include('Axioms/product+2.ax').
fof(mem_c_2Emeasure_2EBorel,axiom,( mem(c_2Emeasure_2EBorel,ty_2Epair_2Eprod(arr(ty_2Eextreal_2Eextreal,bool),arr(arr(ty_2Eextreal_2Eextreal,bool),bool))) )).
fof(mem_c_2Emeasure_2Eadditive,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Eadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) )).
fof(mem_c_2Emeasure_2Ealgebra,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Ealgebra(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ) )).
fof(mem_c_2Emeasure_2Eclosed__cdi,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Eclosed__cdi(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ) )).
fof(mem_c_2Emeasure_2Ecountably__additive,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Ecountably__additive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) )).
fof(mem_c_2Emeasure_2Ecountably__subadditive,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Ecountably__subadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) )).
fof(mem_c_2Emeasure_2Efn__abs,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Efn__abs(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ) )).
fof(mem_c_2Emeasure_2Efn__minus,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Efn__minus(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ) )).
fof(mem_c_2Emeasure_2Efn__plus,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Efn__plus(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))) ) )).
fof(mem_c_2Emeasure_2Eincreasing,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Eincreasing(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) )).
fof(mem_c_2Emeasure_2Eindicator__fn,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Eindicator__fn(A_27a),arr(arr(A_27a,bool),arr(A_27a,ty_2Eextreal_2Eextreal))) ) )).
fof(mem_c_2Emeasure_2Einf__measure,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Einf__measure(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))) ) )).
fof(mem_c_2Emeasure_2Elambda__system,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Elambda__system(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(arr(arr(A_27a,bool),ty_2Erealax_2Ereal),arr(arr(A_27a,bool),bool)))) ) )).
fof(mem_c_2Emeasure_2Em__space,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Em__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(A_27a,bool))) ) )).
fof(mem_c_2Emeasure_2Emeasurable,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Emeasure_2Emeasurable(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),arr(arr(A_27a,A_27b),bool)))) ) ) )).
fof(mem_c_2Emeasure_2Emeasurable__sets,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Emeasurable__sets(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Emeasure_2Emeasure,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Emeasure(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))) ) )).
fof(mem_c_2Emeasure_2Emeasure__preserving,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,A_27b),bool)))) ) ) )).
fof(mem_c_2Emeasure_2Emeasure__space,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Emeasure__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) )).
fof(mem_c_2Emeasure_2Enull__set,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Enull__set(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Emeasure_2Eouter__measure__space,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Eouter__measure__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) )).
fof(mem_c_2Emeasure_2Epos__simple__fn,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Epos__simple__fn(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(arr(ty_2Enum_2Enum,bool),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)))))) ) )).
fof(mem_c_2Emeasure_2Epositive,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Epositive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) )).
fof(mem_c_2Emeasure_2Esigma,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Esigma(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))))) ) )).
fof(mem_c_2Emeasure_2Esigma__algebra,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Esigma__algebra(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)) ) )).
fof(mem_c_2Emeasure_2Esmallest__closed__cdi,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Esmallest__closed__cdi(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))) ) )).
fof(mem_c_2Emeasure_2Espace,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Espace(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(A_27a,bool))) ) )).
fof(mem_c_2Emeasure_2Esubadditive,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Esubadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)) ) )).
fof(mem_c_2Emeasure_2Esubset__class,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Esubset__class(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),bool))) ) )).
fof(mem_c_2Emeasure_2Esubsets,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Emeasure_2Esubsets(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(arr(A_27a,bool),bool))) ) )).
fof(ax_thm_2Emeasure_2Espace__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1y] : ( mem(V1y,arr(arr(A_27a,bool),bool)) => ( ap(c_2Emeasure_2Espace(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0x),V1y)) = V0x ) ) ) ) )).
fof(ax_thm_2Emeasure_2Esubsets__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(A_27a,bool)) => ! [V1y] : ( mem(V1y,arr(arr(A_27a,bool),bool)) => ( ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0x),V1y)) = V1y ) ) ) ) )).
