include('Axioms/product_2.ax').
tff(tp_c_2Emeasure_2EBorel,type,(c_2Emeasure_2EBorel:$i)).
tff(mem_c_2Emeasure_2EBorel,axiom,mem(c_2Emeasure_2EBorel,ty_2Epair_2Eprod(arr(ty_2Eextreal_2Eextreal,bool),arr(arr(ty_2Eextreal_2Eextreal,bool),bool)))).
tff(tp_c_2Emeasure_2Eadditive,type,(c_2Emeasure_2Eadditive:del > $i)).
tff(mem_c_2Emeasure_2Eadditive,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Eadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)))).
tff(tp_c_2Emeasure_2Ealgebra,type,(c_2Emeasure_2Ealgebra:del > $i)).
tff(mem_c_2Emeasure_2Ealgebra,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Ealgebra(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)))).
tff(tp_c_2Emeasure_2Eclosed__cdi,type,(c_2Emeasure_2Eclosed__cdi:del > $i)).
tff(mem_c_2Emeasure_2Eclosed__cdi,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Eclosed__cdi(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)))).
tff(tp_c_2Emeasure_2Ecountably__additive,type,(c_2Emeasure_2Ecountably__additive:del > $i)).
tff(mem_c_2Emeasure_2Ecountably__additive,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Ecountably__additive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)))).
tff(tp_c_2Emeasure_2Ecountably__subadditive,type,(c_2Emeasure_2Ecountably__subadditive:del > $i)).
tff(mem_c_2Emeasure_2Ecountably__subadditive,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Ecountably__subadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)))).
tff(tp_c_2Emeasure_2Efn__abs,type,(c_2Emeasure_2Efn__abs:del > $i)).
tff(mem_c_2Emeasure_2Efn__abs,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Efn__abs(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))))).
tff(tp_c_2Emeasure_2Efn__minus,type,(c_2Emeasure_2Efn__minus:del > $i)).
tff(mem_c_2Emeasure_2Efn__minus,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Efn__minus(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))))).
tff(tp_c_2Emeasure_2Efn__plus,type,(c_2Emeasure_2Efn__plus:del > $i)).
tff(mem_c_2Emeasure_2Efn__plus,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Efn__plus(A_27a),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(A_27a,ty_2Eextreal_2Eextreal))))).
tff(tp_c_2Emeasure_2Eincreasing,type,(c_2Emeasure_2Eincreasing:del > $i)).
tff(mem_c_2Emeasure_2Eincreasing,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Eincreasing(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)))).
tff(tp_c_2Emeasure_2Eindicator__fn,type,(c_2Emeasure_2Eindicator__fn:del > $i)).
tff(mem_c_2Emeasure_2Eindicator__fn,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Eindicator__fn(A_27a),arr(arr(A_27a,bool),arr(A_27a,ty_2Eextreal_2Eextreal))))).
tff(tp_c_2Emeasure_2Einf__measure,type,(c_2Emeasure_2Einf__measure:del > $i)).
tff(mem_c_2Emeasure_2Einf__measure,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Einf__measure(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))).
tff(tp_c_2Emeasure_2Elambda__system,type,(c_2Emeasure_2Elambda__system:del > $i)).
tff(mem_c_2Emeasure_2Elambda__system,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Elambda__system(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(arr(arr(A_27a,bool),ty_2Erealax_2Ereal),arr(arr(A_27a,bool),bool)))))).
tff(tp_c_2Emeasure_2Em__space,type,(c_2Emeasure_2Em__space:del > $i)).
tff(mem_c_2Emeasure_2Em__space,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Em__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(A_27a,bool))))).
tff(tp_c_2Emeasure_2Emeasurable,type,(c_2Emeasure_2Emeasurable:(del * del) > $i)).
tff(mem_c_2Emeasure_2Emeasurable,axiom,( ! [A_27a:del] : ( ! [A_27b:del] : mem(c_2Emeasure_2Emeasurable(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),arr(arr(A_27a,A_27b),bool))))))).
tff(tp_c_2Emeasure_2Emeasurable__sets,type,(c_2Emeasure_2Emeasurable__sets:del > $i)).
tff(mem_c_2Emeasure_2Emeasurable__sets,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Emeasurable__sets(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))))).
tff(tp_c_2Emeasure_2Emeasure,type,(c_2Emeasure_2Emeasure:del > $i)).
tff(mem_c_2Emeasure_2Emeasure,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Emeasure(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))))).
tff(tp_c_2Emeasure_2Emeasure__preserving,type,(c_2Emeasure_2Emeasure__preserving:(del * del) > $i)).
tff(mem_c_2Emeasure_2Emeasure__preserving,axiom,( ! [A_27a:del] : ( ! [A_27b:del] : mem(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,A_27b),bool))))))).
tff(tp_c_2Emeasure_2Emeasure__space,type,(c_2Emeasure_2Emeasure__space:del > $i)).
tff(mem_c_2Emeasure_2Emeasure__space,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Emeasure__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)))).
tff(tp_c_2Emeasure_2Enull__set,type,(c_2Emeasure_2Enull__set:del > $i)).
tff(mem_c_2Emeasure_2Enull__set,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Enull__set(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,bool),bool))))).
tff(tp_c_2Emeasure_2Eouter__measure__space,type,(c_2Emeasure_2Eouter__measure__space:del > $i)).
tff(mem_c_2Emeasure_2Eouter__measure__space,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Eouter__measure__space(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)))).
tff(tp_c_2Emeasure_2Epos__simple__fn,type,(c_2Emeasure_2Epos__simple__fn:del > $i)).
tff(mem_c_2Emeasure_2Epos__simple__fn,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Epos__simple__fn(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),arr(arr(A_27a,ty_2Eextreal_2Eextreal),arr(arr(ty_2Enum_2Enum,bool),arr(arr(ty_2Enum_2Enum,arr(A_27a,bool)),arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)))))))).
tff(tp_c_2Emeasure_2Epositive,type,(c_2Emeasure_2Epositive:del > $i)).
tff(mem_c_2Emeasure_2Epositive,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Epositive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)))).
tff(tp_c_2Emeasure_2Esigma,type,(c_2Emeasure_2Esigma:del > $i)).
tff(mem_c_2Emeasure_2Esigma,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Esigma(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))))))).
tff(tp_c_2Emeasure_2Esigma__algebra,type,(c_2Emeasure_2Esigma__algebra:del > $i)).
tff(mem_c_2Emeasure_2Esigma__algebra,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Esigma__algebra(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),bool)))).
tff(tp_c_2Emeasure_2Esmallest__closed__cdi,type,(c_2Emeasure_2Esmallest__closed__cdi:del > $i)).
tff(mem_c_2Emeasure_2Esmallest__closed__cdi,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Esmallest__closed__cdi(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)))))).
tff(tp_c_2Emeasure_2Espace,type,(c_2Emeasure_2Espace:del > $i)).
