A_27a: del,V0t: $i] :
( mem(V0t,arr(A_27a,bool))
=> ! [V1m: $i] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0t),V2s)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),ap(ap(c_2Epred__set_2EDIFF(A_27a),V2s),V0t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V2s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V0t))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGUNION,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1s,inj__ty_2Enum_2Enum(V2n))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__IN__MSPACE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1A: $i] :
( mem(V1A,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1A))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET__MSPACE,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1m: $i] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0A),ap(c_2Emeasure_2Em__space(A_27a),V1m))) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__EMPTY__MEASURABLE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__MSPACE__MEASURABLE,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__BIGINTER,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )
& ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGINTER,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1s,inj__ty_2Enum_2Enum(V2n))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ).
tff(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE2,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2n))),ap(V1f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))))) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ).
tff(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),ap(V2f,inj__ty_2Enum_2Enum(V3n))))
& ( V1s = ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER2,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))
& ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n)))),ap(V1f,inj__ty_2Enum_2Enum(V2n)))) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ).
tff(lamtp_f3438,type,
f3438: ( del * $i ) > $i ).
tff(lameq_f3438,axiom,
! [A_27a: del,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] : ( ap(f3438(A_27a,V1s),V2t) = ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__RESTRICTED,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),f3438(A_27a,V1s)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(c_2Emeasure_2Emeasure(A_27a),V0m))))) ) ) ) ).
tff(lamtp_f3439,type,
f3439: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3439,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2a: $i] : ( ap(f3439(A_27a,V0m,V1c),V2a) = ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2a)) ) ) ).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__CMUL,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1c))) )
=> p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),f3439(A_27a,V0m,V1c))))) ) ) ).
tff(conj_thm_2Emeasure_2EBIGUNION__IMAGE__Q,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Eextreal_2Eextreal,arr(A_27a,bool)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Eextreal_2Eextreal,arr(A_27a,bool)),c_2Eextreal_2EQ__set),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,arr(A_27a,bool)),V1f),c_2Eextreal_2EQ__set))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ).
tff(lamtp_f3440,type,
f3440: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3440,axiom,
! [V0a: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3440(V0a),inj__ty_2Eextreal_2Eextreal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V1x)),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V0a))) ) ).
tff(lamtp_f3441,type,
f3441: $i ).
tff(lameq_f3441,axiom,
! [V0a: tp__ty_2Eextreal_2Eextreal] : ( ap(f3441,inj__ty_2Eextreal_2Eextreal(V0a)) = ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3440(V0a)) ) ).
tff(ax_thm_2Emeasure_2EBorel__def,axiom,
c_2Emeasure_2EBorel = ap(ap(c_2Emeasure_2Esigma(ty_2Eextreal_2Eextreal),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)),f3441),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))) ).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__BOREL,axiom,
p(ap(c_2Emeasure_2Esigma__algebra(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)) ).
tff(lamtp_f3442,type,
f3442: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3442,axiom,
! [V2c: tp__ty_2Eextreal_2Eextreal,V3x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3442(V2c),inj__ty_2Eextreal_2Eextreal(V3x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V3x)),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V3x)),inj__ty_2Eextreal_2Eextreal(V2c))) ) ).
tff(conj_thm_2Emeasure_2EMEASURABLE__BOREL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,ty_2Eextreal_2Eextreal),V0f),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3442(V2c)))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3443,type,
f3443: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3443,axiom,
! [A_27a: del,V2c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3443(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V3x)),inj__ty_2Eextreal_2Eextreal(V2c))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3443(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3444,type,
f3444: ( del * $i ) > $i ).
tff(lameq_f3444,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x: $i] : ( ap(f3444(A_27a,V0f),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V2x)),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2ENegInf))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__NEGINF,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3444(A_27a,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ).