fof(ax_thm_2Emeasure_2Esubset__class__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1sts] : ( mem(V1sts,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1sts)) <=> ! [V2x] : ( mem(V2x,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1sts)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2x),V0sp)) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Ealgebra__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) <=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( ( ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) & ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Esigma__algebra__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ! [V1c] : ( mem(V1c,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Esigma__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1st] : ( mem(V1st,arr(arr(A_27a,bool),bool)) => ( ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1st) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EBIGINTER(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f3504(A_27a,V0sp,V1st)))) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Em__space__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1sts] : ( mem(V1sts,arr(arr(A_27a,bool),bool)) => ! [V2mu] : ( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ap(c_2Emeasure_2Em__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V0sp ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Emeasurable__sets__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1sts] : ( mem(V1sts,arr(arr(A_27a,bool),bool)) => ! [V2mu] : ( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ap(c_2Emeasure_2Emeasurable__sets(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V1sts ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Emeasure__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1sts] : ( mem(V1sts,arr(arr(A_27a,bool),bool)) => ! [V2mu] : ( mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ap(c_2Emeasure_2Emeasure(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V2mu ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Epositive__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) <=> ( ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) & ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Eadditive__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) <=> ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) ) ) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)) ) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Ecountably__additive__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) <=> ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ( ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ~~ ( V2m = V3n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,V3n))) ) ) ) ) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) => p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Esubadditive__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m)) <=> ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Ecountably__subadditive__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) <=> ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Eincreasing__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) <=> ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Emeasure__space__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Elambda__system__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0gen] : ( mem(V0gen,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gen),V1lam) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3506(A_27a,V0gen,V1lam)) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Eouter__measure__space__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),V0m)) <=> ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Einf__measure__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s) = ap(c_2Ereal_2Einf,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3510(A_27a,V1s,V0m))) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Eclosed__cdi__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) <=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & ( ( ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) & ( ( ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) & ( ( ap(V2f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a) ) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,V3n)),ap(V2f,ap(c_2Enum_2ESUC,V3n)))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) & ! [V4f] : ( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) & ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( ( ~~ ( V5m = V6n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,V5m)),ap(V4f,V6n))) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Esmallest__closed__cdi__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Epred__set_2EBIGINTER(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f3511(A_27a,V0a)))) ) ) ) )).
fof(ax_thm_2Emeasure_2Emeasurable__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,A_27b),arr(A_27a,A_27b)),f3513(A_27b,A_27a,V0a,V1b)) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Emeasure__preserving__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0m1] : ( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1m2] : ( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ( ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,A_27b),arr(A_27a,A_27b)),f3515(A_27b,A_27a,V0m1,V1m2)) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Eindicator__fn__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(c_2Emeasure_2Eindicator__fn(A_27a),V0s) = f3442(A_27a,V0s) ) ) ) )).
fof(ax_thm_2Emeasure_2Epos__simple__fn__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ! [V3a] : ( mem(V3a,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V4x] : ( mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x)) <=> ( ( ! [V5t] : ( mem(V5t,A_27a) => p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),ap(V1f,V5t))) ) ) & ( ( ! [V6t] : ( mem(V6t,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6t),ap(c_2Emeasure_2Em__space(A_27a),V0m))) => ( ap(V1f,V6t) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3516(A_27a,V6t,V3a,V4x)),V2s) ) ) ) ) & ( ( ! [V8i] : ( mem(V8i,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V8i),V2s)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V3a,V8i)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) & ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V2s)) & ( ( ! [V9i] : ( mem(V9i,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V9i),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V4x,V9i))) ) ) ) & ( ( ! [V10i] : ( mem(V10i,ty_2Enum_2Enum) => ! [V11j] : ( mem(V11j,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V10i),V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V11j),V2s)) & ( ~~ ( V10i = V11j ) ) ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3a,V10i)),ap(V3a,V11j))) ) ) ) ) & ( ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3a),V2s)) = ap(c_2Emeasure_2Em__space(A_27a),V0m) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Emeasure_2Enull__set__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Emeasure_2Enull__set(A_27a),V0m),V1s)) <=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) )).
fof(conj_thm_2Emeasure_2ESPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)) = V0a ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__ALT__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) <=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( ( ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) & ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__SPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__FINITE__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1c] : ( mem(V1c,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__INTER__SPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) => ( ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s) = V1s ) & ( ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),ap(c_2Emeasure_2Espace(A_27a),V0a)) = V1s ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & ( p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0g0)),V2l)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ! [V2l1] : ( mem(V2l1,arr(A_27a,bool)) => ! [V3l2] : ( mem(V3l2,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & ( p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l1),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3l2),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2l1),V3l2)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ALGEBRA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) ) => p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,bool)) => ! [V3l1] : ( mem(V3l1,arr(A_27a,bool)) => ! [V4l2] : ( mem(V4l2,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & ( p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2g),ap(c_2Emeasure_2Esubsets(A_27a),V0g0))) & ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V3l1),V4l2)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3l1),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4l2),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) ) ) ) ) ) => ( ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V3l1),V4l2)),V2g)) = ap(ap(c_2Erealax_2Ereal__add,ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V3l1),V2g))),ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V4l2),V2g))) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ! [V2l1] : ( mem(V2l1,A_27b) => ! [V3l2] : ( mem(V3l2,A_27c) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) ) => p(ap(c_2Emeasure_2Eadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) ) ) )))))).