tff(mem_c_2Emeasure_2Espace,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Espace(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(A_27a,bool))))).
tff(tp_c_2Emeasure_2Esubadditive,type,(c_2Emeasure_2Esubadditive:del > $i)).
tff(mem_c_2Emeasure_2Esubadditive,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Esubadditive(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),bool)))).
tff(tp_c_2Emeasure_2Esubset__class,type,(c_2Emeasure_2Esubset__class:del > $i)).
tff(mem_c_2Emeasure_2Esubset__class,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Esubset__class(A_27a),arr(arr(A_27a,bool),arr(arr(arr(A_27a,bool),bool),bool))))).
tff(tp_c_2Emeasure_2Esubsets,type,(c_2Emeasure_2Esubsets:del > $i)).
tff(mem_c_2Emeasure_2Esubsets,axiom,( ! [A_27a:del] : mem(c_2Emeasure_2Esubsets(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),arr(arr(A_27a,bool),bool))))).
tff(ax_thm_2Emeasure_2Espace__def,axiom,( ! [A_27a:del]: ( ! [V0x:$i] : (mem(V0x,arr(A_27a,bool)) => ( ! [V1y:$i] : (mem(V1y,arr(arr(A_27a,bool),bool)) => (ap(c_2Emeasure_2Espace(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0x),V1y)) = V0x))))))).
tff(ax_thm_2Emeasure_2Esubsets__def,axiom,( ! [A_27a:del]: ( ! [V0x:$i] : (mem(V0x,arr(A_27a,bool)) => ( ! [V1y:$i] : (mem(V1y,arr(arr(A_27a,bool),bool)) => (ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0x),V1y)) = V1y))))))).
tff(ax_thm_2Emeasure_2Esubset__class__def,axiom,( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1sts:$i] : (mem(V1sts,arr(arr(A_27a,bool),bool)) => (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1sts)) <=> ( ! [V2x:$i] : (mem(V2x,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1sts)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2x),V0sp)))))))))))).
tff(ax_thm_2Emeasure_2Ealgebra__def,axiom,( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) <=> (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))) & ( ! [V2s:$i] : (mem(V2s,arr(A_27a,bool)) => ( ! [V3t:$i] : (mem(V3t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))))))))))))).
tff(ax_thm_2Emeasure_2Esigma__algebra__def,axiom,( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ( ! [V1c:$i] : (mem(V1c,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))))))))).
tff(ax_thm_2Emeasure_2Esigma__def,axiom,( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1st:$i] : (mem(V1st,arr(arr(A_27a,bool),bool)) => (ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1st) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EBIGINTER(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f3504(A_27a,V0sp,V1st))))))))))).
tff(ax_thm_2Emeasure_2Em__space__def,axiom,( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1sts:$i] : (mem(V1sts,arr(arr(A_27a,bool),bool)) => ( ! [V2mu:$i] : (mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => (ap(c_2Emeasure_2Em__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V0sp))))))))).
tff(ax_thm_2Emeasure_2Emeasurable__sets__def,axiom,( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1sts:$i] : (mem(V1sts,arr(arr(A_27a,bool),bool)) => ( ! [V2mu:$i] : (mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => (ap(c_2Emeasure_2Emeasurable__sets(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V1sts))))))))).
tff(ax_thm_2Emeasure_2Emeasure__def,axiom,( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1sts:$i] : (mem(V1sts,arr(arr(A_27a,bool),bool)) => ( ! [V2mu:$i] : (mem(V2mu,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => (ap(c_2Emeasure_2Emeasure(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0sp),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V1sts),V2mu))) = V2mu))))))))).
tff(ax_thm_2Emeasure_2Epositive__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) <=> ((surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) & ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)))))))))))).
tff(ax_thm_2Emeasure_2Eadditive__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) <=> ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))))))))))))).
tff(ax_thm_2Emeasure_2Ecountably__additive__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) <=> ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((~~ (V2m = V3n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n))))))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))) => p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))))))))))).
tff(ax_thm_2Emeasure_2Esubadditive__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m)) <=> ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))))))))))))).
tff(ax_thm_2Emeasure_2Ecountably__subadditive__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) <=> ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))))))))))).
tff(ax_thm_2Emeasure_2Eincreasing__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) <=> ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))))))))))))).
tff(ax_thm_2Emeasure_2Emeasure__space__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m))))))))).
tff(ax_thm_2Emeasure_2Elambda__system__def,axiom,( ! [A_27a:del]: ( ! [V0gen:$i] : (mem(V0gen,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => (ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gen),V1lam) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3506(A_27a,V0gen,V1lam))))))))).
tff(ax_thm_2Emeasure_2Eouter__measure__space__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),V0m)) <=> (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & (p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m))))))))).
tff(ax_thm_2Emeasure_2Einf__measure__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Einf,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3510(A_27a,V1s,V0m))))))))))).
tff(ax_thm_2Emeasure_2Eclosed__cdi__def,axiom,( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) <=> (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & (( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))))) & (( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) & ((ap(V2f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a)) & ( ! [V3n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3n))),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))))) & ( ! [V4f:$i] : (mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) & ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((~~ (V5m = V6n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5m))),ap(V4f,inj__ty_2Enum_2Enum(V6n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))))))))))))).
tff(ax_thm_2Emeasure_2Esmallest__closed__cdi__def,axiom,( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Epred__set_2EBIGINTER(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),bool)),f3511(A_27a,V0a))))))))).
tff(ax_thm_2Emeasure_2Emeasurable__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => (ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,A_27b),arr(A_27a,A_27b)),f3513(A_27b,A_27a,V0a,V1b)))))))))).
tff(ax_thm_2Emeasure_2Emeasure__preserving__def,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0m1:$i] : (mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1m2:$i] : (mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => (ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2) = ap(c_2Epred__set_2EGSPEC(arr(A_27a,A_27b),arr(A_27a,A_27b)),f3515(A_27b,A_27a,V0m1,V1m2)))))))))).
tff(ax_thm_2Emeasure_2Eindicator__fn__def,axiom,( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,arr(A_27a,bool)) => (ap(c_2Emeasure_2Eindicator__fn(A_27a),V0s) = f3442(A_27a,V0s)))))).
