include('Axioms/measure^2.ax').
thf(tp_ty_2Ereal__topology_2Enet,type,( ty_2Ereal__topology_2Enet: del >del )).
thf(tp_c_2Ereal__topology_2E_2D_2D_3E,type,(c_2Ereal__topology_2E_2D_2D_3E:(del>$i))).
thf(mem_c_2Ereal__topology_2E_2D_2D_3E,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(arr@(arr@A_27a@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@(arr@(ty_2Ereal__topology_2Enet@A_27a)@bool)))))).
thf(stp_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,(tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal : $tType)).
thf(stp_inj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal : (tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > $i))).
thf(stp_surj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,(surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal : ($i > tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal))).
thf(stp_inj_surj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,( ! [X:tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ((surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ (inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X)) = X))).
thf(stp_inj_mem_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,( ! [X:tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : (mem @ (inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X) @ (ty_2Elist_2Elist@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))).
thf(stp_iso_mem_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,( ! [X:$i] : ((mem @ X @ (ty_2Elist_2Elist@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => (X = (inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ (surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X)))))).
thf(tp_c_2Ereal__topology_2ECLOSED__interval,type,(c_2Ereal__topology_2ECLOSED__interval:$i)).
thf(mem_c_2Ereal__topology_2ECLOSED__interval,axiom,(mem@c_2Ereal__topology_2ECLOSED__interval@(arr@(ty_2Elist_2Elist@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2EClosed,type,(c_2Ereal__topology_2EClosed:$i)).
thf(mem_c_2Ereal__topology_2EClosed,axiom,(mem@c_2Ereal__topology_2EClosed@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2EDist,type,(c_2Ereal__topology_2EDist:$i)).
thf(mem_c_2Ereal__topology_2EDist,axiom,(mem@c_2Ereal__topology_2EDist@(arr@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@ty_2Erealax_2Ereal))).
thf(stp_fo_c_2Ereal__topology_2EDist,type,(fo__c_2Ereal__topology_2EDist:(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal))).
thf(stp_eq_fo_c_2Ereal__topology_2EDist,axiom,( ! [X0:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ((inj__ty_2Erealax_2Ereal @ (fo__c_2Ereal__topology_2EDist @ X0)) = (ap @ c_2Ereal__topology_2EDist @ (inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X0))))).
thf(tp_c_2Ereal__topology_2EOPEN__interval,type,(c_2Ereal__topology_2EOPEN__interval:$i)).
thf(mem_c_2Ereal__topology_2EOPEN__interval,axiom,(mem@c_2Ereal__topology_2EOPEN__interval@(arr@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2EOpen,type,(c_2Ereal__topology_2EOpen:$i)).
thf(mem_c_2Ereal__topology_2EOpen,axiom,(mem@c_2Ereal__topology_2EOpen@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(stp_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,type,(tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal : $tType)).
thf(stp_inj_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,type,(inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal : (tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal > $i))).
thf(stp_surj_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,type,(surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal : ($i > tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal))).
thf(stp_inj_surj_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,( ! [X:tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal] : ((surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ X)) = X))).
thf(stp_inj_mem_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,( ! [X:tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal] : (mem @ (inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ X) @ (ty_2Ereal__topology_2Enet@ty_2Erealax_2Ereal)))).
thf(stp_iso_mem_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,( ! [X:$i] : ((mem @ X @ (ty_2Ereal__topology_2Enet@ty_2Erealax_2Ereal)) => (X = (inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ X)))))).
thf(tp_c_2Ereal__topology_2Eat,type,(c_2Ereal__topology_2Eat:$i)).
thf(mem_c_2Ereal__topology_2Eat,axiom,(mem@c_2Ereal__topology_2Eat@(arr@ty_2Erealax_2Ereal@(ty_2Ereal__topology_2Enet@ty_2Erealax_2Ereal)))).
thf(tp_c_2Ereal__topology_2Eat__infinity,type,(c_2Ereal__topology_2Eat__infinity:$i)).
thf(mem_c_2Ereal__topology_2Eat__infinity,axiom,(mem@c_2Ereal__topology_2Eat__infinity@(ty_2Ereal__topology_2Enet@ty_2Erealax_2Ereal))).
thf(tp_c_2Ereal__topology_2Eat__neginfinity,type,(c_2Ereal__topology_2Eat__neginfinity:$i)).
thf(mem_c_2Ereal__topology_2Eat__neginfinity,axiom,(mem@c_2Ereal__topology_2Eat__neginfinity@(ty_2Ereal__topology_2Enet@ty_2Erealax_2Ereal))).
thf(tp_c_2Ereal__topology_2Eat__posinfinity,type,(c_2Ereal__topology_2Eat__posinfinity:$i)).
thf(mem_c_2Ereal__topology_2Eat__posinfinity,axiom,(mem@c_2Ereal__topology_2Eat__posinfinity@(ty_2Ereal__topology_2Enet@ty_2Erealax_2Ereal))).
thf(tp_c_2Ereal__topology_2Eball,type,(c_2Ereal__topology_2Eball:$i)).
thf(mem_c_2Ereal__topology_2Eball,axiom,(mem@c_2Ereal__topology_2Eball@(arr@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Ebetween,type,(c_2Ereal__topology_2Ebetween:$i)).
thf(mem_c_2Ereal__topology_2Ebetween,axiom,(mem@c_2Ereal__topology_2Ebetween@(arr@ty_2Erealax_2Ereal@(arr@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@bool)))).
thf(tp_c_2Ereal__topology_2Ebilinear,type,(c_2Ereal__topology_2Ebilinear:$i)).
thf(mem_c_2Ereal__topology_2Ebilinear,axiom,(mem@c_2Ereal__topology_2Ebilinear@(arr@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@bool))).
thf(tp_c_2Ereal__topology_2Ebounded__def,type,(c_2Ereal__topology_2Ebounded__def:$i)).
thf(mem_c_2Ereal__topology_2Ebounded__def,axiom,(mem@c_2Ereal__topology_2Ebounded__def@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Ecauchy,type,(c_2Ereal__topology_2Ecauchy:$i)).
thf(mem_c_2Ereal__topology_2Ecauchy,axiom,(mem@c_2Ereal__topology_2Ecauchy@(arr@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@bool))).
thf(tp_c_2Ereal__topology_2Ecball,type,(c_2Ereal__topology_2Ecball:$i)).
thf(mem_c_2Ereal__topology_2Ecball,axiom,(mem@c_2Ereal__topology_2Ecball@(arr@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Eclosed__segment,type,(c_2Ereal__topology_2Eclosed__segment:$i)).
thf(mem_c_2Ereal__topology_2Eclosed__segment,axiom,(mem@c_2Ereal__topology_2Eclosed__segment@(arr@(ty_2Elist_2Elist@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Eclosest__point,type,(c_2Ereal__topology_2Eclosest__point:$i)).
thf(mem_c_2Ereal__topology_2Eclosest__point,axiom,(mem@c_2Ereal__topology_2Eclosest__point@(arr@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))).
thf(tp_c_2Ereal__topology_2Eclosure,type,(c_2Ereal__topology_2Eclosure:$i)).
thf(mem_c_2Ereal__topology_2Eclosure,axiom,(mem@c_2Ereal__topology_2Eclosure@(arr@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Ecollinear,type,(c_2Ereal__topology_2Ecollinear:$i)).
thf(mem_c_2Ereal__topology_2Ecollinear,axiom,(mem@c_2Ereal__topology_2Ecollinear@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Ecompact,type,(c_2Ereal__topology_2Ecompact:$i)).
thf(mem_c_2Ereal__topology_2Ecompact,axiom,(mem@c_2Ereal__topology_2Ecompact@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Ecomplete,type,(c_2Ereal__topology_2Ecomplete:$i)).
thf(mem_c_2Ereal__topology_2Ecomplete,axiom,(mem@c_2Ereal__topology_2Ecomplete@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Ecomponents,type,(c_2Ereal__topology_2Ecomponents:$i)).
thf(mem_c_2Ereal__topology_2Ecomponents,axiom,(mem@c_2Ereal__topology_2Ecomponents@(arr@(arr@ty_2Erealax_2Ereal@bool)@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)))).
thf(tp_c_2Ereal__topology_2Econdensation__point__of,type,(c_2Ereal__topology_2Econdensation__point__of:$i)).
thf(mem_c_2Ereal__topology_2Econdensation__point__of,axiom,(mem@c_2Ereal__topology_2Econdensation__point__of@(arr@ty_2Erealax_2Ereal@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)))).
thf(tp_c_2Ereal__topology_2Econnected,type,(c_2Ereal__topology_2Econnected:$i)).
thf(mem_c_2Ereal__topology_2Econnected,axiom,(mem@c_2Ereal__topology_2Econnected@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Econnected__component,type,(c_2Ereal__topology_2Econnected__component:$i)).
thf(mem_c_2Ereal__topology_2Econnected__component,axiom,(mem@c_2Ereal__topology_2Econnected__component@(arr@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))))).
thf(tp_c_2Ereal__topology_2Econtinuous,type,(c_2Ereal__topology_2Econtinuous:(del>$i))).
thf(mem_c_2Ereal__topology_2Econtinuous,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Econtinuous@A_27a)@(arr@(arr@A_27a@ty_2Erealax_2Ereal)@(arr@(ty_2Ereal__topology_2Enet@A_27a)@bool))))).
thf(tp_c_2Ereal__topology_2Econtinuous__on,type,(c_2Ereal__topology_2Econtinuous__on:$i)).
thf(mem_c_2Ereal__topology_2Econtinuous__on,axiom,(mem@c_2Ereal__topology_2Econtinuous__on@(arr@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)))).
thf(tp_c_2Ereal__topology_2Edependent,type,(c_2Ereal__topology_2Edependent:$i)).
thf(mem_c_2Ereal__topology_2Edependent,axiom,(mem@c_2Ereal__topology_2Edependent@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Ediameter,type,(c_2Ereal__topology_2Ediameter:$i)).
thf(mem_c_2Ereal__topology_2Ediameter,axiom,(mem@c_2Ereal__topology_2Ediameter@(arr@(arr@ty_2Erealax_2Ereal@bool)@ty_2Erealax_2Ereal))).
thf(tp_c_2Ereal__topology_2Edim,type,(c_2Ereal__topology_2Edim:$i)).
thf(mem_c_2Ereal__topology_2Edim,axiom,(mem@c_2Ereal__topology_2Edim@(arr@(arr@ty_2Erealax_2Ereal@bool)@ty_2Enum_2Enum))).
thf(tp_c_2Ereal__topology_2Eeuclidean,type,(c_2Ereal__topology_2Eeuclidean:$i)).
thf(mem_c_2Ereal__topology_2Eeuclidean,axiom,(mem@c_2Ereal__topology_2Eeuclidean@(ty_2Etopology_2Etopology@ty_2Erealax_2Ereal))).
thf(tp_c_2Ereal__topology_2Eeventually,type,(c_2Ereal__topology_2Eeventually:(del>$i))).
thf(mem_c_2Ereal__topology_2Eeventually,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Eeventually@A_27a)@(arr@(arr@A_27a@bool)@(arr@(ty_2Ereal__topology_2Enet@A_27a)@bool))))).
thf(tp_c_2Ereal__topology_2Efrom,type,(c_2Ereal__topology_2Efrom:$i)).
thf(mem_c_2Ereal__topology_2Efrom,axiom,(mem@c_2Ereal__topology_2Efrom@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@bool)))).
thf(tp_c_2Ereal__topology_2Efrontier,type,(c_2Ereal__topology_2Efrontier:$i)).
thf(mem_c_2Ereal__topology_2Efrontier,axiom,(mem@c_2Ereal__topology_2Efrontier@(arr@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Efsigma,type,(c_2Ereal__topology_2Efsigma:$i)).
thf(mem_c_2Ereal__topology_2Efsigma,axiom,(mem@c_2Ereal__topology_2Efsigma@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Egdelta,type,(c_2Ereal__topology_2Egdelta:$i)).