tff(lamtp_f3445,type,
f3445: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3445,axiom,
! [A_27a: del,V2c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3445(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2c)),ap(V0f,V3x))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT1,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3445(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3446,type,
f3446: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3446,axiom,
! [A_27a: del,V2c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3446(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V3x)),inj__ty_2Eextreal_2Eextreal(V2c))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT2,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3446(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3447,type,
f3447: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3447,axiom,
! [A_27a: del,V2c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3447(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V2c)),ap(V0f,V3x))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT3,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3447(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3448,type,
f3448: ( del * tp__ty_2Eextreal_2Eextreal * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3448,axiom,
! [A_27a: del,V3d: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c: tp__ty_2Eextreal_2Eextreal,V4x: $i] : ( ap(f3448(A_27a,V3d,V0f,V2c),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2c)),ap(V0f,V4x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V4x)),inj__ty_2Eextreal_2Eextreal(V3d)))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT4,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal,V3d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3448(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3449,type,
f3449: ( del * tp__ty_2Eextreal_2Eextreal * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3449,axiom,
! [A_27a: del,V3d: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c: tp__ty_2Eextreal_2Eextreal,V4x: $i] : ( ap(f3449(A_27a,V3d,V0f,V2c),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V2c)),ap(V0f,V4x))),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V4x)),inj__ty_2Eextreal_2Eextreal(V3d)))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT5,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal,V3d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3449(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3450,type,
f3450: ( del * tp__ty_2Eextreal_2Eextreal * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3450,axiom,
! [A_27a: del,V3d: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c: tp__ty_2Eextreal_2Eextreal,V4x: $i] : ( ap(f3450(A_27a,V3d,V0f,V2c),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2c)),ap(V0f,V4x))),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V4x)),inj__ty_2Eextreal_2Eextreal(V3d)))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT6,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
<=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal,V3d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3450(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3451,type,
f3451: ( del * tp__ty_2Eextreal_2Eextreal * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3451,axiom,
! [A_27a: del,V3d: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c: tp__ty_2Eextreal_2Eextreal,V4x: $i] : ( ap(f3451(A_27a,V3d,V0f,V2c),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V2c)),ap(V0f,V4x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V4x)),inj__ty_2Eextreal_2Eextreal(V3d)))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT7,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal,V3d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3451(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3452,type,
f3452: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3452,axiom,
! [A_27a: del,V2c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3452(A_27a,V2c,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V3x)),inj__ty_2Eextreal_2Eextreal(V2c))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT8,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3452(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3453,type,
f3453: ( del * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3453,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c: tp__ty_2Eextreal_2Eextreal,V3x: $i] : ( ap(f3453(A_27a,V0f,V2c),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V3x)),inj__ty_2Eextreal_2Eextreal(V2c)))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT9,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))
& ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3453(A_27a,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3454,type,
f3454: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3454,axiom,
! [A_27a: del,V4c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5x: $i] : ( ap(f3454(A_27a,V4c,V0f),V5x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V5x),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V4c)),ap(V0f,V5x))) ) ) ).
tff(lamtp_f3455,type,
f3455: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3455,axiom,
! [A_27a: del,V6c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V7x: $i] : ( ap(f3455(A_27a,V6c,V0f),V7x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V7x),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V7x)),inj__ty_2Eextreal_2Eextreal(V6c))) ) ) ).
tff(lamtp_f3456,type,
f3456: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3456,axiom,
! [A_27a: del,V8c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V9x: $i] : ( ap(f3456(A_27a,V8c,V0f),V9x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V9x),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V8c)),ap(V0f,V9x))) ) ) ).
tff(lamtp_f3457,type,
f3457: ( del * tp__ty_2Eextreal_2Eextreal * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3457,axiom,
! [A_27a: del,V11d: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V10c: tp__ty_2Eextreal_2Eextreal,V12x: $i] : ( ap(f3457(A_27a,V11d,V0f,V10c),V12x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V12x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V10c)),ap(V0f,V12x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V12x)),inj__ty_2Eextreal_2Eextreal(V11d)))) ) ) ).
tff(lamtp_f3458,type,
f3458: ( del * tp__ty_2Eextreal_2Eextreal * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3458,axiom,
! [A_27a: del,V14d: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V13c: tp__ty_2Eextreal_2Eextreal,V15x: $i] : ( ap(f3458(A_27a,V14d,V0f,V13c),V15x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V15x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V13c)),ap(V0f,V15x))),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V15x)),inj__ty_2Eextreal_2Eextreal(V14d)))) ) ) ).
tff(lamtp_f3459,type,
f3459: ( del * tp__ty_2Eextreal_2Eextreal * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3459,axiom,
! [A_27a: del,V17d: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V16c: tp__ty_2Eextreal_2Eextreal,V18x: $i] : ( ap(f3459(A_27a,V17d,V0f,V16c),V18x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V18x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V16c)),ap(V0f,V18x))),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V18x)),inj__ty_2Eextreal_2Eextreal(V17d)))) ) ) ).
tff(lamtp_f3460,type,
f3460: ( del * tp__ty_2Eextreal_2Eextreal * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3460,axiom,
! [A_27a: del,V20d: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V19c: tp__ty_2Eextreal_2Eextreal,V21x: $i] : ( ap(f3460(A_27a,V20d,V0f,V19c),V21x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V21x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V19c)),ap(V0f,V21x))),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V21x)),inj__ty_2Eextreal_2Eextreal(V20d)))) ) ) ).