tff(ax_thm_2Emeasure_2Epos__simple__fn__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V2s:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ! [V3a:$i] : (mem(V3a,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ! [V4x:$i] : (mem(V4x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => (p(ap(ap(ap(ap(ap(c_2Emeasure_2Epos__simple__fn(A_27a),V0m),V1f),V2s),V3a),V4x)) <=> (( ! [V5t:$i] : (mem(V5t,A_27a) => p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V1f,V5t))))) & (( ! [V6t:$i] : (mem(V6t,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V6t),ap(c_2Emeasure_2Em__space(A_27a),V0m))) => (surj__ty_2Eextreal_2Eextreal(ap(V1f,V6t)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3516(A_27a,V6t,V3a,V4x)),V2s)))))) & (( ! [V8i:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V8i)),V2s)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V3a,inj__ty_2Enum_2Enum(V8i))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))) & (p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V2s)) & (( ! [V9i:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V9i)),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V4x,inj__ty_2Enum_2Enum(V9i)))))) & (( ! [V10i:tp__ty_2Enum_2Enum] : ( ! [V11j:tp__ty_2Enum_2Enum] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V10i)),V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V11j)),V2s)) & (~~ (V10i = V11j)))) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3a,inj__ty_2Enum_2Enum(V10i))),ap(V3a,inj__ty_2Enum_2Enum(V11j))))))) & (ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3a),V2s)) = ap(c_2Emeasure_2Em__space(A_27a),V0m))))))))))))))))))))).
tff(ax_thm_2Emeasure_2Enull__set__def,axiom,( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Emeasure_2Enull__set(A_27a),V0m),V1s)) <=> (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))))).
tff(conj_thm_2Emeasure_2ESPACE,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)) = V0a)))))).
tff(conj_thm_2Emeasure_2EALGEBRA__ALT__INTER,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) <=> (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))) & ( ! [V2s:$i] : (mem(V2s,arr(A_27a,bool)) => ( ! [V3t:$i] : (mem(V3t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__EMPTY,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__SPACE,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__COMPL,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__UNION,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__INTER,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__DIFF,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__FINITE__UNION,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1c:$i] : (mem(V1c,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & (p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__INTER__SPACE,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => ((ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s) = V1s) & (ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),ap(c_2Emeasure_2Espace(A_27a),V0a)) = V1s)))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__EMPTY,lemma,(( ! [A_27a:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__COMPL,lemma,(( ! [A_27a:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ! [V2l:$i] : (mem(V2l,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & (p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0g0)),V2l)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))))))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INTER,lemma,(( ! [A_27a:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ! [V2l1:$i] : (mem(V2l1,arr(A_27a,bool)) => ( ! [V3l2:$i] : (mem(V3l2,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & (p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2l1),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3l2),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V2l1),V3l2)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))))))))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ALGEBRA,lemma,(( ! [A_27a:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))) => p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : (mem(V2g,arr(A_27a,bool)) => ( ! [V3l1:$i] : (mem(V3l1,arr(A_27a,bool)) => ( ! [V4l2:$i] : (mem(V4l2,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & (p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2g),ap(c_2Emeasure_2Esubsets(A_27a),V0g0))) & (p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V3l1),V4l2)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3l1),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4l2),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))))))) => (surj__ty_2Erealax_2Ereal(ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V3l1),V4l2)),V2g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V3l1),V2g))),ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),V4l2),V2g))))))))))))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ! [V2l1:$i] : (mem(V2l1,A_27b) => ( ! [V3l2:$i] : (mem(V3l2,A_27c) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam))))) => p(ap(c_2Emeasure_2Eadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))))))))))))))))).
tff(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE__SUBADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)))) => p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m)))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__MONO,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) & ((ap(V1f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a)) & ( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2n))),ap(V1f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALT__DISJOINT,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) & ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) & ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((~~ (V2m = V3n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))).
tff(conj_thm_2Emeasure_2ESUBADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ( ! [V3u:$i] : (mem(V3u,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Esubadditive(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))))))))))))))).
tff(conj_thm_2Emeasure_2EADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ( ! [V3u:$i] : (mem(V3u,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) & (V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))))))))))))))).
tff(conj_thm_2Emeasure_2ECOUNTABLY__SUBADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ! [V2s:$i] : (mem(V2s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) & ((V2s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Eseq_2Esuminf,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)))))))))))))).
tff(conj_thm_2Emeasure_2EADDITIVE__SUM,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ! [V2n:tp__ty_2Enum_2Enum] : ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & (p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((~~ (V3m = V4n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V3m))),ap(V1f,inj__ty_2Enum_2Enum(V4n))))))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2n))),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V2n)))))))))))))))).
tff(conj_thm_2Emeasure_2EINCREASING__ADDITIVE__SUMMABLE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & (p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & (p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((~~ (V2m = V3n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n)))))))))))) => p(ap(c_2Eseq_2Esummable,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f))))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__POSITIVE,lemma,(( ! [A_27a:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => (p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) => p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__INCREASING,lemma,(( ! [A_27a:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => (p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) => p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)),V1lam)))))))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__STRONG__SUM,lemma,(( ! [A_27a:del]: ( ! [V0g0:$i] : (mem(V0g0,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : (mem(V2g,arr(A_27a,bool)) => ( ! [V3f:$i] : (mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ( ! [V4n:tp__ty_2Enum_2Enum] : ((p(ap(c_2Emeasure_2Ealgebra(A_27a),V0g0)) & (p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0g0)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0g0)),V1lam)))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2g),ap(c_2Emeasure_2Esubsets(A_27a),V0g0))) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0g0),V1lam)))) & ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((~~ (V5m = V6n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,inj__ty_2Enum_2Enum(V5m))),ap(V3f,inj__ty_2Enum_2Enum(V6n))))))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V4n))),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),V1lam),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,arr(A_27a,bool),arr(A_27a,bool)),f3517(A_27a,V2g)),V3f)))) = surj__ty_2Erealax_2Ereal(ap(V1lam,ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V4n))))),V2g))))))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ALGEBRA,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) => p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)))))))).
tff(conj_thm_2Emeasure_2EOUTER__MEASURE__SPACE__POSITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),V0m)) => p(ap(c_2Emeasure_2Epositive(A_27a),V0m)))))))).
tff(conj_thm_2Emeasure_2ELAMBDA__SYSTEM__CARATHEODORY,lemma,(( ! [A_27a:del]: ( ! [V0gsig:$i] : (mem(V0gsig,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0gsig)) & p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0gsig)),V1lam))))) => ( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam)))) & ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((~~ (V3m = V4n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3m))),ap(V2f,inj__ty_2Enum_2Enum(V4n)))))))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam))) & p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),V1lam),V2f)),ap(V1lam,ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))))))))))))))).
tff(conj_thm_2Emeasure_2ECARATHEODORY__LEMMA,lemma,(( ! [A_27a:del]: ( ! [V0gsig:$i] : (mem(V0gsig,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1lam:$i] : (mem(V1lam,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0gsig)) & p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0gsig)),V1lam))))) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0gsig)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),V0gsig),V1lam)),V1lam)))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__NONEMPTY,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1g:$i] : (mem(V1g,arr(A_27a,bool)) => ( ! [V2s:$i] : (mem(V2s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),V2s))))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3521(A_27a,V1g,V0m)))))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__POS0,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1g:$i] : (mem(V1g,arr(A_27a,bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3521(A_27a,V1g,V0m)))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2x)))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__POS,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1g:$i] : (mem(V1g,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1g),ap(c_2Emeasure_2Em__space(A_27a),V0m))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1g))))))))))).