thf(mem_c_2Ereal__topology_2Egdelta,axiom,(mem@c_2Ereal__topology_2Egdelta@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Ehausdist,type,(c_2Ereal__topology_2Ehausdist:$i)).
thf(mem_c_2Ereal__topology_2Ehausdist,axiom,(mem@c_2Ereal__topology_2Ehausdist@(arr@(ty_2Epair_2Eprod@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@ty_2Erealax_2Ereal))).
thf(tp_c_2Ereal__topology_2Ehomeomorphic,type,(c_2Ereal__topology_2Ehomeomorphic:$i)).
thf(mem_c_2Ereal__topology_2Ehomeomorphic,axiom,(mem@c_2Ereal__topology_2Ehomeomorphic@(arr@(arr@ty_2Erealax_2Ereal@bool)@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)))).
thf(tp_c_2Ereal__topology_2Ehomeomorphism,type,(c_2Ereal__topology_2Ehomeomorphism:$i)).
thf(mem_c_2Ereal__topology_2Ehomeomorphism,axiom,(mem@c_2Ereal__topology_2Ehomeomorphism@(arr@(ty_2Epair_2Eprod@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(arr@(ty_2Epair_2Eprod@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@bool)))).
thf(tp_c_2Ereal__topology_2Ein__direction,type,(c_2Ereal__topology_2Ein__direction:$i)).
thf(mem_c_2Ereal__topology_2Ein__direction,axiom,(mem@c_2Ereal__topology_2Ein__direction@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@(ty_2Ereal__topology_2Enet@ty_2Erealax_2Ereal))))).
thf(tp_c_2Ereal__topology_2Eindependent,type,(c_2Ereal__topology_2Eindependent:$i)).
thf(mem_c_2Ereal__topology_2Eindependent,axiom,(mem@c_2Ereal__topology_2Eindependent@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Einfsum,type,(c_2Ereal__topology_2Einfsum:$i)).
thf(mem_c_2Ereal__topology_2Einfsum,axiom,(mem@c_2Ereal__topology_2Einfsum@(arr@(arr@ty_2Enum_2Enum@bool)@(arr@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@ty_2Erealax_2Ereal)))).
thf(tp_c_2Ereal__topology_2Einterior,type,(c_2Ereal__topology_2Einterior:$i)).
thf(mem_c_2Ereal__topology_2Einterior,axiom,(mem@c_2Ereal__topology_2Einterior@(arr@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Eis__interval,type,(c_2Ereal__topology_2Eis__interval:$i)).
thf(mem_c_2Ereal__topology_2Eis__interval,axiom,(mem@c_2Ereal__topology_2Eis__interval@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Eisnet,type,(c_2Ereal__topology_2Eisnet:(del>(del>$i)))).
thf(mem_c_2Ereal__topology_2Eisnet,axiom,( ! [A_27a:del] : ( ! [A_27b:del] : (mem@(c_2Ereal__topology_2Eisnet@A_27a@A_27b)@(arr@(arr@A_27a@(arr@A_27b@bool))@bool))))).
thf(tp_c_2Ereal__topology_2Elim__def,type,(c_2Ereal__topology_2Elim__def:(del>$i))).
thf(mem_c_2Ereal__topology_2Elim__def,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Elim__def@A_27a)@(arr@(ty_2Ereal__topology_2Enet@A_27a)@(arr@(arr@A_27a@ty_2Erealax_2Ereal)@ty_2Erealax_2Ereal))))).
thf(tp_c_2Ereal__topology_2Elimit__point__of,type,(c_2Ereal__topology_2Elimit__point__of:$i)).
thf(mem_c_2Ereal__topology_2Elimit__point__of,axiom,(mem@c_2Ereal__topology_2Elimit__point__of@(arr@ty_2Erealax_2Ereal@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)))).
thf(tp_c_2Ereal__topology_2Elinear,type,(c_2Ereal__topology_2Elinear:$i)).
thf(mem_c_2Ereal__topology_2Elinear,axiom,(mem@c_2Ereal__topology_2Elinear@(arr@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@bool))).
thf(tp_c_2Ereal__topology_2Elocally,type,(c_2Ereal__topology_2Elocally:$i)).
thf(mem_c_2Ereal__topology_2Elocally,axiom,(mem@c_2Ereal__topology_2Elocally@(arr@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)))).
thf(tp_c_2Ereal__topology_2Emidpoint,type,(c_2Ereal__topology_2Emidpoint:$i)).
thf(mem_c_2Ereal__topology_2Emidpoint,axiom,(mem@c_2Ereal__topology_2Emidpoint@(arr@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@ty_2Erealax_2Ereal))).
thf(stp_fo_c_2Ereal__topology_2Emidpoint,type,(fo__c_2Ereal__topology_2Emidpoint:(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal))).
thf(stp_eq_fo_c_2Ereal__topology_2Emidpoint,axiom,( ! [X0:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ((inj__ty_2Erealax_2Ereal @ (fo__c_2Ereal__topology_2Emidpoint @ X0)) = (ap @ c_2Ereal__topology_2Emidpoint @ (inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X0))))).
thf(tp_c_2Ereal__topology_2Emk__net,type,(c_2Ereal__topology_2Emk__net:(del>$i))).
thf(mem_c_2Ereal__topology_2Emk__net,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Emk__net@A_27a)@(arr@(arr@A_27a@(arr@A_27a@bool))@(ty_2Ereal__topology_2Enet@A_27a))))).
thf(tp_c_2Ereal__topology_2Enetlimit,type,(c_2Ereal__topology_2Enetlimit:(del>$i))).
thf(mem_c_2Ereal__topology_2Enetlimit,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Enetlimit@A_27a)@(arr@(ty_2Ereal__topology_2Enet@A_27a)@A_27a)))).
thf(tp_c_2Ereal__topology_2Enetord,type,(c_2Ereal__topology_2Enetord:(del>$i))).
thf(mem_c_2Ereal__topology_2Enetord,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Enetord@A_27a)@(arr@(ty_2Ereal__topology_2Enet@A_27a)@(arr@A_27a@(arr@A_27a@bool)))))).
thf(tp_c_2Ereal__topology_2Eopen__segment,type,(c_2Ereal__topology_2Eopen__segment:$i)).
thf(mem_c_2Ereal__topology_2Eopen__segment,axiom,(mem@c_2Ereal__topology_2Eopen__segment@(arr@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Epairwise,type,(c_2Ereal__topology_2Epairwise:(del>$i))).
thf(mem_c_2Ereal__topology_2Epairwise,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Epairwise@A_27a)@(arr@(arr@A_27a@(arr@A_27a@bool))@(arr@(arr@A_27a@bool)@bool))))).
thf(tp_c_2Ereal__topology_2Epermutes,type,(c_2Ereal__topology_2Epermutes:(del>$i))).
thf(mem_c_2Ereal__topology_2Epermutes,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Epermutes@A_27a)@(arr@(arr@A_27a@A_27a)@(arr@(arr@A_27a@bool)@bool))))).
thf(stp_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,type,(tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum : $tType)).
thf(stp_inj_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,type,(inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum : (tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum > $i))).
thf(stp_surj_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,type,(surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum : ($i > tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum))).
thf(stp_inj_surj_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,( ! [X:tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum] : ((surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ (inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ X)) = X))).
thf(stp_inj_mem_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,( ! [X:tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum] : (mem @ (inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ X) @ (ty_2Ereal__topology_2Enet@ty_2Enum_2Enum)))).
thf(stp_iso_mem_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,( ! [X:$i] : ((mem @ X @ (ty_2Ereal__topology_2Enet@ty_2Enum_2Enum)) => (X = (inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ (surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ X)))))).
thf(tp_c_2Ereal__topology_2Esequentially,type,(c_2Ereal__topology_2Esequentially:$i)).
thf(mem_c_2Ereal__topology_2Esequentially,axiom,(mem@c_2Ereal__topology_2Esequentially@(ty_2Ereal__topology_2Enet@ty_2Enum_2Enum))).
thf(tp_c_2Ereal__topology_2Esetdist,type,(c_2Ereal__topology_2Esetdist:$i)).
thf(mem_c_2Ereal__topology_2Esetdist,axiom,(mem@c_2Ereal__topology_2Esetdist@(arr@(ty_2Epair_2Eprod@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@ty_2Erealax_2Ereal))).
thf(tp_c_2Ereal__topology_2Espan,type,(c_2Ereal__topology_2Espan:$i)).
thf(mem_c_2Ereal__topology_2Espan,axiom,(mem@c_2Ereal__topology_2Espan@(arr@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Esphere,type,(c_2Ereal__topology_2Esphere:$i)).
thf(mem_c_2Ereal__topology_2Esphere,axiom,(mem@c_2Ereal__topology_2Esphere@(arr@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@bool)))).
thf(tp_c_2Ereal__topology_2Esubspace,type,(c_2Ereal__topology_2Esubspace:$i)).
thf(mem_c_2Ereal__topology_2Esubspace,axiom,(mem@c_2Ereal__topology_2Esubspace@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool))).
thf(tp_c_2Ereal__topology_2Esubtopology,type,(c_2Ereal__topology_2Esubtopology:(del>$i))).
thf(mem_c_2Ereal__topology_2Esubtopology,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Esubtopology@A_27a)@(arr@(ty_2Etopology_2Etopology@A_27a)@(arr@(arr@A_27a@bool)@(ty_2Etopology_2Etopology@A_27a)))))).
thf(tp_c_2Ereal__topology_2Esummable,type,(c_2Ereal__topology_2Esummable:$i)).
thf(mem_c_2Ereal__topology_2Esummable,axiom,(mem@c_2Ereal__topology_2Esummable@(arr@(arr@ty_2Enum_2Enum@bool)@(arr@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@bool)))).
thf(tp_c_2Ereal__topology_2Esums,type,(c_2Ereal__topology_2Esums:$i)).
thf(mem_c_2Ereal__topology_2Esums,axiom,(mem@c_2Ereal__topology_2Esums@(arr@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@(arr@(arr@ty_2Enum_2Enum@bool)@bool))))).
thf(tp_c_2Ereal__topology_2Etrivial__limit,type,(c_2Ereal__topology_2Etrivial__limit:(del>$i))).
thf(mem_c_2Ereal__topology_2Etrivial__limit,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Etrivial__limit@A_27a)@(arr@(ty_2Ereal__topology_2Enet@A_27a)@bool)))).
thf(tp_c_2Ereal__topology_2Euniformly__continuous__on,type,(c_2Ereal__topology_2Euniformly__continuous__on:$i)).
thf(mem_c_2Ereal__topology_2Euniformly__continuous__on,axiom,(mem@c_2Ereal__topology_2Euniformly__continuous__on@(arr@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)))).
thf(tp_c_2Ereal__topology_2Ewithin,type,(c_2Ereal__topology_2Ewithin:(del>$i))).
thf(mem_c_2Ereal__topology_2Ewithin,axiom,( ! [A_27a:del] : (mem@(c_2Ereal__topology_2Ewithin@A_27a)@(arr@(ty_2Ereal__topology_2Enet@A_27a)@(arr@(arr@A_27a@bool)@(ty_2Ereal__topology_2Enet@A_27a)))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__SQUARE__ABS,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y)))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__EQ__SQUARE__ABS,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y)))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__HALF,lemma,(( ! [V0e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V0e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) <=> (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 @ V0e))))) & (( ! [V1e:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) = V1e)) & ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V2e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) = V2e))))).