tff(lamtp_f3461,type,
f3461: ( del * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3461,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V22c: tp__ty_2Eextreal_2Eextreal,V23x: $i] : ( ap(f3461(A_27a,V0f,V22c),V23x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V23x),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V23x)),inj__ty_2Eextreal_2Eextreal(V22c)))) ) ) ).
tff(lamtp_f3462,type,
f3462: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3462,axiom,
! [A_27a: del,V24c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V25x: $i] : ( ap(f3462(A_27a,V24c,V0f),V25x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V25x),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V25x)),inj__ty_2Eextreal_2Eextreal(V24c))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
=> ( ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3443(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V4c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3454(A_27a,V4c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V6c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3455(A_27a,V6c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V8c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3456(A_27a,V8c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V10c: tp__ty_2Eextreal_2Eextreal,V11d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3457(A_27a,V11d,V0f,V10c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V13c: tp__ty_2Eextreal_2Eextreal,V14d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3458(A_27a,V14d,V0f,V13c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V16c: tp__ty_2Eextreal_2Eextreal,V17d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3459(A_27a,V17d,V0f,V16c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V19c: tp__ty_2Eextreal_2Eextreal,V20d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3460(A_27a,V20d,V0f,V19c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V22c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3461(A_27a,V0f,V22c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))
& ! [V24c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3462(A_27a,V24c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ).
tff(lamtp_f3463,type,
f3463: ( del * tp__ty_2Eextreal_2Eextreal * $i ) > $i ).
tff(lameq_f3463,axiom,
! [A_27a: del,V22c: tp__ty_2Eextreal_2Eextreal,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V23x: $i] : ( ap(f3463(A_27a,V22c,V0f),V23x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V23x),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V23x)),inj__ty_2Eextreal_2Eextreal(V22c))) ) ) ).
tff(lamtp_f3464,type,
f3464: ( del * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3464,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V24c: tp__ty_2Eextreal_2Eextreal,V25x: $i] : ( ap(f3464(A_27a,V0f,V24c),V25x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V25x),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V0f,V25x)),inj__ty_2Eextreal_2Eextreal(V24c)))) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL__MEASURE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1m: $i] :
( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))),c_2Emeasure_2EBorel)))
=> ( ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3443(A_27a,V2c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V4c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3454(A_27a,V4c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V6c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3455(A_27a,V6c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V8c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3456(A_27a,V8c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V10c: tp__ty_2Eextreal_2Eextreal,V11d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3457(A_27a,V11d,V0f,V10c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V13c: tp__ty_2Eextreal_2Eextreal,V14d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3458(A_27a,V14d,V0f,V13c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V16c: tp__ty_2Eextreal_2Eextreal,V17d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3459(A_27a,V17d,V0f,V16c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V19c: tp__ty_2Eextreal_2Eextreal,V20d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3460(A_27a,V20d,V0f,V19c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V22c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3463(A_27a,V22c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))
& ! [V24c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3464(A_27a,V0f,V24c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) ) ).
tff(lamtp_f3465,type,
f3465: ( del * $i * $i ) > $i ).
tff(lameq_f3465,axiom,
! [A_27a: del,V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3465(A_27a,V1g,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V3x)),ap(V1g,V3x))) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a: $i] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3465(A_27a,V1g,V0f))),ap(c_2Emeasure_2Espace(A_27a),V2a))),ap(c_2Emeasure_2Esubsets(A_27a),V2a))) ) ) ) ) ).
tff(lamtp_f3466,type,
f3466: ( del * $i * $i ) > $i ).
tff(lameq_f3466,axiom,
! [A_27a: del,V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3466(A_27a,V1g,V0f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V3x)),ap(V1g,V3x))) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a: $i] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3466(A_27a,V1g,V0f))),ap(c_2Emeasure_2Espace(A_27a),V2a))),ap(c_2Emeasure_2Esubsets(A_27a),V2a))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2ESPACE__BOREL,axiom,
ap(c_2Emeasure_2Espace(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel) = c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal) ).