tff(conj_thm_2Emeasure_2EINCREASING,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & (p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t))))))))))))).
tff(conj_thm_2Emeasure_2EADDITIVE__INCREASING,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)))) => p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)))))))).
tff(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)))) => p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)))))))).
tff(conj_thm_2Emeasure_2ECOUNTABLY__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((~~ (V3m = V4n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3m))),ap(V2f,inj__ty_2Enum_2Enum(V4n))))))) & ((V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))) => p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__AGREES,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & (p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__DOWN,lemma,(( ! [A_27a:del]: ( ! [V0m0:$i] : (mem(V0m0,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1m1:$i] : (mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)))) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))) & ((ap(c_2Emeasure_2Emeasure(A_27a),V0m0) = ap(c_2Emeasure_2Emeasure(A_27a),V1m1)) & p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1))))) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m0)))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__SIGMA,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1sts:$i] : (mem(V1sts,arr(arr(A_27a,bool),bool)) => (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1sts)) => p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1sts))))))))))).
tff(conj_thm_2Emeasure_2EPOW__ALGEBRA,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EPOW(A_27a),V0sp))))))))).
tff(conj_thm_2Emeasure_2EPOW__SIGMA__ALGEBRA,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Epred__set_2EPOW(A_27a),V0sp))))))))).
tff(conj_thm_2Emeasure_2EUNIV__SIGMA__ALGEBRA,lemma,(( ! [A_27a:del]: p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),c_2Epred__set_2EUNIV(A_27a)),c_2Epred__set_2EUNIV(arr(A_27a,bool)))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__EMPTY,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & p(ap(c_2Emeasure_2Epositive(A_27a),V0m))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__POSITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & p(ap(c_2Emeasure_2Epositive(A_27a),V0m))) => p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__INCREASING,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & p(ap(c_2Emeasure_2Epositive(A_27a),V0m))) => p(ap(c_2Emeasure_2Eincreasing(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__LE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & (p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3523(A_27a,V1s,V0m))))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)),inj__ty_2Erealax_2Ereal(V2x)))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__CLOSE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Emeasure_2Em__space(A_27a),V0m)))))) => (? [V3f:$i] : (mem(V3f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) & (? [V4l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V3f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V3f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) & (( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((~~ (V5m = V6n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V3f,inj__ty_2Enum_2Enum(V5m))),ap(V3f,inj__ty_2Enum_2Enum(V6n))))))) & (p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V3f)),inj__ty_2Erealax_2Ereal(V4l))) & p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4l)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Einf__measure(A_27a),V0m),V1s)),inj__ty_2Erealax_2Ereal(V2e)))))))))))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__COUNTABLY__SUBADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)))) => p(ap(c_2Emeasure_2Ecountably__subadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))))))))).
tff(conj_thm_2Emeasure_2EINF__MEASURE__OUTER,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)))) => p(ap(c_2Emeasure_2Eouter__measure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__SUBSET,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,arr(arr(A_27a,bool),bool)) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1b)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Esubsets(A_27a),V1b)))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Espace(A_27a),V1b)),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V1b))))))))))).
tff(conj_thm_2Emeasure_2EALGEBRA__SUBSET__LAMBDA__SYSTEM,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & (p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)) & p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),ap(ap(c_2Emeasure_2Elambda__system(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(c_2Emeasure_2Einf__measure(A_27a),V0m)))))))))).
tff(conj_thm_2Emeasure_2ECARATHEODORY,lemma,(( ! [A_27a:del]: ( ! [V0m0:$i] : (mem(V0m0,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0)))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m0)) & p(ap(c_2Emeasure_2Ecountably__additive(A_27a),V0m0)))) => (? [V1m:$i] : (mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) & (( ! [V2s:$i] : (mem(V2s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V2s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m0),V2s)))))) & ((ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)) = ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m0)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m0))) & p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__SUBSET__SUBSETS,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1a:$i] : (mem(V1a,arr(arr(A_27a,bool),bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))))))))))).
tff(conj_thm_2Emeasure_2EIN__SIGMA,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1a:$i] : (mem(V1a,arr(arr(A_27a,bool),bool)) => ( ! [V2x:$i] : (mem(V2x,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V1a)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0p)) <=> (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & (( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))))) & ( ! [V2c:$i] : (mem(V2c,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V2c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2c),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V2c)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__COUNTABLE__UNION,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1c:$i] : (mem(V1c,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & (p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V1c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1c),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1c)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__ENUM,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__COMPL,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)),V1s))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1p:$i] : (mem(V1p,arr(arr(A_27a,bool),bool)) => ( ! [V2a:$i] : (mem(V2a,arr(arr(A_27a,bool),bool)) => ((p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V1p)) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p)) & (( ! [V3s:$i] : (mem(V3s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p))))) & ( ! [V4c:$i] : (mem(V4c,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V4c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V4c),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V4c)),V1p))))))))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__EMPTY,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),c_2Epred__set_2EEMPTY(A_27a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__SUBSET__MEASURABLE__SETS,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,arr(arr(A_27a,bool),bool)) => ( ! [V1m:$i] : (mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V1m)),V0a))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))) & ( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__DISJOINT,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) <=> (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))) & (( ! [V2s:$i] : (mem(V2s,arr(A_27a,bool)) => ( ! [V3t:$i] : (mem(V3t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))))) & ( ! [V4f:$i] : (mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) & ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((~~ (V5m = V6n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5m))),ap(V4f,inj__ty_2Enum_2Enum(V6n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__ALT,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1p:$i] : (mem(V1p,arr(arr(A_27a,bool),bool)) => ( ! [V2a:$i] : (mem(V2a,arr(arr(A_27a,bool),bool)) => ((p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),V1p)) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p)) & (( ! [V3s:$i] : (mem(V3s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p))))) & ( ! [V4f:$i] : (mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p))))))))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)))))))))))).
tff(conj_thm_2Emeasure_2ESPACE__SMALLEST__CLOSED__CDI,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (ap(c_2Emeasure_2Espace(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)) = ap(c_2Emeasure_2Espace(A_27a),V0a))))))).
tff(conj_thm_2Emeasure_2ESMALLEST__CLOSED__CDI,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) & (p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))) & p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))))))))))).
tff(conj_thm_2Emeasure_2ECLOSED__CDI__DUNION,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & (p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),V0p))) & p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))))))))))).
tff(conj_thm_2Emeasure_2ECLOSED__CDI__COMPL,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0p)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))))))))).
tff(conj_thm_2Emeasure_2ECLOSED__CDI__DISJOINT,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) & ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((~~ (V2m = V3n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2m))),ap(V1f,inj__ty_2Enum_2Enum(V3n))))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))))))))).