fof(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE__SUBADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) ) ) => p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m)) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__MONO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) & ( ( ap(V1f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a) ) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,V2n)),ap(V1f,ap(c_2Enum_2ESUC,V2n)))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__DISJOINT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) & ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ~~ ( V2m = V3n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,V3n))) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESUBADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( mem(V3u,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( mem(V3u,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) & ( V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) ) ) ) ) ) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) & ( ( V2s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EADDITIVE__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( ~~ ( V3m = V4n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V3m)),ap(V1f,V4n))) ) ) ) ) ) ) ) => ( ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V2n)),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),ap(c_2Epred__set_2Ecount,V2n)))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EINCREASING__ADDITIVE__SUMMABLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & ( p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ~~ ( V2m = V3n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,V3n))) ) ) ) ) ) ) ) ) => p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__POSITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) => p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) => p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))) ) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0g0] : ( mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,bool)) => ! [V3f] : ( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & ( p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2g),ap(c_2Emeasure_2Esubsets(A_27a),V0g0))) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))) & ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( ( ~~ ( V5m = V6n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,V5m)),ap(V3f,V6n))) ) ) ) ) ) ) ) => ( ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V4n)),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),V1lam),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,arr(A_27a,bool),arr(A_27a,bool)),f3517(A_27a,V2g)),V3f))) = ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),ap(c_2Epred__set_2Ecount,V4n)))),V2g)) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALGEBRA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) => p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) ) )))).
fof(conj_thm_2Emeasure_2EOUTER__MEASURE__SPACE__POSITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),V0m)) => p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) )))).
fof(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__CARATHEODORY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0gsig] : ( mem(V0gsig,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0gsig)) & p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0gsig)),V1lam)))) ) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam)))) & ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( ~~ ( V3m = V4n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,V3m)),ap(V2f,V4n))) ) ) ) ) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam))) & p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),V1lam),V2f)),ap(V1lam,ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ECARATHEODORY__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0gsig] : ( mem(V0gsig,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1lam] : ( mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0gsig)) & p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0gsig)),V1lam)))) ) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam)),V1lam)))) ) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__NONEMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1g] : ( mem(V1g,arr(A_27a,bool)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),V2s)) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3521(A_27a,V1g,V0m)))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__POS0,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1g] : ( mem(V1g,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3521(A_27a,V1g,V0m)))) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2x)) ) ) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__POS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1g] : ( mem(V1g,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1g))) ) ) )))).
fof(conj_thm_2Emeasure_2EINCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EADDITIVE__INCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) ) => p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) ) )))).
fof(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) ) ) => p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) )))).
fof(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( ~~ ( V3m = V4n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,V3m)),ap(V2f,V4n))) ) ) ) ) & ( ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) ) => p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__AGREES,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & ( p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) => ( ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__DOWN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m0] : ( mem(V0m0,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1m1] : ( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))) & ( ( ap(c_2Emeasure_2Emeasure(A_27a),V0m0) = ap(c_2Emeasure_2Emeasure(A_27a),V1m1) ) & p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1)) ) ) ) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m0)) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__SIGMA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1sts] : ( mem(V1sts,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1sts)) => p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1sts))) ) ) )))).
fof(conj_thm_2Emeasure_2EPOW__ALGEBRA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EPOW(A_27a),V0sp)))) )))).
fof(conj_thm_2Emeasure_2EPOW__SIGMA__ALGEBRA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EPOW(A_27a),V0sp)))) )))).
fof(conj_thm_2Emeasure_2EUNIV__SIGMA__ALGEBRA,lemma,( ! [A_27a]: ( ne(A_27a) => p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),c_2Epred__set_2EUNIV(A_27a)),c_2Epred__set_2EUNIV(arr(A_27a,bool)))))))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) => ( ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__POSITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) => p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__INCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) => p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3523(A_27a,V1s,V0m)))) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)),V2x)) ) ) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__CLOSE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) ) => ? [V3f] : ( mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) & ? [V4l] : ( mem(V4l,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) & ( ( ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( ( ~~ ( V5m = V6n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,V5m)),ap(V3f,V6n))) ) ) ) ) & ( p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V3f)),V4l)) & p(ap(ap(c_2Ereal_2Ereal__lte,V4l),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)),V2e))) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__COUNTABLY__SUBADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) ) ) => p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) )))).