tff(lamtp_f3467,type,
f3467: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3467,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3467(V0c),inj__ty_2Eextreal_2Eextreal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V1x)),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V0c))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS1,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3467(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3468,type,
f3468: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3468,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3468(V0c),inj__ty_2Eextreal_2Eextreal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V1x)),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0c)),inj__ty_2Eextreal_2Eextreal(V1x))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS2,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3468(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3469,type,
f3469: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3469,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3469(V0c),inj__ty_2Eextreal_2Eextreal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V1x)),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V0c))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS3,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3469(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3470,type,
f3470: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3470,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3470(V0c),inj__ty_2Eextreal_2Eextreal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V1x)),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V0c)),inj__ty_2Eextreal_2Eextreal(V1x))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS4,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3470(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3471,type,
f3471: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3471,axiom,
! [V1d: tp__ty_2Eextreal_2Eextreal,V0c: tp__ty_2Eextreal_2Eextreal,V2x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3471(V1d,V0c),inj__ty_2Eextreal_2Eextreal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V2x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0c)),inj__ty_2Eextreal_2Eextreal(V2x))),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V2x)),inj__ty_2Eextreal_2Eextreal(V1d)))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS5,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3471(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3472,type,
f3472: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3472,axiom,
! [V1d: tp__ty_2Eextreal_2Eextreal,V0c: tp__ty_2Eextreal_2Eextreal,V2x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3472(V1d,V0c),inj__ty_2Eextreal_2Eextreal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V2x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V0c)),inj__ty_2Eextreal_2Eextreal(V2x))),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2x)),inj__ty_2Eextreal_2Eextreal(V1d)))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS6,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3472(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3473,type,
f3473: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3473,axiom,
! [V1d: tp__ty_2Eextreal_2Eextreal,V0c: tp__ty_2Eextreal_2Eextreal,V2x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3473(V1d,V0c),inj__ty_2Eextreal_2Eextreal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V2x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0c)),inj__ty_2Eextreal_2Eextreal(V2x))),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2x)),inj__ty_2Eextreal_2Eextreal(V1d)))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS7,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3473(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3474,type,
f3474: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3474,axiom,
! [V1d: tp__ty_2Eextreal_2Eextreal,V0c: tp__ty_2Eextreal_2Eextreal,V2x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3474(V1d,V0c),inj__ty_2Eextreal_2Eextreal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V2x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V0c)),inj__ty_2Eextreal_2Eextreal(V2x))),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V2x)),inj__ty_2Eextreal_2Eextreal(V1d)))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS8,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3474(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS9,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V0c)),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3475,type,
f3475: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3475,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3475(V0c),inj__ty_2Eextreal_2Eextreal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V1x)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V0c)))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS10,axiom,
! [V0c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3475(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ).
tff(lamtp_f3476,type,
f3476: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3476,axiom,
! [V2c: tp__ty_2Eextreal_2Eextreal,V3x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3476(V2c),inj__ty_2Eextreal_2Eextreal(V3x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V3x)),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2c)),inj__ty_2Eextreal_2Eextreal(V3x))) ) ).
tff(lamtp_f3477,type,
f3477: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3477,axiom,
! [V4c: tp__ty_2Eextreal_2Eextreal,V5x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3477(V4c),inj__ty_2Eextreal_2Eextreal(V5x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V5x)),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V4c)),inj__ty_2Eextreal_2Eextreal(V5x))) ) ).
tff(lamtp_f3478,type,
f3478: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3478,axiom,
! [V6c: tp__ty_2Eextreal_2Eextreal,V7x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3478(V6c),inj__ty_2Eextreal_2Eextreal(V7x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V7x)),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V7x)),inj__ty_2Eextreal_2Eextreal(V6c))) ) ).
tff(lamtp_f3479,type,
f3479: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3479,axiom,
! [V9d: tp__ty_2Eextreal_2Eextreal,V8c: tp__ty_2Eextreal_2Eextreal,V10x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3479(V9d,V8c),inj__ty_2Eextreal_2Eextreal(V10x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V10x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V8c)),inj__ty_2Eextreal_2Eextreal(V10x))),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V10x)),inj__ty_2Eextreal_2Eextreal(V9d)))) ) ).
tff(lamtp_f3480,type,
f3480: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3480,axiom,
! [V12d: tp__ty_2Eextreal_2Eextreal,V11c: tp__ty_2Eextreal_2Eextreal,V13x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3480(V12d,V11c),inj__ty_2Eextreal_2Eextreal(V13x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V13x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V11c)),inj__ty_2Eextreal_2Eextreal(V13x))),ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V13x)),inj__ty_2Eextreal_2Eextreal(V12d)))) ) ).
tff(lamtp_f3481,type,
f3481: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3481,axiom,
! [V15d: tp__ty_2Eextreal_2Eextreal,V14c: tp__ty_2Eextreal_2Eextreal,V16x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3481(V15d,V14c),inj__ty_2Eextreal_2Eextreal(V16x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V16x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V14c)),inj__ty_2Eextreal_2Eextreal(V16x))),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V16x)),inj__ty_2Eextreal_2Eextreal(V15d)))) ) ).