tff(conj_thm_2Emeasure_2ECLOSED__CDI__INCREASING,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),V0p)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0p)))) & ((ap(V1f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a)) & ( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2n))),ap(V1f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0p))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA1,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA2,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Ealgebra(A_27a),V0a)) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a)))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Esubsets(A_27a),ap(c_2Emeasure_2Esmallest__closed__cdi(A_27a),V0a))))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__LEMMA,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1a:$i] : (mem(V1a,arr(arr(A_27a,bool),bool)) => ( ! [V2p:$i] : (mem(V2p,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a))) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2p)) & p(ap(c_2Emeasure_2Eclosed__cdi(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2p))))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))),V2p)))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT__WEAK,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1p:$i] : (mem(V1p,arr(arr(A_27a,bool),bool)) => ( ! [V2a:$i] : (mem(V2a,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2a))) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p)) & (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),V0sp),V1p)) & (( ! [V3s:$i] : (mem(V3s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),V1p)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p))))) & (( ! [V4f:$i] : (mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1p))) & ((ap(V4f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a)) & ( ! [V5n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5n))),ap(V4f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V5n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p))))) & ( ! [V6f:$i] : (mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V6f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V1p))) & ( ! [V7m:tp__ty_2Enum_2Enum] : ( ! [V8n:tp__ty_2Enum_2Enum] : ((~~ (V7m = V8n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V6f,inj__ty_2Enum_2Enum(V7m))),ap(V6f,inj__ty_2Enum_2Enum(V8n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V6f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p)))))))))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__PROPERTY__DISJOINT,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1p:$i] : (mem(V1p,arr(arr(A_27a,bool),bool)) => ( ! [V2a:$i] : (mem(V2a,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Emeasure_2Ealgebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2a))) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V2a),V1p)) & (( ! [V3s:$i] : (mem(V3s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3s),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3s)),V1p))))) & (( ! [V4f:$i] : (mem(V4f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V4f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))) & ((ap(V4f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a)) & ( ! [V5n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V4f,inj__ty_2Enum_2Enum(V5n))),ap(V4f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V5n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V4f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p))))) & ( ! [V6f:$i] : (mem(V6f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V6f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINTER(arr(A_27a,bool)),V1p),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a)))))) & ( ! [V7m:tp__ty_2Enum_2Enum] : ( ! [V8n:tp__ty_2Enum_2Enum] : ((~~ (V7m = V8n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V6f,inj__ty_2Enum_2Enum(V7m))),ap(V6f,inj__ty_2Enum_2Enum(V8n)))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V6f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),V1p))))))))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V2a))),V1p)))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__COUNTABLY__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((~~ (V3m = V4n)) => p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3m))),ap(V2f,inj__ty_2Enum_2Enum(V4n))))))) & (V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))))) => p(ap(ap(c_2Eseq_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(c_2Emeasure_2Eadditive(A_27a),V0m)))))))).
tff(conj_thm_2Emeasure_2EMEASURE__ADDITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ( ! [V3u:$i] : (mem(V3u,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1s),V2t)) & (V3u = ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V3u)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V2t)))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__COUNTABLE__INCREASING,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ((ap(V2f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = c_2Epred__set_2EEMPTY(A_27a)) & (( ! [V3n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3n))),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))))) & (V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))))) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__REDUCE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),ap(c_2Emeasure_2Emeasure(A_27a),V0m))) = V0m)))))).
tff(conj_thm_2Emeasure_2ESPACE__SIGMA,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1a:$i] : (mem(V1a,arr(arr(A_27a,bool),bool)) => (ap(c_2Emeasure_2Espace(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a)) = V0sp)))))))).
tff(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (( ! [V3n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,inj__ty_2Enum_2Enum(V3n))),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))))) & (V1s = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))))) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__REDUCE,lemma,(( ! [A_27a:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1a:$i] : (mem(V1a,arr(arr(A_27a,bool),bool)) => (ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))) = ap(ap(c_2Emeasure_2Esigma(A_27a),V0sp),V1a))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__SETS__SUBSET__SPACE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),ap(c_2Emeasure_2Em__space(A_27a),V0m))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__DIFF__PROPERTY,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V2f:$i] : (mem(V2f,arr(A_27a,A_27b)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ( ! [V3s:$i] : (mem(V3s,arr(A_27b,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))) => ( ! [V4s:$i] : (mem(V4s,arr(A_27b,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => (ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),ap(ap(c_2Epred__set_2EDIFF(A_27b),ap(c_2Emeasure_2Espace(A_27b),V1b)),V4s)) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V4s))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__BIGUNION__PROPERTY,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V2f:$i] : (mem(V2f,arr(A_27a,A_27b)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ( ! [V3s:$i] : (mem(V3s,arr(A_27b,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))) => ( ! [V4c:$i] : (mem(V4c,arr(arr(A_27b,bool),bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27b,bool)),V4c),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => (ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),ap(c_2Epred__set_2EBIGUNION(A_27b),V4c)) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),V4c))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASUBABLE__BIGUNION__LEMMA,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V2f:$i] : (mem(V2f,arr(A_27a,A_27b)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ( ! [V3s:$i] : (mem(V3s,arr(A_27b,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))) => ( ! [V4c:$i] : (mem(V4c,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V4c)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V4c),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),V4c)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__SIGMA__PREIMAGES,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V2f:$i] : (mem(V2f,arr(A_27a,A_27b)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ( ! [V3s:$i] : (mem(V3s,arr(A_27b,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))) => p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27b,bool),arr(A_27a,bool)),ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f)),ap(c_2Emeasure_2Esubsets(A_27b),V1b)))))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V2f:$i] : (mem(V2f,arr(A_27a,A_27b)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27b),V1b)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Espace(A_27b),V1b)))) & ( ! [V3s:$i] : (mem(V3s,arr(A_27b,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Esubsets(A_27b),V1b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Espace(A_27a),V0a))),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__SIGMA,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,A_27b)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V2b:$i] : (mem(V2b,arr(arr(A_27b,bool),bool)) => ( ! [V3sp:$i] : (mem(V3sp,arr(A_27b,bool)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Emeasure_2Esubset__class(A_27b),V3sp),V2b)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),ap(c_2Emeasure_2Espace(A_27a),V1a)),V3sp))) & ( ! [V4s:$i] : (mem(V4s,arr(A_27b,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4s),V2b)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V0f),V4s)),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),ap(ap(c_2Emeasure_2Esigma(A_27b),V3sp),V2b))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__SUBSET,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Espace(A_27b),V1b)),ap(c_2Emeasure_2Esubsets(A_27b),V1b))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__LIFT,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,A_27b)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V2b:$i] : (mem(V2b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),V2b))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V1a),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Espace(A_27b),V2b)),ap(c_2Emeasure_2Esubsets(A_27b),V2b)))))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURE__PRESERVING,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0m1:$i] : (mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1m2:$i] : (mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ( ! [V2f:$i] : (mem(V2f,arr(A_27a,A_27b)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2))) <=> (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V2f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2))))) & (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)) & (p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2)) & ( ! [V3s:$i] : (mem(V3s,arr(A_27b,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3s),ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