fof(conj_thm_2Emeasure_2EINF__MEASURE__OUTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) ) ) => p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,arr(arr(A_27a,bool),bool)) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1b)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Esubsets(A_27a),V1b))) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Espace(A_27a),V1b)),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V1b))) ) ) )))).
fof(conj_thm_2Emeasure_2EALGEBRA__SUBSET__LAMBDA__SYSTEM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & ( p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m)))) ) )))).
fof(conj_thm_2Emeasure_2ECARATHEODORY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m0] : ( mem(V0m0,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m0)) & p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m0)) ) ) => ? [V1m] : ( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) & ( ( ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0))) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V2s) = ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m0),V2s) ) ) ) ) & ( ( ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)) = ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)) ) & p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__SUBSET__SUBSETS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,arr(arr(A_27a,bool),bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)))) ) )))).
fof(conj_thm_2Emeasure_2EIN__SIGMA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,arr(arr(A_27a,bool),bool)) => ! [V2x] : ( mem(V2x,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1a)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)))) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0p)) <=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & ( ( ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) & ! [V2c] : ( mem(V2c,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V2c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2c),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V2c)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__COUNTABLE__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1c] : ( mem(V1c,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ENUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)),V1s)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1p] : ( mem(V1p,arr(arr(A_27a,bool),bool)) => ! [V2a] : ( mem(V2a,arr(arr(A_27a,bool),bool)) => ( ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V1p)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p)) & ( ( ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) ) ) & ! [V4c] : ( mem(V4c,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V4c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V4c),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V4c)),V1p)) ) ) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__SUBSET__MEASURABLE__SETS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,arr(arr(A_27a,bool),bool)) => ! [V1m] : ( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V1m)),V0a))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( ( ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) & ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__DISJOINT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( ( ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) & ( ( ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ! [V3t] : ( mem(V3t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) & ! [V4f] : ( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) & ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( ( ~~ ( V5m = V6n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,V5m)),ap(V4f,V6n))) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__ALT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1p] : ( mem(V1p,arr(arr(A_27a,bool),bool)) => ! [V2a] : ( mem(V2a,arr(arr(A_27a,bool),bool)) => ( ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V1p)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p)) & ( ( ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) ) ) & ! [V4f] : ( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESPACE__SMALLEST__CLOSED__CDI,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ap(c_2Emeasure_2Espace(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)) = ap(c_2Emeasure_2Espace(A_27a),V0a) ) )))).
fof(conj_thm_2Emeasure_2ESMALLEST__CLOSED__CDI,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) & ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))) & p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) )))).
fof(conj_thm_2Emeasure_2ECLOSED__CDI__DUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) ) )))).
fof(conj_thm_2Emeasure_2ECLOSED__CDI__COMPL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) )))).
fof(conj_thm_2Emeasure_2ECLOSED__CDI__DISJOINT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) & ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ~~ ( V2m = V3n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,V2m)),ap(V1f,V3n))) ) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) )))).
fof(conj_thm_2Emeasure_2ECLOSED__CDI__INCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) & ( ( ap(V1f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a) ) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,V2n)),ap(V1f,ap(c_2Enum_2ESUC,V2n)))) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA1,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,arr(arr(A_27a,bool),bool)) => ! [V2p] : ( mem(V2p,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2p)) & p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2p))) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))),V2p)) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__WEAK,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1p] : ( mem(V1p,arr(arr(A_27a,bool),bool)) => ! [V2a] : ( mem(V2a,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2a))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p)) & ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p)) & ( ( ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V1p)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) ) ) & ( ( ! [V4f] : ( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1p))) & ( ( ap(V4f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a) ) & ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V4f,V5n)),ap(V4f,ap(c_2Enum_2ESUC,V5n)))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) ) & ! [V6f] : ( mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V6f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1p))) & ! [V7m] : ( mem(V7m,ty_2Enum_2Enum) => ! [V8n] : ( mem(V8n,ty_2Enum_2Enum) => ( ( ~~ ( V7m = V8n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V6f,V7m)),ap(V6f,V8n))) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V6f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) ) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1p] : ( mem(V1p,arr(arr(A_27a,bool),bool)) => ! [V2a] : ( mem(V2a,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2a))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p)) & ( ( ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p)) ) ) ) & ( ( ! [V4f] : ( mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))) & ( ( ap(V4f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a) ) & ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V4f,V5n)),ap(V4f,ap(c_2Enum_2ESUC,V5n)))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) ) & ! [V6f] : ( mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V6f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))) & ! [V7m] : ( mem(V7m,ty_2Enum_2Enum) => ! [V8n] : ( mem(V8n,ty_2Enum_2Enum) => ( ( ~~ ( V7m = V8n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V6f,V7m)),ap(V6f,V8n))) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V6f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)) ) ) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__COUNTABLY__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( ~~ ( V3m = V4n ) ) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,V3m)),ap(V2f,V4n))) ) ) ) ) & ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) => p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__ADDITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( mem(V3u,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) & ( V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) ) ) ) ) ) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__COUNTABLE__INCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ( ap(V2f,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(A_27a) ) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,V3n)),ap(V2f,ap(c_2Enum_2ESUC,V3n)))) ) ) & ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) ) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__REDUCE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),ap(c_2Emeasure_2Emeasure(A_27a),V0m))) = V0m ) )))).