tff(lamtp_f3482,type,
f3482: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3482,axiom,
! [V18d: tp__ty_2Eextreal_2Eextreal,V17c: tp__ty_2Eextreal_2Eextreal,V19x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3482(V18d,V17c),inj__ty_2Eextreal_2Eextreal(V19x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V19x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V17c)),inj__ty_2Eextreal_2Eextreal(V19x))),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V19x)),inj__ty_2Eextreal_2Eextreal(V18d)))) ) ).
tff(lamtp_f3483,type,
f3483: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3483,axiom,
! [V21c: tp__ty_2Eextreal_2Eextreal,V22x: tp__ty_2Eextreal_2Eextreal] : ( ap(f3483(V21c),inj__ty_2Eextreal_2Eextreal(V22x)) = ap(ap(c_2Epair_2E_2C(ty_2Eextreal_2Eextreal,bool),inj__ty_2Eextreal_2Eextreal(V22x)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V22x)),inj__ty_2Eextreal_2Eextreal(V21c)))) ) ).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS,axiom,
( ! [V0c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3467(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V2c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3476(V2c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V4c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3477(V4c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V6c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3478(V6c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V8c: tp__ty_2Eextreal_2Eextreal,V9d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3479(V9d,V8c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V11c: tp__ty_2Eextreal_2Eextreal,V12d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3480(V12d,V11c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V14c: tp__ty_2Eextreal_2Eextreal,V15d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3481(V15d,V14c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V17c: tp__ty_2Eextreal_2Eextreal,V18d: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3482(V18d,V17c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V20c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V20c)),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))
& ! [V21c: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3483(V21c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CONST,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1k: tp__ty_2Eextreal_2Eextreal,V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V2f,V3x)) = V1k ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__INDICATOR,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1A: $i] :
( mem(V1A,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1A),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V2f,V3x)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V1A),V3x)) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CMUL,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3z: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V2g,V4x)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V3z))),ap(V1f,V4x))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ABS,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V2g,V3x)) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__abs,ap(V1f,V3x))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SQR,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V2g,V3x)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__pow,ap(V1f,V3x)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ADD,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3h: $i] :
( mem(V3h,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V3h,V4x)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V4x)),ap(V2g,V4x))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V3h),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SUB,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3h: $i] :
( mem(V3h,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V3h,V4x)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__sub,ap(V1f,V4x)),ap(V2g,V4x))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V3h),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3h: $i] :
( mem(V3h,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( ( surj__ty_2Eextreal_2Eextreal(ap(V1f,V4x)) != fo__c_2Eextreal_2ENegInf )
& ( surj__ty_2Eextreal_2Eextreal(ap(V1f,V4x)) != fo__c_2Eextreal_2EPosInf )
& ( surj__ty_2Eextreal_2Eextreal(ap(V2g,V4x)) != fo__c_2Eextreal_2ENegInf )
& ( surj__ty_2Eextreal_2Eextreal(ap(V2g,V4x)) != fo__c_2Eextreal_2EPosInf ) ) ) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V3h,V5x)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V5x)),ap(V2g,V5x))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V3h),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
tff(lamtp_f3484,type,
f3484: ( del * del * $i * $i ) > $i ).
tff(lameq_f3484,axiom,
! [A_27a: del,A_27b: del,V1f: $i] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ! [V6i: $i] : ( ap(f3484(A_27a,A_27b,V1f,V5x),V6i) = ap(ap(V1f,V6i),V5x) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__SUM,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s))
& p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& ! [V4i: $i] :
( mem(V4i,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4i),V3s))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V1f,V4i)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) )
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Espace(A_27a),V0a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V2g,V5x)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),f3484(A_27a,A_27b,V1f,V5x)),V3s)) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ) ).
tff(lamtp_f3485,type,
f3485: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3485,axiom,
! [A_27a: del,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1z: tp__ty_2Erealax_2Ereal,V3x: $i] : ( ap(f3485(A_27a,V2s,V1z),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1z))),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V2s),V3x)) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__CMUL__INDICATOR,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1z: tp__ty_2Erealax_2Ereal,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3485(A_27a,V2s,V1z)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ).
tff(lamtp_f3486,type,
f3486: ( del * $i * $i ) > $i ).
tff(lameq_f3486,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3x: $i] : ( ap(f3486(A_27a,V1f,V2s),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V3x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V2s),V3x)) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL__INDICATOR,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3486(A_27a,V1f,V2s)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(lamtp_f3487,type,
f3487: ( del * $i * $i ) > $i ).