m1),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),V2f),V3s)),ap(c_2Emeasure_2Em__space(A_27a),V0m1)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27b),V1m2),V3s)))))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__LIFT,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0m1:$i] : (mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1m2:$i] : (mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ( ! [V2a:$i] : (mem(V2a,arr(arr(A_27b,bool),bool)) => ( ! [V3f:$i] : (mem(V3f,arr(A_27a,A_27b)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)) & (p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2)) & ((ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2) = ap(c_2Emeasure_2Esubsets(A_27b),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),V2a))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27b),V1m2))))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2)))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__SUBSET,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0m1:$i] : (mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1m2:$i] : (mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ( ! [V2a:$i] : (mem(V2a,arr(arr(A_27b,bool),bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)) & (p(ap(c_2Emeasure_2Emeasure__space(A_27b),V1m2)) & (ap(c_2Emeasure_2Emeasurable__sets(A_27b),V1m2) = ap(c_2Emeasure_2Esubsets(A_27b),ap(ap(c_2Emeasure_2Esigma(A_27b),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),V2a))))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27b),V1m2)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27b),V1m2))))),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2)))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__I,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27a)),c_2Ecombin_2EI(A_27a)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27a),V0a),V0a))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__COMP,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,A_27b)) => ( ! [V1g:$i] : (mem(V1g,arr(A_27b,A_27c)) => ( ! [V2a:$i] : (mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V3b:$i] : (mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V4c:$i] : (mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27b,A_27c),V3b),V4c)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONG,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,A_27b)) => ( ! [V1g:$i] : (mem(V1g,arr(A_27b,A_27c)) => ( ! [V2a:$i] : (mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V3b:$i] : (mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V4c:$i] : (mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b))) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27c),V4c)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Epred__set_2EFUNSET(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V3b)),ap(c_2Emeasure_2Espace(A_27c),V4c)))) & ( ! [V5x:$i] : (mem(V5x,arr(A_27c,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V5x),ap(c_2Emeasure_2Esubsets(A_27c),V4c))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V5x)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Emeasure_2Espace(A_27a),V2a)))),ap(c_2Emeasure_2Esubsets(A_27b),V3b))))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__COMP__STRONGER,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,A_27b)) => ( ! [V1g:$i] : (mem(V1g,arr(A_27b,A_27c)) => ( ! [V2a:$i] : (mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V3b:$i] : (mem(V3b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V4c:$i] : (mem(V4c,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool))) => ( ! [V5t:$i] : (mem(V5t,arr(A_27b,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V2a),V3b))) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27c),V4c)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27b,A_27c)),V1g),ap(ap(c_2Epred__set_2EFUNSET(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V3b)),ap(c_2Emeasure_2Espace(A_27c),V4c)))) & (p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Emeasure_2Espace(A_27a),V2a))),V5t)) & ( ! [V6s:$i] : (mem(V6s,arr(A_27c,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27c,bool)),V6s),ap(c_2Emeasure_2Esubsets(A_27c),V4c))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),ap(ap(c_2Epred__set_2EPREIMAGE(A_27b,A_27c),V1g),V6s)),V5t)),ap(c_2Emeasure_2Esubsets(A_27b),V3b)))))))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V0f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V2a),V4c))))))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__UP__LIFT,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1a:$i] : (mem(V1a,arr(arr(A_27a,bool),bool)) => ( ! [V2b:$i] : (mem(V2b,arr(arr(A_27a,bool),bool)) => ( ! [V3c:$i] : (mem(V3c,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V4f:$i] : (mem(V4f,arr(A_27a,A_27b)) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V4f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)),V3c))) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b))) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2b)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V4f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)),V3c)))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__UP__SUBSET,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0sp:$i] : (mem(V0sp,arr(A_27a,bool)) => ( ! [V1a:$i] : (mem(V1a,arr(arr(A_27a,bool),bool)) => ( ! [V2b:$i] : (mem(V2b,arr(arr(A_27a,bool),bool)) => ( ! [V3c:$i] : (mem(V3c,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ((p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V1a),V2b)) & p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V1a)),V3c)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0sp),V2b)),V3c)))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__UP__SIGMA,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1b:$i] : (mem(V1b,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1b)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))),V1b))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__LIFT,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,arr(arr(A_27a,bool),bool)) => ( ! [V1m1:$i] : (mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V2m2:$i] : (mem(V2m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ( ! [V3f:$i] : (mem(V3f,arr(A_27a,A_27b)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V0a),ap(c_2Emeasure_2Emeasure(A_27a),V1m1)))),V2m2))) & (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V3f),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V1m1),V2m2)))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SUBSET,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0a:$i] : (mem(V0a,arr(arr(A_27a,bool),bool)) => ( ! [V1m1:$i] : (mem(V1m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V2m2:$i] : (mem(V2m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m1)) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0a),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1))) & p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m1)))))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V1m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V0a),ap(c_2Emeasure_2Emeasure(A_27a),V1m1)))),V2m2)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V1m1),V2m2)))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__PRESERVING__UP__SIGMA,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0m1:$i] : (mem(V0m1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1m2:$i] : (mem(V1m2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(arr(A_27b,bool),bool),arr(arr(A_27b,bool),ty_2Erealax_2Ereal)))) => ( ! [V2a:$i] : (mem(V2a,arr(arr(A_27a,bool),bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m1)) & (ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m1) = ap(c_2Emeasure_2Esubsets(A_27a),ap(ap(c_2Emeasure_2Esigma(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),V2a)))) => p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,A_27b)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m1)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),V2a),ap(c_2Emeasure_2Emeasure(A_27a),V0m1)))),V1m2)),ap(ap(c_2Emeasure_2Emeasure__preserving(A_27a,A_27b),V0m1),V1m2)))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__PROD__SIGMA,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1a1:$i] : (mem(V1a1,ty_2Epair_2Eprod(arr(A_27b,bool),arr(arr(A_27b,bool),bool))) => ( ! [V2a2:$i] : (mem(V2a2,ty_2Epair_2Eprod(arr(A_27c,bool),arr(arr(A_27c,bool),bool))) => ( ! [V3f:$i] : (mem(V3f,arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,ty_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2EFST(A_27b,A_27c)),V3f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),V0a),V1a1))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,ty_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2ESND(A_27b,A_27c)),V3f)),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27c),V0a),V2a2))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),V3f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),V0a),ap(ap(c_2Emeasure_2Esigma(ty_2Epair_2Eprod(A_27b,A_27c)),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27c),ap(c_2Emeasure_2Espace(A_27b),V1a1)),ap(c_2Emeasure_2Espace(A_27c),V2a2))),ap(ap(c_2Eutil__prob_2Eprod__sets(A_27b,A_27c),ap(c_2Emeasure_2Esubsets(A_27b),V1a1)),ap(c_2Emeasure_2Esubsets(A_27c),V2a2)))))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__RANGE__REDUCE,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,A_27b)) => ( ! [V2s:$i] : (mem(V2s,arr(A_27b,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),V2s),ap(c_2Epred__set_2EPOW(A_27b),V2s))))) & (~~ (V2s = c_2Epred__set_2EEMPTY(A_27b))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EINTER(A_27b),V2s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))))))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,A_27b)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),c_2Epred__set_2EUNIV(A_27b)),ap(c_2Epred__set_2EPOW(A_27b),c_2Epred__set_2EUNIV(A_27b))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURABLE__POW__TO__POW__IMAGE,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,A_27b)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m) = ap(c_2Epred__set_2EPOW(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,A_27b)),V1f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),arr(arr(A_27b,bool),bool)),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m))),ap(c_2Epred__set_2EPOW(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),ap(c_2Emeasure_2Em__space(A_27a),V0m)))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET,lemma,(( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,arr(A_27a,bool)) => ( ! [V1s_27:$i] : (mem(V1s_27,arr(A_27a,bool)) => ( ! [V2m:$i] : (mem(V2m,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s_27),V0s)) & p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V0s),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V0s)),V2m))))) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s_27),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),V2m)))))))))))))).