fof(conj_thm_2Emeasure_2ESPACE__SIGMA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,arr(arr(A_27a,bool),bool)) => ( ap(c_2Emeasure_2Espace(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)) = V0sp ) ) )))).
fof(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,V3n)),ap(V2f,ap(c_2Enum_2ESUC,V3n)))) ) ) & ( V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__REDUCE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,arr(arr(A_27a,bool),bool)) => ( ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))) = ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURABLE__SETS__SUBSET__SPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURABLE__DIFF__PROPERTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) => ! [V4s] : ( mem(V4s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),ap(ap(c_2Epred__set_2EDIFF(A_27b),ap(c_2Emeasure_2Espace(A_27b),V1b)),V4s)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V4s)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__BIGUNION__PROPERTY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) => ! [V4c] : ( mem(V4c,arr(arr(A_27b,bool),bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27b,bool)),V4c),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => ( ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),ap(c_2Epred__set_2EBIGUNION(A_27b),V4c)) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),V4c)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASUBABLE__BIGUNION__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) => ! [V4c] : ( mem(V4c,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V4c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V4c),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V4c)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))) ) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__SIGMA__PREIMAGES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) => p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b))))) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Espace(A_27a),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__SIGMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V2b] : ( mem(V2b,arr(arr(A_27b,bool),bool)) => ! [V3sp] : ( mem(V3sp,arr(A_27b,bool)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27b),V3sp),V2b)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V1a)),V3sp))) & ! [V4s] : ( mem(V4s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4s),V2b)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V4s)),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),ap(ap(c_2Emeasure_2Esigma(A_27b),V3sp),V2b)))) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Espace(A_27b),V1b)),ap(c_2Emeasure_2Esubsets(A_27b),V1b))))) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__LIFT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V2b] : ( mem(V2b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),V2b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Espace(A_27b),V2b)),ap(c_2Emeasure_2Esubsets(A_27b),V2b))))) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EIN__MEASURE__PRESERVING,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0m1] : ( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1m2] : ( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) <=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2))))) & ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)) & ( p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2)) & ! [V3s] : ( mem(V3s,arr(A_27b,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2))) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m1),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Em__space(A_27a),V0m1))) = ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m2),V3s) ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__LIFT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0m1] : ( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1m2] : ( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ! [V2a] : ( mem(V2a,arr(arr(A_27b,bool),bool)) => ! [V3f] : ( mem(V3f,arr(A_27a,A_27b)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)) & ( p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2)) & ( ( ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2) = ap(c_2Emeasure_2Esubsets(A_27b),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),V2a)) ) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27b),V1m2)))))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0m1] : ( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1m2] : ( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ! [V2a] : ( mem(V2a,arr(arr(A_27b,bool),bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)) & ( p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2)) & ( ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2) = ap(c_2Emeasure_2Esubsets(A_27b),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),V2a)) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27b),V1m2))))),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__I,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27a)),c_2Ecombin_2EI(A_27a)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27a),V0a),V0a))) ) )))).