thf(conj_thm_2Ereal__topology_2EFINITE__SUBSET__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@V2t)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V1s)))) <=> (? [V3s_27:$i] : ((mem@V3s_27@(arr@A_27a@bool)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3s_27)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V3s_27)@V1s)) & (V2t = (ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V3s_27)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EEXISTS__FINITE__SUBSET__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@(arr@A_27a@bool)@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => ((? [V3t:$i] : ((mem@V3t@(arr@A_27a@bool)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V3t)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27b@A_27a)@V1f)@V2s))) & (p@(ap@V0P@V3t)))))) <=> (? [V4t:$i] : ((mem@V4t@(arr@A_27b@bool)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V4t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@V4t)@V2s)) & (p@(ap@V0P@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27b@A_27a)@V1f)@V4t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EFORALL__FINITE__SUBSET__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@(arr@A_27a@bool)@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@A_27a)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => (( ! [V3t:$i] : ((mem@V3t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V3t)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27b@A_27a)@V1f)@V2s)))) => (p@(ap@V0P@V3t))))) <=> ( ! [V4t:$i] : ((mem@V4t@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V4t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@V4t)@V2s))) => (p@(ap@V0P@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27b@A_27a)@V1f)@V4t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EFORALL__IN__GSPEC,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [A_27d:del]: ( ! [A_27e:del]: ( ! [A_27f:del]: ( ! [A_27g:del]: ( ! [V0Q:$i] : ((mem@V0Q@(arr@A_27b@bool)) => (( ! [V1P:$i] : ((mem@V1P@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@A_27b)) => (( ! [V3z:$i] : ((mem@V3z@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3z)@(ap@(c_2Epred__set_2EGSPEC@A_27b@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@(ap@V2f@V4x))@(ap@V1P@V4x))))))) => (p@(ap@V0Q@V3z))))) <=> ( ! [V5x:$i] : ((mem@V5x@A_27a) => ((p@(ap@V1P@V5x)) => (p@(ap@V0Q@(ap@V2f@V5x))))))))))) & (( ! [V6P:$i] : ((mem@V6P@(arr@A_27c@(arr@A_27d@bool))) => ( ! [V7f:$i] : ((mem@V7f@(arr@A_27c@(arr@A_27d@A_27b))) => (( ! [V8z:$i] : ((mem@V8z@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V8z)@(ap@(c_2Epred__set_2EGSPEC@A_27b@(ty_2Epair_2Eprod@A_27c@A_27d))@(ap@(c_2Epair_2EUNCURRY@A_27c@A_27d@(ty_2Epair_2Eprod@A_27b@bool))@(lam@A_27c@(^ [V9x:$i] : (lam@A_27d@(^ [V10y:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@(ap@(ap@V7f@V9x)@V10y))@(ap@(ap@V6P@V9x)@V10y)))))))))) => (p@(ap@V0Q@V8z))))) <=> ( ! [V11x:$i] : ((mem@V11x@A_27c) => ( ! [V12y:$i] : ((mem@V12y@A_27d) => ((p@(ap@(ap@V6P@V11x)@V12y)) => (p@(ap@V0Q@(ap@(ap@V7f@V11x)@V12y))))))))))))) & ( ! [V13P:$i] : ((mem@V13P@(arr@A_27e@(arr@A_27f@(arr@A_27g@bool)))) => ( ! [V14f:$i] : ((mem@V14f@(arr@A_27e@(arr@A_27f@(arr@A_27g@A_27b)))) => (( ! [V15z:$i] : ((mem@V15z@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V15z)@(ap@(c_2Epred__set_2EGSPEC@A_27b@(ty_2Epair_2Eprod@A_27e@(ty_2Epair_2Eprod@A_27f@A_27g)))@(ap@(c_2Epair_2EUNCURRY@A_27e@(ty_2Epair_2Eprod@A_27f@A_27g)@(ty_2Epair_2Eprod@A_27b@bool))@(lam@A_27e@(^ [V16w:$i] : (ap@(c_2Epair_2EUNCURRY@A_27f@A_27g@(ty_2Epair_2Eprod@A_27b@bool))@(lam@A_27f@(^ [V17x:$i] : (lam@A_27g@(^ [V18y:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@(ap@(ap@(ap@V14f@V16w)@V17x)@V18y))@(ap@(ap@(ap@V13P@V16w)@V17x)@V18y))))))))))))) => (p@(ap@V0Q@V15z))))) <=> ( ! [V19w:$i] : ((mem@V19w@A_27e) => ( ! [V20x:$i] : ((mem@V20x@A_27f) => ( ! [V21y:$i] : ((mem@V21y@A_27g) => ((p@(ap@(ap@(ap@V13P@V19w)@V20x)@V21y)) => (p@(ap@V0Q@(ap@(ap@(ap@V14f@V19w)@V20x)@V21y))))))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EEXISTS__IN__GSPEC,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [A_27d:del]: ( ! [A_27e:del]: ( ! [A_27f:del]: ( ! [A_27g:del]: ( ! [V0Q:$i] : ((mem@V0Q@(arr@A_27b@bool)) => (( ! [V1P:$i] : ((mem@V1P@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@A_27b)) => ((? [V3z:$i] : ((mem@V3z@A_27b) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3z)@(ap@(c_2Epred__set_2EGSPEC@A_27b@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@(ap@V2f@V4x))@(ap@V1P@V4x))))))) & (p@(ap@V0Q@V3z))))) <=> (? [V5x:$i] : ((mem@V5x@A_27a) & ((p@(ap@V1P@V5x)) & (p@(ap@V0Q@(ap@V2f@V5x))))))))))) & (( ! [V6P:$i] : ((mem@V6P@(arr@A_27c@(arr@A_27d@bool))) => ( ! [V7f:$i] : ((mem@V7f@(arr@A_27c@(arr@A_27d@A_27b))) => ((? [V8z:$i] : ((mem@V8z@A_27b) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V8z)@(ap@(c_2Epred__set_2EGSPEC@A_27b@(ty_2Epair_2Eprod@A_27c@A_27d))@(ap@(c_2Epair_2EUNCURRY@A_27c@A_27d@(ty_2Epair_2Eprod@A_27b@bool))@(lam@A_27c@(^ [V9x:$i] : (lam@A_27d@(^ [V10y:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@(ap@(ap@V7f@V9x)@V10y))@(ap@(ap@V6P@V9x)@V10y)))))))))) & (p@(ap@V0Q@V8z))))) <=> (? [V11x:$i] : ((mem@V11x@A_27c) & (? [V12y:$i] : ((mem@V12y@A_27d) & ((p@(ap@(ap@V6P@V11x)@V12y)) & (p@(ap@V0Q@(ap@(ap@V7f@V11x)@V12y))))))))))))) & ( ! [V13P:$i] : ((mem@V13P@(arr@A_27e@(arr@A_27f@(arr@A_27g@bool)))) => ( ! [V14f:$i] : ((mem@V14f@(arr@A_27e@(arr@A_27f@(arr@A_27g@A_27b)))) => ((? [V15z:$i] : ((mem@V15z@A_27b) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V15z)@(ap@(c_2Epred__set_2EGSPEC@A_27b@(ty_2Epair_2Eprod@A_27e@(ty_2Epair_2Eprod@A_27f@A_27g)))@(ap@(c_2Epair_2EUNCURRY@A_27e@(ty_2Epair_2Eprod@A_27f@A_27g)@(ty_2Epair_2Eprod@A_27b@bool))@(lam@A_27e@(^ [V16w:$i] : (ap@(c_2Epair_2EUNCURRY@A_27f@A_27g@(ty_2Epair_2Eprod@A_27b@bool))@(lam@A_27f@(^ [V17x:$i] : (lam@A_27g@(^ [V18y:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@(ap@(ap@(ap@V14f@V16w)@V17x)@V18y))@(ap@(ap@(ap@V13P@V16w)@V17x)@V18y))))))))))))) & (p@(ap@V0Q@V15z))))) <=> (? [V19w:$i] : ((mem@V19w@A_27e) & (? [V20x:$i] : ((mem@V20x@A_27f) & (? [V21y:$i] : ((mem@V21y@A_27g) & ((p@(ap@(ap@(ap@V13P@V19w)@V20x)@V21y)) & (p@(ap@V0Q@(ap@(ap@(ap@V14f@V19w)@V20x)@V21y))))))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EEMPTY__BIGUNION,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@(arr@A_27a@bool)@bool)) => (((ap@(c_2Epred__set_2EBIGUNION@A_27a)@V0s) = (c_2Epred__set_2EEMPTY@A_27a)) <=> ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V1t)@V0s)) => (V1t = (c_2Epred__set_2EEMPTY@A_27a)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LT__RNEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1y)))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Enum_2Enum)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => (? [V2a:tp__ty_2Enum_2Enum] : ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@V0f@V3x))@(inj__ty_2Enum_2Enum @ V2a)))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__BOUNDS__LT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1k:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1k)))@(inj__ty_2Erealax_2Ereal @ V0x))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1k)))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1k))))))).
thf(conj_thm_2Ereal__topology_2EBIGUNION__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@bool))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((ap@(c_2Epred__set_2EBIGUNION@A_27b)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@(arr@A_27b@bool))@V0f)@V1s)) = (ap@(c_2Epred__set_2EGSPEC@A_27b@A_27b)@(lam@A_27b@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@V2y)@(ap@(c_2Ebool_2E_3F@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V2y)@(ap@V0f@V3x)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBIGINTER__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@bool))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((ap@(c_2Epred__set_2EBIGINTER@A_27b)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@(arr@A_27b@bool))@V0f)@V1s)) = (ap@(c_2Epred__set_2EGSPEC@A_27b@A_27b)@(lam@A_27b@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@V2y)@(ap@(c_2Ebool_2E_21@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V2y)@(ap@V0f@V3x)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__LMUL1,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z))))))))).
thf(conj_thm_2Ereal__topology_2ELE__EXISTS,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))) <=> (? [V2d:tp__ty_2Enum_2Enum] : (V1n = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V2d))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDS__LINEAR,lemma,( ! [V0A:tp__ty_2Enum_2Enum] : ( ! [V1B:tp__ty_2Enum_2Enum] : ( ! [V2C:tp__ty_2Enum_2Enum] : (( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@c_2Earithmetic_2E_2A@(inj__ty_2Enum_2Enum @ V0A))@(inj__ty_2Enum_2Enum @ V3n)))@(ap@(ap@c_2Earithmetic_2E_2B@(ap@(ap@c_2Earithmetic_2E_2A@(inj__ty_2Enum_2Enum @ V1B))@(inj__ty_2Enum_2Enum @ V3n)))@(inj__ty_2Enum_2Enum @ V2C))))) <=> (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0A))@(inj__ty_2Enum_2Enum @ V1B)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDS__LINEAR__0,lemma,( ! [V0A:tp__ty_2Enum_2Enum] : ( ! [V1B:tp__ty_2Enum_2Enum] : (( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(ap@c_2Earithmetic_2E_2A@(inj__ty_2Enum_2Enum @ V0A))@(inj__ty_2Enum_2Enum @ V2n)))@(inj__ty_2Enum_2Enum @ V1B)))) <=> (V0A = fo__c_2Enum_2E0))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__BETWEEN,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))).
thf(conj_thm_2Ereal__topology_2EWLOG__LE,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@bool))) => ((( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) <=> (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V2n))@(inj__ty_2Enum_2Enum @ V1m)))))) & ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))))))) => ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__POWERSET,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) => (p@(ap@(c_2Epred__set_2EFINITE@(arr@A_27a@bool))@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V1t:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@V1t)@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V1t)@V0s)))))))))))).
thf(conj_thm_2Ereal__topology_2ELE__ADD,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Ereal__topology_2ELE__ADDR,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1n))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Ereal__topology_2EADD__SUB2,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2D@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))@(inj__ty_2Enum_2Enum @ V0m))) = V1n)))).
thf(conj_thm_2Ereal__topology_2EADD__SUBR2,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V0m))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))) = fo__c_2Enum_2E0)))).
thf(conj_thm_2Ereal__topology_2EADD__SUBR,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V1n))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n)))) = fo__c_2Enum_2E0)))).