tff(lameq_f3487,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2x: $i] : ( ap(f3487(A_27a,V1f,V0a),V2x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(V1f,V2x)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V2x)) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MUL__INDICATOR__EQ,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
<=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3487(A_27a,V1f,V0a)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__POS__SIMPLE__FN,axiom,
! [A_27a: del,V0m: $i] :
( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ? [V2s: $i] :
( mem(V2s,arr(ty_2Enum_2Enum,bool))
& ? [V3a: $i] :
( mem(V3a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
& ? [V4x: $i] :
( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x)) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),c_2Emeasure_2EBorel))) ) ) ) ).
tff(lamtp_f3488,type,
f3488: ( del * $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3488,axiom,
! [A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0n: tp__ty_2Enum_2Enum,V4x: $i] : ( ap(f3488(A_27a,V2f,V0n),V4x) = ap(ap(c_2Eextreal_2Eextreal__pow,ap(V2f,V4x)),inj__ty_2Enum_2Enum(V0n)) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__POW,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1a: $i] :
( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel)))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(c_2Emeasure_2Espace(A_27a),V1a)))
=> ( ( surj__ty_2Eextreal_2Eextreal(ap(V2f,V3x)) != fo__c_2Eextreal_2ENegInf )
& ( surj__ty_2Eextreal_2Eextreal(ap(V2f,V3x)) != fo__c_2Eextreal_2EPosInf ) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3488(A_27a,V2f,V0n)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) ) ) ) ).
tff(lamtp_f3489,type,
f3489: ( del * $i * $i ) > $i ).
tff(lameq_f3489,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3489(A_27a,V1f,V2g),V3x) = ap(ap(c_2Eextreal_2Eextreal__max,ap(V1f,V3x)),ap(V2g,V3x)) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MAX,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V2g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),f3489(A_27a,V1f,V2g)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(lamtp_f3490,type,
f3490: ( del * $i * $i ) > $i ).
tff(lameq_f3490,axiom,
! [A_27a: del,V0fn: $i] :
( mem(V0fn,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7n: tp__ty_2Enum_2Enum] : ( ap(f3490(A_27a,V0fn,V6x),inj__ty_2Enum_2Enum(V7n)) = ap(ap(V0fn,inj__ty_2Enum_2Enum(V7n)),V6x) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__MONO__SUP,axiom,
! [A_27a: del,V0fn: $i] :
( mem(V0fn,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Eextreal_2Eextreal)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a: $i] :
( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V2a))
& ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(V0fn,inj__ty_2Enum_2Enum(V3n))),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel)))
& ! [V4n: tp__ty_2Enum_2Enum,V5x: $i] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),ap(c_2Emeasure_2Espace(A_27a),V2a)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(V0fn,inj__ty_2Enum_2Enum(V4n)),V5x)),ap(ap(V0fn,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V4n))),V5x))) ) )
& ! [V6x: $i] :
( mem(V6x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),ap(c_2Emeasure_2Espace(A_27a),V2a)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V1f,V6x)) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3490(A_27a,V0fn,V6x)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ) ) )
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(lamtp_f3491,type,
f3491: ( del * $i ) > $i ).
tff(lameq_f3491,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x: $i] : ( ap(f3491(A_27a,V0f),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,V1x))),ap(V0f,V1x)),ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ).
tff(ax_thm_2Emeasure_2Efn__plus__def,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ap(c_2Emeasure_2Efn__plus(A_27a),V0f) = f3491(A_27a,V0f) ) ) ).
tff(lamtp_f3492,type,
f3492: ( del * $i ) > $i ).
tff(lameq_f3492,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x: $i] : ( ap(f3492(A_27a,V0f),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,V1x)),ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))),ap(c_2Eextreal_2Eextreal__ainv,ap(V0f,V1x))),ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ).
tff(ax_thm_2Emeasure_2Efn__minus__def,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ap(c_2Emeasure_2Efn__minus(A_27a),V0f) = f3492(A_27a,V0f) ) ) ).
tff(lamtp_f3493,type,
f3493: ( del * $i ) > $i ).
tff(lameq_f3493,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x: $i] : ( ap(f3493(A_27a,V0f),V1x) = ap(c_2Eextreal_2Eextreal__abs,ap(V0f,V1x)) ) ) ).
tff(ax_thm_2Emeasure_2Efn__abs__def,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ap(c_2Emeasure_2Efn__abs(A_27a),V0f) = f3493(A_27a,V0f) ) ) ).
tff(conj_thm_2Emeasure_2EFN__PLUS__POS,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V0g),V1x))) ) ) ).