fof(conj_thm_2Emeasure_2EMEASURABLE__COMP,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27c)) => ! [V2a] : ( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V3b] : ( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V4c] : ( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27b,A_27c),V3b),V4c))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) )))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONG,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27c)) => ! [V2a] : ( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V3b] : ( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V4c] : ( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b))) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27c),V4c)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Epred__set_2EFUNSET(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V3b)),ap(c_2Emeasure_2Espace(A_27c),V4c)))) & ! [V5x] : ( mem(V5x,arr(A_27c,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V5x),ap(c_2Emeasure_2Esubsets(A_27c),V4c))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V5x)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Emeasure_2Espace(A_27a),V2a)))),ap(c_2Emeasure_2Esubsets(A_27b),V3b))) ) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) )))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONGER,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1g] : ( mem(V1g,arr(A_27b,A_27c)) => ! [V2a] : ( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V3b] : ( mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V4c] : ( mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool))) => ! [V5t] : ( mem(V5t,arr(A_27b,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b))) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27c),V4c)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Epred__set_2EFUNSET(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V3b)),ap(c_2Emeasure_2Espace(A_27c),V4c)))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Emeasure_2Espace(A_27a),V2a))),V5t)) & ! [V6s] : ( mem(V6s,arr(A_27c,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V6s),ap(c_2Emeasure_2Esubsets(A_27c),V4c))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V6s)),V5t)),ap(c_2Emeasure_2Esubsets(A_27b),V3b))) ) ) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))) ) ) ) ) ) ) )))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__UP__LIFT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,arr(arr(A_27a,bool),bool)) => ! [V2b] : ( mem(V2b,arr(arr(A_27a,bool),bool)) => ! [V3c] : ( mem(V3c,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V4f] : ( mem(V4f,arr(A_27a,A_27b)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V4f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)),V3c))) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b))) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2b)) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V4f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)),V3c))) ) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__UP__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0sp] : ( mem(V0sp,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,arr(arr(A_27a,bool),bool)) => ! [V2b] : ( mem(V2b,arr(arr(A_27a,bool),bool)) => ! [V3c] : ( mem(V3c,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2b)) & p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b))) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)),V3c)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)),V3c))) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__UP__SIGMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1b] : ( mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))),V1b))) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__LIFT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,arr(arr(A_27a,bool),bool)) => ! [V1m1] : ( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V2m2] : ( mem(V2m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ! [V3f] : ( mem(V3f,arr(A_27a,A_27b)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V0a),ap(c_2Emeasure_2Emeasure(A_27a),V1m1)))),V2m2))) & ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V1m1),V2m2))) ) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0a] : ( mem(V0a,arr(arr(A_27a,bool),bool)) => ! [V1m1] : ( mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V2m2] : ( mem(V2m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))) & p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V0a),ap(c_2Emeasure_2Emeasure(A_27a),V1m1)))),V2m2)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V1m1),V2m2))) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SIGMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0m1] : ( mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1m2] : ( mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ! [V2a] : ( mem(V2a,arr(arr(A_27a,bool),bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)) & ( ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1) = ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),V2a)) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27a),V0m1)))),V1m2)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__PROD__SIGMA,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1a1] : ( mem(V1a1,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ! [V2a2] : ( mem(V2a2,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool))) => ! [V3f] : ( mem(V3f,arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,ty_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2EFST(A_27b,A_27c)),V3f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1a1))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,ty_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2ESND(A_27b,A_27c)),V3f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V0a),V2a2))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),V3f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V0a),ap(ap(c_2Emeasure_2Esigma(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V1a1)),ap(c_2Emeasure_2Espace(A_27c),V2a2))),ap(ap(c_2Eutil__prob_2Eprod__sets(A_27b,A_27c),ap(c_2Emeasure_2Esubsets(A_27b),V1a1)),ap(c_2Emeasure_2Esubsets(A_27c),V2a2)))))) ) ) ) ) )))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__RANGE__REDUCE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),V2s),ap(c_2Epred__set_2EPOW(A_27b),V2s))))) & ( ~~ ( V2s = c_2Epred__set_2EEMPTY(A_27b) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))))))) ) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),c_2Epred__set_2EUNIV(A_27b)),ap(c_2Epred__set_2EPOW(A_27b),c_2Epred__set_2EUNIV(A_27b)))))) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))))))) ) ) ))))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1s_27] : ( mem(V1s_27,arr(A_27a,bool)) => ! [V2m] : ( mem(V2m,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s_27),V0s)) & p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0s),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V0s)),V2m)))) ) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s_27),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),V2m)))) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESTRONG__MEASURE__SPACE__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s_27] : ( mem(V1s_27,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s_27),ap(c_2Emeasure_2Em__space(A_27a),V0s))) & ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0s))) ) ) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s_27),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),ap(c_2Emeasure_2Emeasure(A_27a),V0s))))) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__REAL__SUM__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & ( ( ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) ) ) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3524(A_27a,V0m)),V1s) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__POW,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ( ap(ap(c_2Emeasure_2Esigma(A_27a),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s)) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s)) ) )))).