thf(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE__EQ,lemma,( ! [V0R:$i] : ((mem@V0R@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@bool))) => ((( ! [V1x:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V1x))@(inj__ty_2Enum_2Enum @ V1x)))) & ( ! [V2x:tp__ty_2Enum_2Enum] : ( ! [V3y:tp__ty_2Enum_2Enum] : ( ! [V4z:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V2x))@(inj__ty_2Enum_2Enum @ V3y))) & (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V3y))@(inj__ty_2Enum_2Enum @ V4z)))) => (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V2x))@(inj__ty_2Enum_2Enum @ V4z)))))))) => (( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))) => (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n)))))) <=> ( ! [V7n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V7n))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V7n)))))))))).
thf(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE,lemma,( ! [V0R:$i] : ((mem@V0R@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@bool))) => ((( ! [V1x:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V1x))@(inj__ty_2Enum_2Enum @ V1x)))) & (( ! [V2x:tp__ty_2Enum_2Enum] : ( ! [V3y:tp__ty_2Enum_2Enum] : ( ! [V4z:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V2x))@(inj__ty_2Enum_2Enum @ V3y))) & (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V3y))@(inj__ty_2Enum_2Enum @ V4z)))) => (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V2x))@(inj__ty_2Enum_2Enum @ V4z))))))) & ( ! [V5n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V5n))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V5n))))))) => ( ! [V6m:tp__ty_2Enum_2Enum] : ( ! [V7n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V6m))@(inj__ty_2Enum_2Enum @ V7n))) => (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V6m))@(inj__ty_2Enum_2Enum @ V7n)))))))))).
thf(conj_thm_2Ereal__topology_2ELAMBDA__PAIR,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@(arr@A_27b@A_27c))) => ((ap@(c_2Epair_2EUNCURRY@A_27a@A_27b@A_27c)@(lam@A_27a@(^ [V1x:$i] : (lam@A_27b@(^ [V2y:$i] : (ap@(ap@V0P@V1x)@V2y)))))) = (lam@(ty_2Epair_2Eprod@A_27a@A_27b)@(^ [V3p:$i] : (ap@(ap@V0P@(ap@(c_2Epair_2EFST@A_27a@A_27b)@V3p))@(ap@(c_2Epair_2ESND@A_27a@A_27b)@V3p))))))))))).
thf(conj_thm_2Ereal__topology_2ENOT__EQ,lemma,( ! [A_27a:del]: ( ! [V0a:$i] : ((mem@V0a@A_27a) => ( ! [V1b:$i] : ((mem@V1b@A_27a) => ((~~ (V0a = V1b)) <=> (~~ (V0a = V1b))))))))).
thf(conj_thm_2Ereal__topology_2EABS__LE__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Ereal__topology_2EREAL__OF__NUM__GE,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V0m)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V1n)))) <=> (p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Ereal__topology_2EPOWERSET__CLAUSES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: (((ap@(c_2Epred__set_2EGSPEC@(arr@A_27b@bool)@(arr@A_27b@bool))@(lam@(arr@A_27b@bool)@(^ [V0s:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27b@bool)@bool)@V0s)@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@V0s)@(c_2Epred__set_2EEMPTY@A_27b)))))) = (ap@(ap@(c_2Epred__set_2EINSERT@(arr@A_27b@bool))@(c_2Epred__set_2EEMPTY@A_27b))@(c_2Epred__set_2EEMPTY@(arr@A_27b@bool)))) & ( ! [V1a:$i] : ((mem@V1a@A_27a) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ((ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V3s:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@V3s)@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V3s)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1a)@V2t)))))) = (ap@(ap@(c_2Epred__set_2EUNION@(arr@A_27a@bool))@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V4s:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@V4s)@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V4s)@V2t))))))@(ap@(ap@(c_2Epred__set_2EIMAGE@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V5s:$i] : (ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1a)@V5s))))@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V6s:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@V6s)@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V6s)@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LT__LCANCEL__IMP,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))))))).
thf(conj_thm_2Ereal__topology_2ESIMPLE__IMAGE__GEN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27a@bool)) => ((ap@(c_2Epred__set_2EGSPEC@A_27b@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@(ap@V0f@V2x))@(ap@V1P@V2x))))) = (ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@(ap@(c_2Epred__set_2EGSPEC@A_27a@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27a@bool)@V3x)@(ap@V1P@V3x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__BIGUNION,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@(arr@A_27a@bool)@bool)) => ( ! [V1g:$i] : ((mem@V1g@(arr@(arr@A_27a@bool)@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@A_27a@bool))@V0f)@V1g)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@V0f))@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@V1g)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LT__POW2,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Epow@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V0n)))))).
thf(conj_thm_2Ereal__topology_2EFUN__IN__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => (p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@(ap@V0f@V2x))@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V1s))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__ANTISYM__EQ,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V0s)@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V1t)@V0s))) <=> (V0s = V1t)))))))).
thf(conj_thm_2Ereal__topology_2EDIFF__BIGINTER,lemma,( ! [A_27a:del]: ( ! [V0u:$i] : ((mem@V0u@(arr@A_27a@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@(arr@A_27a@bool)@bool)) => ((ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@V0u)@(ap@(c_2Epred__set_2EBIGINTER@A_27a)@V1s)) = (ap@(c_2Epred__set_2EBIGUNION@A_27a)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V2t:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@V0u)@V2t))@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V2t)@V1s))))))))))))).
thf(conj_thm_2Ereal__topology_2EBIGINTER__BIGUNION,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@(arr@A_27a@bool)@bool)) => ((ap@(c_2Epred__set_2EBIGINTER@A_27a)@V0s) = (ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@(c_2Epred__set_2EUNIV@A_27a))@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V1t:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@(c_2Epred__set_2EUNIV@A_27a))@V1t))@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V1t)@V0s)))))))))))).
thf(conj_thm_2Ereal__topology_2EBIGUNION__BIGINTER,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@(arr@A_27a@bool)@bool)) => ((ap@(c_2Epred__set_2EBIGUNION@A_27a)@V0s) = (ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@(c_2Epred__set_2EUNIV@A_27a))@(ap@(c_2Epred__set_2EBIGINTER@A_27a)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V1t:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@(c_2Epred__set_2EUNIV@A_27a))@V1t))@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V1t)@V0s)))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__POW__1__LE,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__POW__LE__1,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V1x))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LT__INV2,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((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 @ V0x))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0x)))))))).
thf(conj_thm_2Ereal__topology_2EREAL__INV__1__LE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (((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 @ V0x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0x))))))).
thf(conj_thm_2Ereal__topology_2ESUM__GP__BASIC,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V1n)))@(lam@ty_2Enum_2Enum@(^ [V2i:$i] : (ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@V2i)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V1n))))))))).
thf(conj_thm_2Ereal__topology_2ESUM__GP__MULTIPLIED,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@V3i)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Enum_2Enum @ V1m)))@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2n))))))))))).
thf(conj_thm_2Ereal__topology_2ESUM__GP,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@V3i))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V2n))@(inj__ty_2Enum_2Enum @ V1m)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@c_2Ereal_2Ereal__of__num@(ap@(ap@c_2Earithmetic_2E_2D@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V1m))))@(ap@(ap@c_2Ereal_2E_2F@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Enum_2Enum @ V1m)))@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2n)))))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V0x))))))))))).
thf(stp_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,type,(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o : $tType)).
thf(stp_inj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,type,(inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o : (tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o > $i))).
thf(stp_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,type,(surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o : ($i > tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o))).
thf(stp_inj_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,( ! [X:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o] : ((surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ (inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ X)) = X))).
thf(stp_inj_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,( ! [X:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o] : (mem @ (inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ X) @ (ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool)))).
thf(stp_iso_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,( ! [X:$i] : ((mem @ X @ (ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool)) => (X = (inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ (surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o @ X)))))).
thf(conj_thm_2Ereal__topology_2ESUMS__SYM,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V4y:$i] : (lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V4y)@V5x))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4y)@V1t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V0s))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUM__ABS__TRIANGLE,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V0s)@(lam@A_27a@(^ [V3a:$i] : (ap@c_2Ereal_2Eabs@(ap@V1f@V3a))))))@(inj__ty_2Erealax_2Ereal @ V2b)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V0s)@V1f)))@(inj__ty_2Erealax_2Ereal @ V2b))))))))))).
thf(conj_thm_2Ereal__topology_2EIMAGE__SING,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1a:$i] : ((mem@V1a@A_27a) => ((ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1a)@(c_2Epred__set_2EEMPTY@A_27a))) = (ap@(ap@(c_2Epred__set_2EINSERT@A_27b)@(ap@V0f@V1a))@(c_2Epred__set_2EEMPTY@A_27b)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__WLOG__LE,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))) <=> (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V1x)))))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y))) => (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y))))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V6y))))))))).
thf(ax_thm_2Ereal__topology_2Epairwise,axiom,( ! [A_27a:del]: ( ! [V0r:$i] : ((mem@V0r@(arr@A_27a@(arr@A_27a@bool))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27a)@V0r)@V1s)) <=> ( ! [V2x:$i] : ((mem@V2x@A_27a) => ( ! [V3y:$i] : ((mem@V3y@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3y)@V1s)) & (~ (V2x = V3y)))) => (p@(ap@(ap@V0r@V2x)@V3y)))))))))))))).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__EMPTY,lemma,( ! [A_27a:del]: ( ! [V0r:$i] : ((mem@V0r@(arr@A_27a@(arr@A_27a@bool))) => ((p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27a)@V0r)@(c_2Epred__set_2EEMPTY@A_27a))) <=> $true))))).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__SING,lemma,( ! [A_27a:del]: ( ! [V0r:$i] : ((mem@V0r@(arr@A_27a@(arr@A_27a@bool))) => ( ! [V1x:$i] : ((mem@V1x@A_27a) => ((p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27a)@V0r)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1x)@(c_2Epred__set_2EEMPTY@A_27a)))) <=> $true))))))).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__MONO,lemma,( ! [A_27a:del]: ( ! [V0r:$i] : ((mem@V0r@(arr@A_27a@(arr@A_27a@bool))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27a)@V0r)@V1s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2t)@V1s))) => (p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27a)@V0r)@V2t))))))))))).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__INSERT,lemma,( ! [A_27a:del]: ( ! [V0r:$i] : ((mem@V0r@(arr@A_27a@(arr@A_27a@bool))) => ( ! [V1x:$i] : ((mem@V1x@A_27a) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27a)@V0r)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1x)@V2s))) <=> (( ! [V3y:$i] : ((mem@V3y@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3y)@V2s)) & (~~ (V3y = V1x))) => ((p@(ap@(ap@V0r@V1x)@V3y)) & (p@(ap@(ap@V0r@V3y)@V1x)))))) & (p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27a)@V0r)@V2s)))))))))))).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27b@bool)) => ( ! [V1r:$i] : ((mem@V1r@(arr@A_27a@(arr@A_27a@bool))) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27b@A_27a)) => ((p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27a)@V1r)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27b@A_27a)@V2f)@V0s))) <=> (p@(ap@(ap@(c_2Ereal__topology_2Epairwise@A_27b)@(lam@A_27b@(^ [V3x:$i] : (lam@A_27b@(^ [V4y:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@A_27a)@(ap@V2f@V3x))@(ap@V2f@V4y))))@(ap@(ap@V1r@(ap@V2f@V3x))@(ap@V2f@V4y))))))))@V0s)))))))))))).
thf(ax_thm_2Ereal__topology_2Epermutes,axiom,( ! [A_27a:del]: ( ! [V0p:$i] : ((mem@V0p@(arr@A_27a@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Epermutes@A_27a)@V0p)@V1s)) <=> (( ! [V2x:$i] : ((mem@V2x@A_27a) => ((~ (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s))) => ((ap@V0p@V2x) = V2x)))) & ( ! [V3y:$i] : ((mem@V3y@A_27a) => (p@(ap@(c_2Ebool_2E_3F_21@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Emin_2E_3D@A_27a)@(ap@V0p@V4x))@V3y))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPERMUTES__IMAGE,lemma,( ! [A_27a:del]: ( ! [V0p:$i] : ((mem@V0p@(arr@A_27a@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Epermutes@A_27a)@V0p)@V1s)) => ((ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27a)@V0p)@V1s) = V1s)))))))).