tff(conj_thm_2Emeasure_2ESTRONG__MEASURE__SPACE__SUBSET,lemma,(( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s_27:$i] : (mem(V1s_27,arr(A_27a,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s_27),ap(c_2Emeasure_2Em__space(A_27a),V0s))) & (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0s))))) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s_27),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Epred__set_2EPOW(A_27a),V1s_27)),ap(c_2Emeasure_2Emeasure(A_27a),V0s))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__REAL__SUM__IMAGE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & (( ! [V2x:$i] : (mem(V2x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))))) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3524(A_27a,V0m)),V1s))))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__POW,lemma,(( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,arr(A_27a,bool)) => (ap(ap(c_2Emeasure_2Esigma(A_27a),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s)) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),V0s),ap(c_2Epred__set_2EPOW(A_27a),V0s)))))))).
tff(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces,lemma,(( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1m:$i] : (mem(V1m,arr(arr(A_27a,bool),ty_2Erealax_2Ereal)) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0s)) & (p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0s))) & (p(ap(c_2Emeasure_2Epositive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))) & p(ap(c_2Emeasure_2Eadditive(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m))))))) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Espace(A_27a),V0s)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Esubsets(A_27a),V0s)),V1m)))))))))))).
tff(conj_thm_2Emeasure_2Efinite__additivity__sufficient__for__finite__spaces2,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Emeasure_2Em__space(A_27a),V0m))) & (p(ap(c_2Emeasure_2Epositive(A_27a),V0m)) & p(ap(c_2Emeasure_2Eadditive(A_27a),V0m))))) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)))))))).
tff(conj_thm_2Emeasure_2EIMAGE__SING,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,A_27b)) => ( ! [V1x:$i] : (mem(V1x,A_27a) => (ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))) = ap(ap(c_2Epred__set_2EINSERT(A_27b),ap(V0f,V1x)),c_2Epred__set_2EEMPTY(A_27b))))))))))).
tff(conj_thm_2Emeasure_2ESUBSET__BIGINTER,lemma,(( ! [A_27a:del]: ( ! [V0X:$i] : (mem(V0X,arr(A_27a,bool)) => ( ! [V1P:$i] : (mem(V1P,arr(arr(A_27a,bool),bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),ap(c_2Epred__set_2EBIGINTER(A_27a),V1P))) <=> ( ! [V2Y:$i] : (mem(V2Y,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2Y),V1P)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0X),V2Y))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__INCREASING,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(c_2Emeasure_2Eincreasing(A_27a),V0m)))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__POSITIVE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(c_2Emeasure_2Epositive(A_27a),V0m)))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__INTER,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__UNION,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__DIFF,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__COMPL__SUBSET,lemma,(( ! [A_27a:del]: ( ! [V0t:$i] : (mem(V0t,arr(A_27a,bool)) => ( ! [V1m:$i] : (mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V2s:$i] : (mem(V2s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0t),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0t),V2s))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),ap(ap(c_2Epred__set_2EDIFF(A_27a),V2s),V0t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V2s)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V1m),V0t)))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGUNION,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1s,inj__ty_2Enum_2Enum(V2n))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__IN__MSPACE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1A:$i] : (mem(V1A,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) => ( ! [V2x:$i] : (mem(V2x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1A)) => p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(c_2Emeasure_2Em__space(A_27a),V0m)))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__SUBSET__MSPACE,lemma,(( ! [A_27a:del]: ( ! [V0A:$i] : (mem(V0A,arr(A_27a,bool)) => ( ! [V1m:$i] : (mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V1m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V0A),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0A),ap(c_2Emeasure_2Em__space(A_27a),V1m))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__EMPTY__MEASURABLE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__MSPACE__MEASURABLE,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__FN__BIGINTER,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) => (p(ap(ap(c_2Emeasure_2Esubset__class(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),c_2Epred__set_2EEMPTY(A_27a)),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) & (( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Esubsets(A_27a),V0a))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Emeasure_2Espace(A_27a),V0a)),V1s)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))))) & ( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Esubsets(A_27a),V0a)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__BIGINTER,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & ( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V1s,inj__ty_2Enum_2Enum(V2n))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))))))))))).
tff(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE2,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V2n))),ap(V1f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n)))))))) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))))))))))).
tff(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ( ! [V2f:$i] : (mem(V2f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V2f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & (( ! [V3n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V2f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),ap(V2f,inj__ty_2Enum_2Enum(V3n))))) & (V1s = ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V2f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))))) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V2f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),V1s))))))))))))).
tff(conj_thm_2Emeasure_2EMONOTONE__CONVERGENCE__BIGINTER2,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Enum_2Enum,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Enum_2Enum,arr(A_27a,bool)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) & ( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n)))),ap(V1f,inj__ty_2Enum_2Enum(V2n))))))) => p(ap(ap(c_2Eseq_2E_2D_2D_3E,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,arr(A_27a,bool)),ap(c_2Emeasure_2Emeasure(A_27a),V0m)),V1f)),ap(ap(c_2Emeasure_2Emeasure(A_27a),V0m),ap(c_2Epred__set_2EBIGINTER(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,arr(A_27a,bool)),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__RESTRICTED,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1s),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)))) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),V1s),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),f3525(A_27a,V1s)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m))),ap(c_2Emeasure_2Emeasure(A_27a),V0m))))))))))))).
tff(conj_thm_2Emeasure_2EMEASURE__SPACE__CMUL,lemma,(( ! [A_27a:del]: ( ! [V0m:$i] : (mem(V0m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m)) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1c)))) => p(ap(c_2Emeasure_2Emeasure__space(A_27a),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))),ap(c_2Emeasure_2Em__space(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V0m)),f3526(A_27a,V0m,V1c)))))))))))).