fof(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1m] : ( mem(V1m,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0s)) & ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0s))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))) & p(ap(c_2Emeasure_2Eadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))) ) ) ) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))) ) ) )))).
fof(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))) & ( p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) ) ) ) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) ) )))).
fof(conj_thm_2Emeasure_2EIMAGE__SING,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1x] : ( mem(V1x,A_27a) => ( ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))) = ap(ap(c_2Epred__set_2EINSERT(A_27b),ap(V0f,V1x)),c_2Epred__set_2EEMPTY(A_27b)) ) ) ))))).
fof(conj_thm_2Emeasure_2ESUBSET__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0X] : ( mem(V0X,arr(A_27a,bool)) => ! [V1P] : ( mem(V1P,arr(arr(A_27a,bool),bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),ap(c_2Epred__set_2EBIGINTER(A_27a),V1P))) <=> ! [V2Y] : ( mem(V2Y,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2Y),V1P)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),V2Y)) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__INCREASING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__POSITIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__INTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__COMPL__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0t] : ( mem(V0t,arr(A_27a,bool)) => ! [V1m] : ( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0t),V2s)) ) ) ) => ( ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),ap(ap(c_2Epred__set_2EDIFF(A_27a),V2s),V0t)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V2s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V0t)) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1s,V2n)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__IN__MSPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1A] : ( mem(V1A,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) => ! [V2x] : ( mem(V2x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1A)) => p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m))) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET__MSPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0A] : ( mem(V0A,arr(A_27a,bool)) => ! [V1m] : ( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0A),ap(c_2Emeasure_2Em__space(A_27a),V1m))) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__EMPTY__MEASURABLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__MSPACE__MEASURABLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) )))).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) => ( p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & ( ( ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) & ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1s,V2n)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) ) )))).
fof(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,V2n)),ap(V1f,ap(c_2Enum_2ESUC,V2n)))) ) ) ) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) )))).
fof(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,ap(c_2Enum_2ESUC,V3n))),ap(V2f,V3n))) ) ) & ( V1s = ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,ap(c_2Enum_2ESUC,V2n))),ap(V1f,V2n))) ) ) ) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__RESTRICTED,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) ) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),f3525(A_27a,V1s)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(c_2Emeasure_2Emeasure(A_27a),V0m))))) ) ) )))).
fof(conj_thm_2Emeasure_2EMEASURE__SPACE__CMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0m] : ( mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1c)) ) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),f3526(A_27a,V0m,V1c))))) ) ) )))).
fof(conj_thm_2Emeasure_2EBIGUNION__IMAGE__Q,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ! [V1f] : ( mem(V1f,arr(ty_2Eextreal_2Eextreal,arr(A_27a,bool))) => ( ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Eextreal_2Eextreal,arr(A_27a,bool)),c_2Eextreal_2EQ__set),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,arr(A_27a,bool)),V1f),c_2Eextreal_2EQ__set))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) ) ) )))).
fof(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)),f3528),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))) )).
fof(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__BOREL,lemma,(p(ap(c_2Emeasure_2Esigma__algebra(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))).