thf(conj_thm_2Ereal__topology_2EPERMUTES__INJECTIVE,lemma,( ! [A_27a:del]: ( ! [V0p:$i] : ((mem@V0p@(arr@A_27a@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Epermutes@A_27a)@V0p)@V1s)) => ( ! [V2x:$i] : ((mem@V2x@A_27a) => ( ! [V3y:$i] : ((mem@V3y@A_27a) => (((ap@V0p@V2x) = (ap@V0p@V3y)) <=> (V2x = V3y))))))))))))).
thf(conj_thm_2Ereal__topology_2EEXISTS__IN__INSERT,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1a:$i] : ((mem@V1a@A_27a) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((? [V3x:$i] : ((mem@V3x@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1a)@V2s))) & (p@(ap@V0P@V3x))))) <=> ((p@(ap@V0P@V1a)) | (? [V4x:$i] : ((mem@V4x@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2s)) & (p@(ap@V0P@V4x))))))))))))))).
thf(conj_thm_2Ereal__topology_2EDEPENDENT__CHOICE__FIXED,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Enum_2Enum@(arr@A_27a@bool))) => ( ! [V1R:$i] : ((mem@V1R@(arr@ty_2Enum_2Enum@(arr@A_27a@(arr@A_27a@bool)))) => ( ! [V2a:$i] : ((mem@V2a@A_27a) => (((p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@V2a)) & ( ! [V3n:tp__ty_2Enum_2Enum] : ( ! [V4x:$i] : ((mem@V4x@A_27a) => ((p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V3n))@V4x)) => (? [V5y:$i] : ((mem@V5y@A_27a) & ((p@(ap@(ap@V0P@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V3n)))@V5y)) & (p@(ap@(ap@(ap@V1R@(inj__ty_2Enum_2Enum @ V3n))@V4x)@V5y)))))))))) => (? [V6f:$i] : ((mem@V6f@(arr@ty_2Enum_2Enum@A_27a)) & (((ap@V6f@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)) = V2a) & (( ! [V7n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V7n))@(ap@V6f@(inj__ty_2Enum_2Enum @ V7n))))) & ( ! [V8n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(ap@V1R@(inj__ty_2Enum_2Enum @ V8n))@(ap@V6f@(inj__ty_2Enum_2Enum @ V8n)))@(ap@V6f@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V8n))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EDEPENDENT__CHOICE,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Enum_2Enum@(arr@A_27a@bool))) => ( ! [V1R:$i] : ((mem@V1R@(arr@ty_2Enum_2Enum@(arr@A_27a@(arr@A_27a@bool)))) => (((? [V2a:$i] : ((mem@V2a@A_27a) & (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@V2a)))) & ( ! [V3n:tp__ty_2Enum_2Enum] : ( ! [V4x:$i] : ((mem@V4x@A_27a) => ((p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V3n))@V4x)) => (? [V5y:$i] : ((mem@V5y@A_27a) & ((p@(ap@(ap@V0P@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V3n)))@V5y)) & (p@(ap@(ap@(ap@V1R@(inj__ty_2Enum_2Enum @ V3n))@V4x)@V5y)))))))))) => (? [V6f:$i] : ((mem@V6f@(arr@ty_2Enum_2Enum@A_27a)) & (( ! [V7n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V7n))@(ap@V6f@(inj__ty_2Enum_2Enum @ V7n))))) & ( ! [V8n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(ap@V1R@(inj__ty_2Enum_2Enum @ V8n))@(ap@V6f@(inj__ty_2Enum_2Enum @ V8n)))@(ap@V6f@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V8n)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBIGUNION__MONO__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@A_27b@bool))) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@(arr@A_27b@bool))) => (( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V0s)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@(ap@V1f@V3x))@(ap@V2g@V3x)))))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@(ap@(c_2Epred__set_2EBIGUNION@A_27b)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@(arr@A_27b@bool))@V1f)@V0s)))@(ap@(c_2Epred__set_2EBIGUNION@A_27b)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@(arr@A_27b@bool))@V2g)@V0s)))))))))))))).
thf(conj_thm_2Ereal__topology_2EBIGUNION__MONO,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@(arr@A_27a@bool)@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@(arr@A_27a@bool)@bool)) => (( ! [V2x:$i] : ((mem@V2x@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V2x)@V0s)) => (? [V3y:$i] : ((mem@V3y@(arr@A_27a@bool)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V3y)@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2x)@V3y)))))))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@V0s))@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__WLOG__LT,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V1x)))) & (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))) <=> (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V2x)))))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5y))) => (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5y)))))))) => ( ! [V6x:tp__ty_2Erealax_2Ereal] : ( ! [V7y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V6x))@(inj__ty_2Erealax_2Ereal @ V7y))))))))).
thf(ax_thm_2Ereal__topology_2Edist,axiom,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))))))).
thf(conj_thm_2Ereal__topology_2EDIST__REFL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))).
thf(conj_thm_2Ereal__topology_2EDIST__SYM,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V0x)))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z))))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ALT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z))))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z)))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__EQ__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (V0x = V1y))))).
thf(conj_thm_2Ereal__topology_2EDIST__POS__LT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((~~ (V0x = V1y)) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__NZ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((~~ (V0x = V1y)) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__LE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))))@(inj__ty_2Erealax_2Ereal @ V3e))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(inj__ty_2Erealax_2Ereal @ V3e))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__LT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))))@(inj__ty_2Erealax_2Ereal @ V3e))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(inj__ty_2Erealax_2Ereal @ V3e))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__L,lemma,( ! [V0e:tp__ty_2Erealax_2Ereal] : ( ! [V1x1:tp__ty_2Erealax_2Ereal] : ( ! [V2x2:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x1))@(inj__ty_2Erealax_2Ereal @ V3y))))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V0e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x2))@(inj__ty_2Erealax_2Ereal @ V3y))))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V0e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x1))@(inj__ty_2Erealax_2Ereal @ V2x2))))@(inj__ty_2Erealax_2Ereal @ V0e))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__R,lemma,( ! [V0e:tp__ty_2Erealax_2Ereal] : ( ! [V1x1:tp__ty_2Erealax_2Ereal] : ( ! [V2x2:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V1x1))))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V0e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V2x2))))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V0e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x1))@(inj__ty_2Erealax_2Ereal @ V2x2))))@(inj__ty_2Erealax_2Ereal @ V0e))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ADD,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1x_27:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3y_27:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2y)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x_27))@(inj__ty_2Erealax_2Ereal @ V3y_27)))))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1x_27))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V3y_27))))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__MUL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V1y))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2c)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ADD__HALF,lemma,( ! [V0e:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2x_27:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ( ! [V4y_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2x_27))))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V0e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V4y_27))))@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V0e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V2x_27))@(inj__ty_2Erealax_2Ereal @ V4y_27)))))@(inj__ty_2Erealax_2Ereal @ V0e)))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__LE__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (V0x = V1y))))).
thf(conj_thm_2Ereal__topology_2EDIST__POS__LE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))))))).
thf(conj_thm_2Ereal__topology_2EDIST__EQ,lemma,( ! [V0w:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3z:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0w))@(inj__ty_2Erealax_2Ereal @ V1x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V3z))))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Epow@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0w))@(inj__ty_2Erealax_2Ereal @ V1x))))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Epow@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V3z))))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))) & ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x))))))).
thf(conj_thm_2Ereal__topology_2EREAL__CHOOSE__DIST,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1e))) => (? [V2y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2y)))) = V1e)))))).
thf(ax_thm_2Ereal__topology_2Elinear,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) <=> (( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2y))))))) & ( ! [V3c:tp__ty_2Erealax_2Ereal] : ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3c))@(inj__ty_2Erealax_2Ereal @ V4x)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3c))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__SCALING,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@V1x))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__CMUL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@V0f@V2x))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__NEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V0f@V1x)))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__ADD,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@c_2Ereal__topology_2Elinear@V1g))) => (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@V2x))@(ap@V1g@V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__SUB,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@c_2Ereal__topology_2Elinear@V1g))) => (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@V2x))@(ap@V1g@V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@c_2Ereal__topology_2Elinear@V1g))) => (p@(ap@c_2Ereal__topology_2Elinear@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__ID,lemma,(p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : V0x))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__ZERO,lemma,(p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__NEGATION,lemma,(p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (ap@c_2Erealax_2Ereal__neg@V0x)))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__SUM,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ( ! [V2a:$i] : ((mem@V2a@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2a)@V1s)) => (p@(ap@c_2Ereal__topology_2Elinear@(ap@V0f@V2a))))))) => (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@(lam@A_27a@(^ [V4a:$i] : (ap@(ap@V0f@V4a)@V3x))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__MUL__COMPONENT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1v:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V0f@V2x))@(inj__ty_2Erealax_2Ereal @ V1v))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__0,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__CMUL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V2x)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__NEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Erealax_2Ereal__neg@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__ADD,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2y))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__SUB,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2y))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__SUM,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@V1g))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1g))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__SUM__MUL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2c:$i] : ((mem@V2c@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3v:$i] : ((mem@V3v@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@(lam@A_27a@(^ [V4i:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2c@V4i))@(ap@V3v@V4i))))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V1s)@(lam@A_27a@(^ [V5i:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2c@V5i))@(ap@V0f@(ap@V3v@V5i))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__BOUNDED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (? [V1B:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1B))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__BOUNDED__POS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (? [V1B:tp__ty_2Erealax_2Ereal] : ((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 @ V1B))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1B))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2ESYMMETRIC__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V2x)))@V1s)))) & (p@(ap@c_2Ereal__topology_2Elinear@V0f))) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V3x)))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))))).
thf(ax_thm_2Ereal__topology_2Ebilinear,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ((p@(ap@c_2Ereal__topology_2Ebilinear@V0f)) <=> (( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))@V2y)))))) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Elinear@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@V0f@V4x)@(inj__ty_2Erealax_2Ereal @ V3y)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__SWAP,lemma,( ! [V0op:$i] : ((mem@V0op@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ((p@(ap@c_2Ereal__topology_2Ebilinear@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@V0op@V2y)@V1x))))))) <=> (p@(ap@c_2Ereal__topology_2Ebilinear@V0op)))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LADD,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3z:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))@(inj__ty_2Erealax_2Ereal @ V3z))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V3z)))@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V3z)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RADD,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3z:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V3z)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V3z)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LMUL,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V3y))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RMUL,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V3y)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LNEG,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V2y))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Erealax_2Ereal__neg@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RNEG,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V2y)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Erealax_2Ereal__neg@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LZERO,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RZERO,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__LSUB,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3z:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))@(inj__ty_2Erealax_2Ereal @ V3z))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V3z)))@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V3z)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__RSUB,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3z:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V3z)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V3z)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__SUM,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27b@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3g:$i] : ((mem@V3g@(arr@A_27b@ty_2Erealax_2Ereal)) => ( ! [V4h:$i] : ((mem@V4h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => (((p@(ap@c_2Ereal__topology_2Ebilinear@V4h)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & (p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V1t)))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V4h@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V0s)@V2f))@(ap@(ap@(c_2Eiterate_2ESum@A_27b)@V1t)@V3g))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@(ty_2Epair_2Eprod@A_27a@A_27b))@(ap@(ap@(c_2Epred__set_2ECROSS@A_27a@A_27b)@V0s)@V1t))@(ap@(c_2Epair_2EUNCURRY@A_27a@A_27b@ty_2Erealax_2Ereal)@(lam@A_27a@(^ [V5i:$i] : (lam@A_27b@(^ [V6j:$i] : (ap@(ap@V4h@(ap@V2f@V5i))@(ap@V3g@V6j)))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__BOUNDED,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => (? [V1B:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1B))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x))))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V3y)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__BOUNDED__POS,lemma,( ! [V0h:$i] : ((mem@V0h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ((p@(ap@c_2Ereal__topology_2Ebilinear@V0h)) => (? [V1B:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1B))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@V0h@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1B))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x))))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V3y))))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__SUC,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V2h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n)))@(lam@ty_2Enum_2Enum@(^ [V5k:$i] : (ap@(ap@V2h@(ap@V0f@V5k))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2B@V5k)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V1g@V5k))))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n)))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V2h@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V4n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V4n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))@(ap@(ap@V2h@(ap@V0f@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V1g@(inj__ty_2Enum_2Enum @ V3m)))))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n)))@(lam@ty_2Enum_2Enum@(^ [V6k:$i] : (ap@(ap@V2h@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@V6k)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V0f@V6k)))@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2B@V6k)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__PRE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@c_2Ereal__topology_2Ebilinear@V2h)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n)))@(lam@ty_2Enum_2Enum@(^ [V5k:$i] : (ap@(ap@V2h@(ap@V0f@V5k))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1g@V5k))@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2D@V5k)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n)))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V2h@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V4n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V1g@(inj__ty_2Enum_2Enum @ V4n))))@(ap@(ap@V2h@(ap@V0f@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V3m))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n)))@(lam@ty_2Enum_2Enum@(^ [V6k:$i] : (ap@(ap@V2h@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@V6k)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V0f@V6k)))@(ap@V1g@V6k)))))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))))))).