tff(conj_thm_2Emeasure_2EBIGUNION__IMAGE__Q,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ( ! [V1f:$i] : (mem(V1f,arr(ty_2Eextreal_2Eextreal,arr(A_27a,bool))) => ((p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V0a)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,arr(A_27a,bool))),V1f),ap(ap(c_2Epred__set_2EFUNSET(ty_2Eextreal_2Eextreal,arr(A_27a,bool)),c_2Eextreal_2EQ__set),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,arr(A_27a,bool)),V1f),c_2Eextreal_2EQ__set))),ap(c_2Emeasure_2Esubsets(A_27a),V0a))))))))))).
tff(ax_thm_2Emeasure_2EBorel__def,axiom,(c_2Emeasure_2EBorel = ap(ap(c_2Emeasure_2Esigma(ty_2Eextreal_2Eextreal),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)),f3528),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal))))).
tff(conj_thm_2Emeasure_2ESIGMA__ALGEBRA__BOREL,lemma,(p(ap(c_2Emeasure_2Esigma__algebra(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))).
tff(conj_thm_2Emeasure_2EMEASURABLE__BOREL,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(ap(c_2Epred__set_2EPREIMAGE(A_27a,ty_2Eextreal_2Eextreal),V0f),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3529(V2c)))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3530(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__NEGINF,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3531(A_27a,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT1,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3532(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT2,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3533(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT3,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3534(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT4,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : ( ! [V3d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3535(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT5,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : ( ! [V3d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3536(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT6,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) <=> (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : ( ! [V3d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3537(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT7,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : ( ! [V3d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3538(A_27a,V3d,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT8,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3539(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALT9,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => (p(ap(c_2Emeasure_2Esigma__algebra(A_27a),V1a)) & (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Epred__set_2EFUNSET(A_27a,ty_2Eextreal_2Eextreal),ap(c_2Emeasure_2Espace(A_27a),V1a)),c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))) & ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3540(A_27a,V0f,V2c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1a:$i] : (mem(V1a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V1a),c_2Emeasure_2EBorel))) => (( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3530(A_27a,V2c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))) & (( ! [V4c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3541(A_27a,V4c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))) & (( ! [V6c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3542(A_27a,V6c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))) & (( ! [V8c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3543(A_27a,V8c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))) & (( ! [V10c:tp__ty_2Eextreal_2Eextreal] : ( ! [V11d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3544(A_27a,V11d,V0f,V10c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))) & (( ! [V13c:tp__ty_2Eextreal_2Eextreal] : ( ! [V14d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3545(A_27a,V14d,V0f,V13c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))) & (( ! [V16c:tp__ty_2Eextreal_2Eextreal] : ( ! [V17d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3546(A_27a,V17d,V0f,V16c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))) & (( ! [V19c:tp__ty_2Eextreal_2Eextreal] : ( ! [V20d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3547(A_27a,V20d,V0f,V19c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))) & (( ! [V22c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3548(A_27a,V0f,V22c))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a)))) & ( ! [V24c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3549(A_27a,V24c,V0f))),ap(c_2Emeasure_2Espace(A_27a),V1a))),ap(c_2Emeasure_2Esubsets(A_27a),V1a))))))))))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__ALL__MEASURE,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1m:$i] : (mem(V1m,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),ty_2Erealax_2Ereal)))) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),arr(arr(A_27a,bool),bool)),ap(c_2Emeasure_2Em__space(A_27a),V1m)),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))),c_2Emeasure_2EBorel))) => (( ! [V2c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3530(A_27a,V2c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))) & (( ! [V4c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3541(A_27a,V4c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))) & (( ! [V6c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3542(A_27a,V6c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))) & (( ! [V8c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3543(A_27a,V8c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))) & (( ! [V10c:tp__ty_2Eextreal_2Eextreal] : ( ! [V11d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3544(A_27a,V11d,V0f,V10c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))))) & (( ! [V13c:tp__ty_2Eextreal_2Eextreal] : ( ! [V14d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3545(A_27a,V14d,V0f,V13c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))))) & (( ! [V16c:tp__ty_2Eextreal_2Eextreal] : ( ! [V17d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3546(A_27a,V17d,V0f,V16c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))))) & (( ! [V19c:tp__ty_2Eextreal_2Eextreal] : ( ! [V20d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3547(A_27a,V20d,V0f,V19c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))))) & (( ! [V22c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3550(A_27a,V22c,V0f))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m)))) & ( ! [V24c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3551(A_27a,V0f,V24c))),ap(c_2Emeasure_2Em__space(A_27a),V1m))),ap(c_2Emeasure_2Emeasurable__sets(A_27a),V1m))))))))))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LT,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1g:$i] : (mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V2a:$i] : (mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3552(A_27a,V1g,V0f))),ap(c_2Emeasure_2Espace(A_27a),V2a))),ap(c_2Emeasure_2Esubsets(A_27a),V2a))))))))))))).
tff(conj_thm_2Emeasure_2EIN__MEASURABLE__BOREL__LE,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V1g:$i] : (mem(V1g,arr(A_27a,ty_2Eextreal_2Eextreal)) => ( ! [V2a:$i] : (mem(V2a,ty_2Epair_2Eprod(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) => ((p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V0f),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel))) & p(ap(ap(c_2Ebool_2EIN(arr(A_27a,ty_2Eextreal_2Eextreal)),V1g),ap(ap(c_2Emeasure_2Emeasurable(A_27a,ty_2Eextreal_2Eextreal),V2a),c_2Emeasure_2EBorel)))) => p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3553(A_27a,V1g,V0f))),ap(c_2Emeasure_2Espace(A_27a),V2a))),ap(c_2Emeasure_2Esubsets(A_27a),V2a))))))))))))).
tff(conj_thm_2Emeasure_2ESPACE__BOREL,lemma,((ap(c_2Emeasure_2Espace(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel) = c_2Epred__set_2EUNIV(ty_2Eextreal_2Eextreal)))).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS1,lemma,(( ! [V0c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3554(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))))).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS2,lemma,(( ! [V0c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3555(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))))).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS3,lemma,(( ! [V0c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3556(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))))).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS4,lemma,(( ! [V0c:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3557(V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel)))))).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS5,lemma,(( ! [V0c:tp__ty_2Eextreal_2Eextreal] : ( ! [V1d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3558(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))))))).
tff(conj_thm_2Emeasure_2EBOREL__MEASURABLE__SETS6,conjecture,(( ! [V0c:tp__ty_2Eextreal_2Eextreal] : ( ! [V1d:tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Ebool_2EIN(arr(ty_2Eextreal_2Eextreal,bool)),ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3559(V1d,V0c))),ap(c_2Emeasure_2Esubsets(ty_2Eextreal_2Eextreal),c_2Emeasure_2EBorel))))))).