tff(conj_thm_2Emeasure_2EFN__MINUS__POS,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V0g),V1x))) ) ) ).
tff(conj_thm_2Emeasure_2EFN__PLUS__POS__ID,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0g,V1x))) )
=> ( ap(c_2Emeasure_2Efn__plus(A_27a),V0g) = V0g ) ) ) ).
tff(conj_thm_2Emeasure_2EFN__MINUS__POS__ZERO,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0g,V1x))) )
=> ( ap(c_2Emeasure_2Efn__minus(A_27a),V0g) = k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(lamtp_f3494,type,
f3494: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3494,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2x: $i] : ( ap(f3494(A_27a,V0f,V1c),V2x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(V0f,V2x)) ) ) ).
tff(lamtp_f3495,type,
f3495: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3495,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V3x: $i] : ( ap(f3495(A_27a,V0f,V1c),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V0f),V3x)) ) ) ).
tff(lamtp_f3496,type,
f3496: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3496,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V4x: $i] : ( ap(f3496(A_27a,V0f,V1c),V4x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(V0f,V4x)) ) ) ).
tff(lamtp_f3497,type,
f3497: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3497,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V5x: $i] : ( ap(f3497(A_27a,V0f,V1c),V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c)))),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V0f),V5x)) ) ) ).
tff(conj_thm_2Emeasure_2EFN__PLUS__CMUL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1c)))
=> ( ap(c_2Emeasure_2Efn__plus(A_27a),f3494(A_27a,V0f,V1c)) = f3495(A_27a,V0f,V1c) ) )
& ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1c)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
=> ( ap(c_2Emeasure_2Efn__plus(A_27a),f3496(A_27a,V0f,V1c)) = f3497(A_27a,V0f,V1c) ) ) ) ) ).
tff(lamtp_f3498,type,
f3498: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3498,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V3x: $i] : ( ap(f3498(A_27a,V0f,V1c),V3x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V0f),V3x)) ) ) ).
tff(lamtp_f3499,type,
f3499: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3499,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V5x: $i] : ( ap(f3499(A_27a,V0f,V1c),V5x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c)))),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V0f),V5x)) ) ) ).
tff(conj_thm_2Emeasure_2EFN__MINUS__CMUL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1c)))
=> ( ap(c_2Emeasure_2Efn__minus(A_27a),f3494(A_27a,V0f,V1c)) = f3498(A_27a,V0f,V1c) ) )
& ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1c)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
=> ( ap(c_2Emeasure_2Efn__minus(A_27a),f3496(A_27a,V0f,V1c)) = f3499(A_27a,V0f,V1c) ) ) ) ) ).
tff(lamtp_f3500,type,
f3500: ( del * $i * $i ) > $i ).
tff(lameq_f3500,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ( ap(f3500(A_27a,V0f,V1g),V3x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Emeasure_2EFN__PLUS__ADD__LE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Emeasure_2Efn__plus(A_27a),f3500(A_27a,V0f,V1g)),V2x)),ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Emeasure_2Efn__plus(A_27a),V0f),V2x)),ap(ap(c_2Emeasure_2Efn__plus(A_27a),V1g),V2x)))) ) ) ) ).
tff(conj_thm_2Emeasure_2EFN__MINUS__ADD__LE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Emeasure_2Efn__minus(A_27a),f3500(A_27a,V0f,V1g)),V2x)),ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Emeasure_2Efn__minus(A_27a),V0f),V2x)),ap(ap(c_2Emeasure_2Efn__minus(A_27a),V1g),V2x)))) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__FN__PLUS,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(c_2Emeasure_2Efn__plus(A_27a),V1f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__FN__MINUS,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(c_2Emeasure_2Efn__minus(A_27a),V1f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__PLUS__MINUS,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
<=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(c_2Emeasure_2Efn__plus(A_27a),V1f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel)))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),ap(c_2Emeasure_2Efn__minus(A_27a),V1f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V0a),c_2Emeasure_2EBorel))) ) ) ) ) ).
tff(lamtp_f3501,type,
f3501: ( del * $i * $i ) > $i ).
tff(lameq_f3501,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B: $i] :
( mem(V1B,arr(A_27a,bool))
=> ! [V2t: $i] : ( ap(f3501(A_27a,V0A,V1B),V2t) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V0A),V2t)),ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),V1B),V2t)) ) ) ) ).
tff(lamtp_f3502,type,
f3502: ( del * $i * $i ) > $i ).
tff(lameq_f3502,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B: $i] :
( mem(V1B,arr(A_27a,bool))
=> ! [V3t: $i] : ( ap(f3502(A_27a,V0A,V1B),V3t) = ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0A),V1B)),V3t) ) ) ) ).