thf(ax_thm_2Ereal__topology_2Esubspace,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) <=> ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0s)) & (( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))@V0s))))) & ( ! [V3c:tp__ty_2Erealax_2Ereal] : ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3c))@(inj__ty_2Erealax_2Ereal @ V4x)))@V0s))))))))))).
thf(ax_thm_2Ereal__topology_2Espan,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Espan@V0s) = (ap@(ap@(c_2Etopology_2Ehull@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Esubspace)@V0s))))).
thf(ax_thm_2Ereal__topology_2Edependent,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Edependent@V0s)) <=> (? [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V0s)@(inj__ty_2Erealax_2Ereal @ V1a))))))))))).
thf(ax_thm_2Ereal__topology_2Eindependent,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eindependent@V0s)) <=> (~ (p@(ap@c_2Ereal__topology_2Edependent@V0s))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__UNIV,lemma,(p@(ap@c_2Ereal__topology_2Esubspace@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__IMP__NONEMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) => (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__0,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__ADD,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@V2s)))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@V2s)))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__MUL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V2s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0x)))@V2s)))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0x)))@V1s))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SUB,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@V2s)))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@V2s)))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SUM,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) & ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2t)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2t)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1f@V3x))@V0s))))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2t)@V1f))@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@c_2Ereal__topology_2Esubspace@V1s))) => (p@(ap@c_2Ereal__topology_2Esubspace@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__LINEAR__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@c_2Ereal__topology_2Esubspace@V1s))) => (p@(ap@c_2Ereal__topology_2Esubspace@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V2x))@V1s))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__TRIVIAL,lemma,(p@(ap@c_2Ereal__topology_2Esubspace@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) & (p@(ap@c_2Ereal__topology_2Esubspace@V1t))) => (p@(ap@c_2Ereal__topology_2Esubspace@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__BIGINTER,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) => (p@(ap@c_2Ereal__topology_2Esubspace@V1s))))) => (p@(ap@c_2Ereal__topology_2Esubspace@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__0__SUBSPACE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@c_2Ereal__topology_2Esubspace@V1s))) => (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))))) => (V2x = V3y)))) <=> ( ! [V4x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (V4x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__UNION__CHAIN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) & ((p@(ap@c_2Ereal__topology_2Esubspace@V1t)) & (p@(ap@c_2Ereal__topology_2Esubspace@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))))) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) | (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__SPAN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Espan@(ap@c_2Ereal__topology_2Espan@V0s)) = (ap@c_2Ereal__topology_2Espan@V0s))))).
thf(conj_thm_2Ereal__topology_2ESPAN__MONO,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Espan@V0s))@(ap@c_2Ereal__topology_2Espan@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SPAN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@c_2Ereal__topology_2Esubspace@(ap@c_2Ereal__topology_2Espan@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESPAN__CLAUSES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V2s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Ereal__topology_2Espan@V2s))))))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2Espan@V0s))) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : ( ! [V5s:$i] : ((mem@V5s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal__topology_2Espan@V5s))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@(ap@c_2Ereal__topology_2Espan@V5s)))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y)))@(ap@c_2Ereal__topology_2Espan@V5s)))))))) & ( ! [V6x:tp__ty_2Erealax_2Ereal] : ( ! [V7c:tp__ty_2Erealax_2Ereal] : ( ! [V8s:$i] : ((mem@V8s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@(ap@c_2Ereal__topology_2Espan@V8s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V7c))@(inj__ty_2Erealax_2Ereal @ V6x)))@(ap@c_2Ereal__topology_2Espan@V8s)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__INDUCT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1h:$i] : ((mem@V1h@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1h)))) & (p@(ap@c_2Ereal__topology_2Esubspace@V1h))) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal__topology_2Espan@V0s))) => (p@(ap@V1h@(inj__ty_2Erealax_2Ereal @ V3x))))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__EMPTY,lemma,((ap@c_2Ereal__topology_2Espan@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__NONZERO,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eindependent@V0s)) => (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EDEPENDENT__MONO,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Edependent@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))) => (p@(ap@c_2Ereal__topology_2Edependent@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__SUBSPACE,lemma,( ! [V0b:$i] : ((mem@V0b@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0b)@V1s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1s)@(ap@c_2Ereal__topology_2Espan@V0b))) & (p@(ap@c_2Ereal__topology_2Esubspace@V1s)))) => ((ap@c_2Ereal__topology_2Espan@V0b) = V1s))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__INDUCT__ALT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1h:$i] : ((mem@V1h@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@V1h@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) & (p@(ap@V1h@(inj__ty_2Erealax_2Ereal @ V4y)))) => (p@(ap@V1h@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3x)))@(inj__ty_2Erealax_2Ereal @ V4y))))))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(ap@c_2Ereal__topology_2Espan@V0s))) => (p@(ap@V1h@(inj__ty_2Erealax_2Ereal @ V5x))))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__SUPERSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Espan@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__INC,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Espan@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESPAN__UNION__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Espan@V0s))@(ap@c_2Ereal__topology_2Espan@V1t)))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__UNIV,lemma,((ap@c_2Ereal__topology_2Espan@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2ESPAN__0,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2Espan@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESPAN__ADD,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Espan@V2s))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Espan@V2s)))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Ereal__topology_2Espan@V2s))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__MUL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Espan@V2s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal__topology_2Espan@V2s))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__MUL__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((~~ (V1c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal__topology_2Espan@V2s))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Espan@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Espan@V1s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal__topology_2Espan@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__NEG__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal__topology_2Espan@V1s))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Espan@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__SUB,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Espan@V2s))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Espan@V2s)))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Ereal__topology_2Espan@V2s))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__SUM,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2t)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2t)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1f@V3x))@(ap@c_2Ereal__topology_2Espan@V0s))))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2t)@V1f))@(ap@c_2Ereal__topology_2Espan@V0s)))))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__ADD__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Espan@V0s))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))@(ap@c_2Ereal__topology_2Espan@V0s))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@(ap@c_2Ereal__topology_2Espan@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__EQ__SELF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Espan@V0s) = V0s) <=> (p@(ap@c_2Ereal__topology_2Esubspace@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESPAN__SUBSET__SUBSPACE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@c_2Ereal__topology_2Esubspace@V1t))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Espan@V0s))@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ESURJECTIVE__IMAGE__EQ,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => ((( ! [V3y:$i] : ((mem@V3y@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3y)@V2t)) => (? [V4x:$i] : ((mem@V4x@A_27a) & ((ap@V0f@V4x) = V3y)))))) & ( ! [V5x:$i] : ((mem@V5x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@(ap@V0f@V5x))@V2t)) <=> (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V1s)))))) => ((ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V1s) = V2t))))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s))) => ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1a))@V2x))))@V0s) = V0s)))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) => (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1a))@V2x))))@V0s) = V0s) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SUMS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) & (p@(ap@c_2Ereal__topology_2Esubspace@V1t))) => (p@(ap@c_2Ereal__topology_2Esubspace@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t)) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@(ap@c_2Ereal__topology_2Espan@V0s)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@(ap@c_2Ereal__topology_2Espan@V1t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y))))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1y))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y)))@(inj__ty_2Erealax_2Ereal @ V0x)))))))).
thf(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__ABS__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y))))) <=> (((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1y))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y)))@(inj__ty_2Erealax_2Ereal @ V0x)))) | ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1y))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Ereal__neg@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y))))@(inj__ty_2Erealax_2Ereal @ V0x))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__EQ__LINV,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0x))) = V0x) <=> (V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Ereal__topology_2EREAL__EQ__RINV,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0x)))) <=> (V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Ereal__topology_2EABS__TRIANGLE__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y))))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1y))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y)))@(inj__ty_2Erealax_2Ereal @ V0x)))))))).
thf(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z))))@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))))))))).
thf(ax_thm_2Ereal__topology_2Ecollinear,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecollinear@V0s)) <=> (? [V1u:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s))) => (? [V4c:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V4c))@(inj__ty_2Erealax_2Ereal @ V1u))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecollinear@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))) => (p@(ap@c_2Ereal__topology_2Ecollinear@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__EMPTY,lemma,(p@(ap@c_2Ereal__topology_2Ecollinear@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__SING,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__2,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__SMALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V0s))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) => (p@(ap@c_2Ereal__topology_2Ecollinear@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__3,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2z))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) <=> (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V2z))@(inj__ty_2Erealax_2Ereal @ V1y)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__LEMMA,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) <=> ((V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | ((V1y = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | (? [V2c:tp__ty_2Erealax_2Ereal] : (V1y = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V0x))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__LEMMA__ALT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) <=> ((V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | (? [V2c:tp__ty_2Erealax_2Ereal] : (V1y = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V0x)))))))))).
thf(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQUAL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y))))) <=> (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2EMUL__CAUCHY__SCHWARZ__EQUAL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Epow@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V1y))))) <=> (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__3__EXPAND,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) <=> ((V0a = V2c) | (? [V3u:tp__ty_2Erealax_2Ereal] : (V1b = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3u))@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V3u)))@(inj__ty_2Erealax_2Ereal @ V2c)))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__TRIPLES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((~~ (V1a = V2b)) => ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V0s)))) <=> ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__4__3,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = V1b)) => ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3d))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) <=> ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3d))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__3__TRANS,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3d))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & (~~ (V1b = V2c)))) => (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3d))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))).
thf(ax_thm_2Ereal__topology_2Ebetween,axiom,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V0x))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2b))))))))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__REFL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) & ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) & (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V0a))))))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__REFL__EQ,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V0a)))) <=> (V1x = V0a))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__SYM,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) <=> (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a))))))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__ANTISYM,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c)))) & (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2c))))) => (V0a = V1b)))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__TRANS,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c)))) & (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V3d))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2c))))) => (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V3d))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c)))))))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__TRANS__2,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c)))) & (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V3d))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) => (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))))))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__ABS,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2x)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2x))))@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a)))))))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__IMP__COLLINEAR,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) => (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__BETWEEN__CASES,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) <=> ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c)))) | ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V0a)))) | (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__BETWEEN,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))) => (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__1,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@c_2Ereal__topology_2Ecollinear@V0s))))).
thf(ax_thm_2Ereal__topology_2Emidpoint,axiom,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__REFL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V0x)))) = V0x))).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__SYM,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a)))))))).