fof(conj_thm_2Emeasure_2EMEASURABLE__BOREL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,ty_2Eextreal_2Eextreal),V0f),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3529(V2c)))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3530(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__NEGINF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3531(A_27a,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT1,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3532(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3533(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT3,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3534(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT4,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => ! [V3d] : ( mem(V3d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3535(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT5,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => ! [V3d] : ( mem(V3d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3536(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT6,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => ! [V3d] : ( mem(V3d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3537(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT7,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => ! [V3d] : ( mem(V3d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3538(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT8,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3539(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT9,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => ( p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3540(A_27a,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1a] : ( mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => ( ( ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3530(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) & ( ( ! [V4c] : ( mem(V4c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3541(A_27a,V4c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) & ( ( ! [V6c] : ( mem(V6c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3542(A_27a,V6c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) & ( ( ! [V8c] : ( mem(V8c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3543(A_27a,V8c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) & ( ( ! [V10c] : ( mem(V10c,ty_2Eextreal_2Eextreal) => ! [V11d] : ( mem(V11d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3544(A_27a,V11d,V0f,V10c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) & ( ( ! [V13c] : ( mem(V13c,ty_2Eextreal_2Eextreal) => ! [V14d] : ( mem(V14d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3545(A_27a,V14d,V0f,V13c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) & ( ( ! [V16c] : ( mem(V16c,ty_2Eextreal_2Eextreal) => ! [V17d] : ( mem(V17d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3546(A_27a,V17d,V0f,V16c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) & ( ( ! [V19c] : ( mem(V19c,ty_2Eextreal_2Eextreal) => ! [V20d] : ( mem(V20d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3547(A_27a,V20d,V0f,V19c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) & ( ( ! [V22c] : ( mem(V22c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3548(A_27a,V0f,V22c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) & ! [V24c] : ( mem(V24c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3549(A_27a,V24c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))) ) ) ) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL__MEASURE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1m] : ( mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))),c_2Emeasure_2EBorel))) => ( ( ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3530(A_27a,V2c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) & ( ( ! [V4c] : ( mem(V4c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3541(A_27a,V4c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) & ( ( ! [V6c] : ( mem(V6c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3542(A_27a,V6c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) & ( ( ! [V8c] : ( mem(V8c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3543(A_27a,V8c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) & ( ( ! [V10c] : ( mem(V10c,ty_2Eextreal_2Eextreal) => ! [V11d] : ( mem(V11d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3544(A_27a,V11d,V0f,V10c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) & ( ( ! [V13c] : ( mem(V13c,ty_2Eextreal_2Eextreal) => ! [V14d] : ( mem(V14d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3545(A_27a,V14d,V0f,V13c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) & ( ( ! [V16c] : ( mem(V16c,ty_2Eextreal_2Eextreal) => ! [V17d] : ( mem(V17d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3546(A_27a,V17d,V0f,V16c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) & ( ( ! [V19c] : ( mem(V19c,ty_2Eextreal_2Eextreal) => ! [V20d] : ( mem(V20d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3547(A_27a,V20d,V0f,V19c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) & ( ( ! [V22c] : ( mem(V22c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3550(A_27a,V22c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) & ! [V24c] : ( mem(V24c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3551(A_27a,V0f,V24c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) ) ) ) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V2a] : ( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3552(A_27a,V1g,V0f))),ap(c_2Emeasure_2Espace(A_27a),V2a))),ap(c_2Emeasure_2Esubsets(A_27a),V2a))) ) ) ) )))).
fof(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal)) => ! [V2a] : ( mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) ) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3553(A_27a,V1g,V0f))),ap(c_2Emeasure_2Espace(A_27a),V2a))),ap(c_2Emeasure_2Esubsets(A_27a),V2a))) ) ) ) )))).
fof(conj_thm_2Emeasure_2ESPACE__BOREL,lemma,(( ap(c_2Emeasure_2Espace(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel) = c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS1,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3554(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS2,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3555(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS3,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3556(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS4,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3557(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS5,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => ! [V1d] : ( mem(V1d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3558(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS6,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => ! [V1d] : ( mem(V1d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3559(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS7,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => ! [V1d] : ( mem(V1d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3560(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS8,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => ! [V1d] : ( mem(V1d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3561(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS9,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),V0c),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS10,lemma,(! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3562(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ))).
fof(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS,conjecture,(( ( ! [V0c] : ( mem(V0c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3554(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) & ( ( ! [V2c] : ( mem(V2c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3563(V2c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) & ( ( ! [V4c] : ( mem(V4c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3564(V4c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) & ( ( ! [V6c] : ( mem(V6c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3565(V6c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) & ( ( ! [V8c] : ( mem(V8c,ty_2Eextreal_2Eextreal) => ! [V9d] : ( mem(V9d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3566(V9d,V8c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ) & ( ( ! [V11c] : ( mem(V11c,ty_2Eextreal_2Eextreal) => ! [V12d] : ( mem(V12d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3567(V12d,V11c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ) & ( ( ! [V14c] : ( mem(V14c,ty_2Eextreal_2Eextreal) => ! [V15d] : ( mem(V15d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3568(V15d,V14c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ) & ( ( ! [V17c] : ( mem(V17c,ty_2Eextreal_2Eextreal) => ! [V18d] : ( mem(V18d,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3569(V18d,V17c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ) & ( ( ! [V20c] : ( mem(V20c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Eextreal_2Eextreal),V20c),c_2Epred__set_2EEMPTY(ty_2Eextreal_2Eextreal))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) & ! [V21c] : ( mem(V21c,ty_2Eextreal_2Eextreal) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3570(V21c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))) ) ) ) ) ) ) ) ) ) ))).