tff(conj_thm_2Emeasure_2EINDICATOR__FN__MUL__INTER,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B: $i] :
( mem(V1B,arr(A_27a,bool))
=> ( f3501(A_27a,V0A,V1B) = f3502(A_27a,V0A,V1B) ) ) ) ).
tff(lamtp_f3503,type,
f3503: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3503,axiom,
! [A_27a: del,V2b: $i] :
( mem(V2b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5a: $i] :
( mem(V5a,arr(A_27a,bool))
=> ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7i: tp__ty_2Enum_2Enum] : ( ap(f3503(A_27a,V2b,V5a,V6x),inj__ty_2Enum_2Enum(V7i)) = ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V5a),ap(V2b,inj__ty_2Enum_2Enum(V7i)))),V6x) ) ) ) ) ).
tff(lamtp_f3504,type,
f3504: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3504,axiom,
! [A_27a: del,V2b: $i] :
( mem(V2b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V5a: $i] :
( mem(V5a,arr(A_27a,bool))
=> ! [V0r: $i] :
( mem(V0r,arr(ty_2Enum_2Enum,bool))
=> ! [V6x: $i] : ( ap(f3504(A_27a,V2b,V5a,V0r),V6x) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3503(A_27a,V2b,V5a,V6x)),V0r) ) ) ) ) ).
tff(conj_thm_2Emeasure_2Eindicator__fn__split,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,arr(ty_2Enum_2Enum,bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2b: $i] :
( mem(V2b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0r))
& ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2b),V0r)) = V1s )
& ! [V3i: tp__ty_2Enum_2Enum,V4j: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3i)),V0r))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4j)),V0r))
& ( V3i != V4j ) )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2b,inj__ty_2Enum_2Enum(V3i))),ap(V2b,inj__ty_2Enum_2Enum(V4j)))) ) )
=> ! [V5a: $i] :
( mem(V5a,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V5a),V1s))
=> ( ap(c_2Emeasure_2Eindicator__fn(A_27a),V5a) = f3504(A_27a,V2b,V5a,V0r) ) ) ) ) ) ) ) ).
tff(lamtp_f3505,type,
f3505: ( del * $i * $i ) > $i ).
tff(lameq_f3505,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V4i: tp__ty_2Enum_2Enum] : ( ap(f3505(A_27a,V0a,V1x),inj__ty_2Enum_2Enum(V4i)) = ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(V0a,inj__ty_2Enum_2Enum(V4i))),V1x) ) ) ) ).
tff(conj_thm_2Emeasure_2Eindicator__fn__suminf,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V2m != V3n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0a,inj__ty_2Enum_2Enum(V2m))),ap(V0a,inj__ty_2Enum_2Enum(V3n)))) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,f3505(A_27a,V0a,V1x))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Emeasure_2Eindicator__fn(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V0a),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1x)) ) ) ) ) ).
tff(lamtp_f3506,type,
f3506: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3506,axiom,
! [A_27a: del,V6a: $i] :
( mem(V6a,arr(A_27a,bool))
=> ! [V1b: $i] :
( mem(V1b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2m: $i] :
( mem(V2m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ! [V7i: tp__ty_2Enum_2Enum] : ( ap(f3506(A_27a,V6a,V1b,V2m),inj__ty_2Enum_2Enum(V7i)) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V2m),ap(ap(c_2Epred__set_2EINTER(A_27a),V6a),ap(V1b,inj__ty_2Enum_2Enum(V7i)))) ) ) ) ) ).
tff(conj_thm_2Emeasure_2Emeasure__split,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,arr(ty_2Enum_2Enum,bool))
=> ! [V1b: $i] :
( mem(V1b,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2m: $i] :
( mem(V2m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V2m))
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0r))
& ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1b),V0r)) = ap(c_2Emeasure_2Em__space(A_27a),V2m) )
& ! [V3i: tp__ty_2Enum_2Enum,V4j: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3i)),V0r))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4j)),V0r))
& ( V3i != V4j ) )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1b,inj__ty_2Enum_2Enum(V3i))),ap(V1b,inj__ty_2Enum_2Enum(V4j)))) )
& ! [V5i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V5i)),V0r))
=> p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1b,inj__ty_2Enum_2Enum(V5i))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V2m))) ) )
=> ! [V6a: $i] :
( mem(V6a,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V6a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V2m)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V2m),V6a)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Enum_2Enum),f3506(A_27a,V6a,V1b,V2m)),V0r)) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------