thf(conj_thm_2Ereal__topology_2EDIST__MIDPOINT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2E_2F@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2E_2F@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(inj__ty_2Erealax_2Ereal @ V0a)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2E_2F@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(inj__ty_2Erealax_2Ereal @ V1b)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2E_2F@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))))).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__EQ__ENDPOINT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) = V0a) <=> (V0a = V1b)) & ((((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) = V1b) <=> (V0a = V1b)) & (((V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) <=> (V0a = V1b)) & ((V1b = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) <=> (V0a = V1b)))))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__MIDPOINT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) & (p@(ap@(ap@c_2Ereal__topology_2Ebetween@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a)))))))).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1a)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2b))))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__MIDPOINT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__COLLINEAR,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = V2c)) => ((V1b = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2c))))) <=> ((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c)))))))))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__MONO,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Eindependent@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))) => (p@(ap@c_2Ereal__topology_2Eindependent@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__BREAKDOWN,lemma,( ! [V0b:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0b))@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(ap@c_2Ereal__topology_2Espan@V1s)))) => (? [V3k:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V2a))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3k))@(inj__ty_2Erealax_2Ereal @ V0b))))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V1s)@(inj__ty_2Erealax_2Ereal @ V0b)))))))))))).
thf(conj_thm_2Ereal__topology_2EIN__SPAN__INSERT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@V2s)))) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Espan@V2s))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__INSERT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eindependent@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V1s))) <=> (p@(ap@(ap@(ap@(c_2Ebool_2ECOND@bool)@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V1s))@(ap@c_2Ereal__topology_2Eindependent@V1s))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@c_2Ereal__topology_2Eindependent@V1s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Espan@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__EMPTY,lemma,(p@(ap@c_2Ereal__topology_2Eindependent@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__SING,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Eindependent@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) <=> (~~ (V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__STDBASIS,lemma,(p@(ap@c_2Ereal__topology_2Eindependent@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0i:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V0i)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@V0i))@(ap@(ap@c_2Ereal_2Ereal__lte@V0i)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))))).
thf(conj_thm_2Ereal__topology_2ESPANNING__SUBSET__INDEPENDENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & ((p@(ap@c_2Ereal__topology_2Eindependent@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Espan@V1t))))) => (V0s = V1t))))))).
thf(conj_thm_2Ereal__topology_2EIN__SPAN__DELETE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Espan@V2s))) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V2s)@(inj__ty_2Erealax_2Ereal @ V1b))))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V2s)@(inj__ty_2Erealax_2Ereal @ V1b)))))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__TRANS,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Espan@V2s))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2s))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Espan@V2s))))))))).
thf(conj_thm_2Ereal__topology_2EEXCHANGE__LEMMA,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t)) & ((p@(ap@c_2Ereal__topology_2Eindependent@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Espan@V1t))))) => (? [V2t_27:$i] : ((mem@V2t_27@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@ty_2Erealax_2Ereal)@V2t_27)@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V1t))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V2t_27)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t_27)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Espan@V2t_27)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECARD__STDBASIS,lemma,((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) = (surj__ty_2Enum_2Enum @ (ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__SPAN__BOUND,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t)) & ((p@(ap@c_2Ereal__topology_2Eindependent@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Espan@V1t))))) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V0s))@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__BOUND,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eindependent@V0s)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V0s))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))).
thf(conj_thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET__EXTEND,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1v:$i] : ((mem@V1v@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1v)) & (p@(ap@c_2Ereal__topology_2Eindependent@V0s))) => (? [V2b:$i] : ((mem@V2b@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V2b)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2b)@V1v)) & ((p@(ap@c_2Ereal__topology_2Eindependent@V2b)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1v)@(ap@c_2Ereal__topology_2Espan@V2b)))))))))))))).
thf(conj_thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET,lemma,( ! [V0v:$i] : ((mem@V0v@(arr@ty_2Erealax_2Ereal@bool)) => (? [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1b)@V0v)) & ((p@(ap@c_2Ereal__topology_2Eindependent@V1b)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0v)@(ap@c_2Ereal__topology_2Espan@V1b)))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__BREAKDOWN__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V2s)))) <=> (? [V3k:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3k))@(inj__ty_2Erealax_2Ereal @ V1a))))@(ap@c_2Ereal__topology_2Espan@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND__LEMMA,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1b)) => ((p@(ap@c_2Ereal__topology_2Eindependent@V1b)) => (? [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal__topology_2Espan@V1b))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@(ap@c_2Ereal__topology_2Espan@V1b)))) => ((surj__ty_2Erealax_2Ereal @ (ap@V2g@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@V2g@(inj__ty_2Erealax_2Ereal @ V3x)))@(ap@V2g@(inj__ty_2Erealax_2Ereal @ V4y)))))))) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(ap@c_2Ereal__topology_2Espan@V1b))) => ((surj__ty_2Erealax_2Ereal @ (ap@V2g@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V6c))@(inj__ty_2Erealax_2Ereal @ V5x)))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V6c))@(ap@V2g@(inj__ty_2Erealax_2Ereal @ V5x)))))))) & ( ! [V7x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@V1b)) => ((surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V7x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V7x))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eindependent@V1b)) => (? [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@c_2Ereal__topology_2Elinear@V2g)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1b)) => ((surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__KERNEL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@c_2Ereal__topology_2Esubspace@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V1x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__EQ__0__SPAN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1b)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal__topology_2Espan@V1b))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__EQ__0,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2s)@(ap@c_2Ereal__topology_2Espan@V1b))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1b)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2b:$i] : ((mem@V2b@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ((p@(ap@c_2Ereal__topology_2Elinear@V1g)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3s)@(ap@c_2Ereal__topology_2Espan@V2b))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V2b)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V4x))))))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V3s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V5x)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__EQ__STDBASIS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ((p@(ap@c_2Ereal__topology_2Elinear@V1g)) & ( ! [V2i:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V2i))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2i))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2i))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V2i)))))))) => (V0f = V1g))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__LEFT__INVERSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2y)))) => (V1x = V2y))))) => (? [V3g:$i] : ((mem@V3g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@c_2Ereal__topology_2Elinear@V3g)) & ((ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V3g)@V0f) = (lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : V4x)))))))))).
thf(ax_thm_2Ereal__topology_2Edim,axiom,( ! [V0v:$i] : ((mem@V0v@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Enum_2Enum @ (ap@c_2Ereal__topology_2Edim@V0v)) = (surj__ty_2Enum_2Enum @ (ap@(c_2Emin_2E_40@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V1n:$i] : (ap@(c_2Ebool_2E_3F@(arr@ty_2Erealax_2Ereal@bool))@(lam@(arr@ty_2Erealax_2Ereal@bool)@(^ [V2b:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2b)@V0v))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@c_2Ereal__topology_2Eindependent@V2b))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0v)@(ap@c_2Ereal__topology_2Espan@V2b)))@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@ty_2Erealax_2Ereal)@V2b)@V1n))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBASIS__EXISTS,lemma,( ! [V0v:$i] : ((mem@V0v@(arr@ty_2Erealax_2Ereal@bool)) => (? [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1b)@V0v)) & ((p@(ap@c_2Ereal__topology_2Eindependent@V1b)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0v)@(ap@c_2Ereal__topology_2Espan@V1b))) & (p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@ty_2Erealax_2Ereal)@V1b)@(ap@c_2Ereal__topology_2Edim@V0v))))))))))).
thf(conj_thm_2Ereal__topology_2ECARD__GE__DIM__INDEPENDENT,lemma,( ! [V0v:$i] : ((mem@V0v@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1b)@V0v)) & ((p@(ap@c_2Ereal__topology_2Eindependent@V1b)) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Ereal__topology_2Edim@V0v))@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V1b))))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0v)@(ap@c_2Ereal__topology_2Espan@V1b))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__EXPLICIT,lemma,( ! [V0p:$i] : ((mem@V0p@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Espan@V0p) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1y)@(ap@(c_2Ebool_2E_3F@(arr@ty_2Erealax_2Ereal@bool))@(lam@(arr@ty_2Erealax_2Ereal@bool)@(^ [V2s:$i] : (ap@(c_2Ebool_2E_3F@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(lam@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(^ [V3u:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V2s))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2s)@V0p))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Erealax_2Ereal)@V2s)@(lam@ty_2Erealax_2Ereal@(^ [V4v:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V3u@V4v))@V4v)))))@V1y))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EDEPENDENT__EXPLICIT,lemma,( ! [V0p:$i] : ((mem@V0p@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Edependent@V0p)) <=> (? [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1s)@V0p)) & ((? [V3v:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3v))@V1s)) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V2u@(inj__ty_2Erealax_2Ereal @ V3v))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))) & ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@ty_2Erealax_2Ereal)@V1s)@(lam@ty_2Erealax_2Ereal@(^ [V4v:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2u@V4v))@V4v))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE__GEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Eindependent@V1s)) & ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Espan@V1s))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(ap@c_2Ereal__topology_2Espan@V1s))) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))))) => (V2x = V3y)))))) => (p@(ap@c_2Ereal__topology_2Eindependent@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Eindependent@V1s)) & ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y)))))) => (p@(ap@c_2Ereal__topology_2Eindependent@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => ((ap@c_2Ereal__topology_2Espan@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Espan@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__IMP__SURJECTIVE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2y)))) => (V1x = V2y))))) => ( ! [V3y:tp__ty_2Erealax_2Ereal] : (? [V4x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = V3y))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW__POS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ((p@(ap@c_2Ereal__topology_2Elinear@V1g)) & ((ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f) = (c_2Ecombin_2EI@ty_2Erealax_2Ereal)))) => (? [V2B:tp__ty_2Erealax_2Ereal] : ((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 @ V2B))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2B))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V3x))))@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ((p@(ap@c_2Ereal__topology_2Elinear@V1g)) & ((ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f) = (c_2Ecombin_2EI@ty_2Erealax_2Ereal)))) => (? [V2B:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2B))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V3x))))@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__BOUNDED__BELOW__POS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2y)))) => (V1x = V2y))))) => (? [V3B:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3B))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V4x)))@(inj__ty_2Erealax_2Ereal @ V3B)))@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))))))))))).
thf(conj_thm_2Ereal__topology_2EINDEPENDENT__CARD__LE__DIM,lemma,( ! [V0v:$i] : ((mem@V0v@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1b)@V0v)) & (p@(ap@c_2Ereal__topology_2Eindependent@V1b))) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1b)) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V1b))@(ap@c_2Ereal__topology_2Edim@V0v)))))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__CARD__GE__DIM,lemma,( ! [V0v:$i] : ((mem@V0v@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0v)@(ap@c_2Ereal__topology_2Espan@V1b))) & (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1b))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Ereal__topology_2Edim@V0v))@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V1b))))))))).
thf(conj_thm_2Ereal__topology_2EBASIS__CARD__EQ__DIM,lemma,( ! [V0v:$i] : ((mem@V0v@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1b)@V0v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0v)@(ap@c_2Ereal__topology_2Espan@V1b))) & (p@(ap@c_2Ereal__topology_2Eindependent@V1b)))) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1b)) & ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V1b)) = (surj__ty_2Enum_2Enum @ (ap@c_2Ereal__topology_2Edim@V0v)))))))))).
thf(conj_thm_2Ereal__topology_2EBASIS__HAS__SIZE__DIM,lemma,( ! [V0v:$i] : ((mem@V0v@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Eindependent@V1b)) & ((ap@c_2Ereal__topology_2Espan@V1b) = V0v)) => (p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@ty_2Erealax_2Ereal)@V1b)@(ap@c_2Ereal__topology_2Edim@V0v))))))))).
thf(conj_thm_2Ereal__topology_2EDIM__UNIQUE,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1v:$i] : ((mem@V1v@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2b:$i] : ((mem@V2b@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2b)@V1v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1v)@(ap@c_2Ereal__topology_2Espan@V2b))) & ((p@(ap@c_2Ereal__topology_2Eindependent@V2b)) & (p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@ty_2Erealax_2Ereal)@V2b)@(inj__ty_2Enum_2Enum @ V0n)))))) => ((surj__ty_2Enum_2Enum @ (ap@c_2Ereal__topology_2Edim@V1v)) = V0n)))))))).
thf(conj_thm_2Ereal__topology_2EDIM__LE__CARD,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Ereal__topology_2Edim@V0s))@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V0s))))))).
thf(conj_thm_2Ereal__topology_2ESPAN__STDBASIS,lemma,((ap@c_2Ereal__topology_2Espan@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0i:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V0i)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@V0i))@(ap@(ap@c_2Ereal_2Ereal__lte@V0i)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2EHAS__SIZE__STDBASIS,lemma,(p@(ap@(ap@(c_2Ecardinal_2EHAS__SIZE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0i:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V0i)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@V0i))@(ap@(ap@c_2Ereal_2Ereal__lte@V0i)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))).
thf(conj_thm_2Ereal__topology_2EDIM__UNIV,lemma,((surj__ty_2Enum_2Enum @ (ap@c_2Ereal__topology_2Edim@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))) = (surj__ty_2Enum_2Enum @ (ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))).
thf(conj_thm_2Ereal__topology_2EDIM__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Ereal__topology_2Edim@V0s))@(ap@c_2Ereal__topology_2Edim@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EDIM__SUBSET__UNIV,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Ereal__topology_2Edim@V0s))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))).
thf(ax_thm_2Ereal__topology_2Esubtopology,axiom,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ((ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1u) = (ap@(c_2Etopology_2Etopology@A_27a)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V2s:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@(ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V2s)@V1u))@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@V0top)@V2s))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__IMAGE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@A_27b)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27b@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27b)@V1s)@(ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V2t))) <=> (? [V3u:$i] : ((mem@V3u@(arr@A_27a@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V3u)@V2t)) & (V1s = (ap@(ap@(c_2Epred__set_2EIMAGE@A_27a@A_27b)@V0f)@V3u))))))))))))))).
thf(conj_thm_2Ereal__topology_2EINTER__BIGUNION,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: (( ! [V0s:$i] : ((mem@V0s@(arr@(arr@A_27a@bool)@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => ((ap@(ap@(c_2Epred__set_2EINTER@A_27a)@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@V0s))@V1t) = (ap@(c_2Epred__set_2EBIGUNION@A_27a)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@(ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V2x)@V1t))@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V2x)@V0s))))))))))) & ( ! [V3s:$i] : ((mem@V3s@(arr@(arr@A_27b@bool)@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@A_27b@bool)) => ((ap@(ap@(c_2Epred__set_2EINTER@A_27b)@V4t)@(ap@(c_2Epred__set_2EBIGUNION@A_27b)@V3s)) = (ap@(c_2Epred__set_2EBIGUNION@A_27b)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27b@bool)@(arr@A_27b@bool))@(lam@(arr@A_27b@bool)@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27b@bool)@bool)@(ap@(ap@(c_2Epred__set_2EINTER@A_27b)@V4t)@V5x))@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27b@bool))@V5x)@V3s))))))))))))))).
thf(conj_thm_2Ereal__topology_2EISTOPLOGY__SUBTOPOLOGY,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => (p@(ap@(c_2Etopology_2Eistopology@A_27a)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V2s:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@(ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V2s)@V1u))@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@V0top)@V2s))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1u))@V2s)) <=> (? [V3t:$i] : ((mem@V3t@(arr@A_27a@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@V0top)@V3t)) & (V2s = (ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V3t)@V1u)))))))))))))).
thf(conj_thm_2Ereal__topology_2ETOPSPACE__SUBTOPOLOGY,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ((ap@(c_2Etopology_2Etopspace@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1u)) = (ap@(ap@(c_2Epred__set_2EINTER@A_27a)@(ap@(c_2Etopology_2Etopspace@A_27a)@V0top))@V1u)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1u))@V2s)) <=> (? [V3t:$i] : ((mem@V3t@(arr@A_27a@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@A_27a)@V0top)@V3t)) & (V2s = (ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V3t)@V1u)))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__EMPTY,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@(c_2Epred__set_2EEMPTY@A_27a)))@V1s)) <=> (V1s = (c_2Epred__set_2EEMPTY@A_27a))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__EMPTY,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@(c_2Epred__set_2EEMPTY@A_27a)))@V1s)) <=> (V1s = (c_2Epred__set_2EEMPTY@A_27a))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__REFL,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1u))@V1u)) <=> (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V1u)@(ap@(c_2Etopology_2Etopspace@A_27a)@V0top)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__REFL,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1u:$i] : ((mem@V1u@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1u))@V1u)) <=> (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V1u)@(ap@(c_2Etopology_2Etopspace@A_27a)@V0top)))))))))).
thf(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__SUPERSET,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@(ap@(c_2Etopology_2Etopspace@A_27a)@V0top))@V1s)) => ((ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1s) = V0top)))))))).
thf(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__TOPSPACE,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ((ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@(ap@(c_2Etopology_2Etopspace@A_27a)@V0top)) = V0top))))).
thf(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__UNIV,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ((ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@(c_2Epred__set_2EUNIV@A_27a)) = V0top))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__IMP__SUBSET,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1s))@V2t)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2t)@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__IMP__SUBSET,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V1s))@V2t)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V2t)@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__UNION,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V2t))@V1s)) & (p@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V3u))@V1s))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V2t)@V3u)))@V1s))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__UNION,lemma,( ! [A_27a:del]: ( ! [V0top:$i] : ((mem@V0top@(ty_2Etopology_2Etopology@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V2t))@V1s)) & (p@(ap@(ap@(c_2Etopology_2Eclosed__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@V3u))@V1s))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@A_27a)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@A_27a)@V0top)@(ap@(ap@(c_2Epred__set_2EUNION@A_27a)@V2t)@V3u)))@V1s))))))))))))).
thf(ax_thm_2Ereal__topology_2Eopen__def,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (? [V2e:tp__ty_2Erealax_2Ereal] : ((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))) & ( ! [V3x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x_27))@(inj__ty_2Erealax_2Ereal @ V1x))))@(inj__ty_2Erealax_2Ereal @ V2e))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x_27))@V0s)))))))))))).
thf(ax_thm_2Ereal__topology_2Eclosed__def,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) <=> (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s))))))).
thf(ax_thm_2Ereal__topology_2Eeuclidean,axiom,((surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ c_2Ereal__topology_2Eeuclidean) = (surj__c_ty_2Etopology_2Etopology_ty_2Erealax_2Ereal @ (ap@(c_2Etopology_2Etopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2EOpen)))).
thf(conj_thm_2Ereal__topology_2EOPEN__EMPTY,lemma,(p@(ap@c_2Ereal__topology_2EOpen@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EOPEN__UNIV,lemma,(p@(ap@c_2Ereal__topology_2EOpen@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__BIGUNION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) => (p@(ap@c_2Ereal__topology_2EOpen@V1s))))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0f))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__EXISTS__IN,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1Q:$i] : ((mem@V1Q@(arr@A_27a@(arr@ty_2Erealax_2Ereal@bool))) => (( ! [V2a:$i] : ((mem@V2a@A_27a) => ((p@(ap@V0P@V2a)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@V1Q@V2a)@V3x)))))))))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(c_2Ebool_2E_3F@A_27a)@(lam@A_27a@(^ [V5a:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@V0P@V5a))@(ap@(ap@V1Q@V5a)@V4x)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__EXISTS,lemma,( ! [A_27a:del]: ( ! [V0Q:$i] : ((mem@V0Q@(arr@A_27a@(arr@ty_2Erealax_2Ereal@bool))) => (( ! [V1a:$i] : ((mem@V1a@A_27a) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@V0Q@V1a)@V2x))))))))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(c_2Ebool_2E_3F@A_27a)@(lam@A_27a@(^ [V4a:$i] : (ap@(ap@V0Q@V4a)@V3x))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s)))))).
thf(conj_thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN,lemma,((ap@(c_2Etopology_2Etopspace@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN__SUBTOPOLOGY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@(c_2Etopology_2Etopspace@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s)) = V0s)))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__REFL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V0s))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__REFL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V0s))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) <=> (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s)))))).
thf(conj_thm_2Ereal__topology_2EOPEN__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SUB__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (? [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V2t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t)@V0s)))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__EMPTY,lemma,(p@(ap@c_2Ereal__topology_2EClosed@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECLOSED__UNIV,lemma,(p@(ap@c_2Ereal__topology_2EClosed@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECLOSED__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__BIGINTER,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) => (p@(ap@c_2Ereal__topology_2EClosed@V1s))))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f))))))).
thf(conj_thm_2Ereal__topology_2EBIGINTER__GSPEC,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [A_27d:del]: ( ! [A_27e:del]: ( ! [A_27f:del]: ( ! [A_27g:del]: ( ! [A_27h:del]: ( ! [A_27i:del]: (( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@A_27b@bool))) => ((ap@(c_2Epred__set_2EBIGINTER@A_27b)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27b@bool)@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27b@bool)@bool)@(ap@V1f@V2x))@(ap@V0P@V2x)))))) = (ap@(c_2Epred__set_2EGSPEC@A_27b@A_27b)@(lam@A_27b@(^ [V3a:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@V3a)@(ap@(c_2Ebool_2E_21@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V0P@V4x))@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3a)@(ap@V1f@V4x))))))))))))))) & (( ! [V5P:$i] : ((mem@V5P@(arr@A_27c@(arr@A_27d@bool))) => ( ! [V6f:$i] : ((mem@V6f@(arr@A_27c@(arr@A_27d@(arr@A_27e@bool)))) => ((ap@(c_2Epred__set_2EBIGINTER@A_27e)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27e@bool)@(ty_2Epair_2Eprod@A_27c@A_27d))@(ap@(c_2Epair_2EUNCURRY@A_27c@A_27d@(ty_2Epair_2Eprod@(arr@A_27e@bool)@bool))@(lam@A_27c@(^ [V7x:$i] : (lam@A_27d@(^ [V8y:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27e@bool)@bool)@(ap@(ap@V6f@V7x)@V8y))@(ap@(ap@V5P@V7x)@V8y))))))))) = (ap@(c_2Epred__set_2EGSPEC@A_27e@A_27e)@(lam@A_27e@(^ [V9a:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27e@bool)@V9a)@(ap@(c_2Ebool_2E_21@A_27c)@(lam@A_27c@(^ [V10x:$i] : (ap@(c_2Ebool_2E_21@A_27d)@(lam@A_27d@(^ [V11y:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@V5P@V10x)@V11y))@(ap@(ap@(c_2Ebool_2EIN@A_27e)@V9a)@(ap@(ap@V6f@V10x)@V11y)))))))))))))))))) & ( ! [V12P:$i] : ((mem@V12P@(arr@A_27f@(arr@A_27g@(arr@A_27h@bool)))) => ( ! [V13f:$i] : ((mem@V13f@(arr@A_27f@(arr@A_27g@(arr@A_27h@(arr@A_27i@bool))))) => ((ap@(c_2Epred__set_2EBIGINTER@A_27i)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27i@bool)@(ty_2Epair_2Eprod@A_27f@(ty_2Epair_2Eprod@A_27g@A_27h)))@(ap@(c_2Epair_2EUNCURRY@A_27f@(ty_2Epair_2Eprod@A_27g@A_27h)@(ty_2Epair_2Eprod@(arr@A_27i@bool)@bool))@(lam@A_27f@(^ [V14x:$i] : (ap@(c_2Epair_2EUNCURRY@A_27g@A_27h@(ty_2Epair_2Eprod@(arr@A_27i@bool)@bool))@(lam@A_27g@(^ [V15y:$i] : (lam@A_27h@(^ [V16z:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27i@bool)@bool)@(ap@(ap@(ap@V13f@V14x)@V15y)@V16z))@(ap@(ap@(ap@V12P@V14x)@V15y)@V16z)))))))))))) = (ap@(c_2Epred__set_2EGSPEC@A_27i@A_27i)@(lam@A_27i@(^ [V17a:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27i@bool)@V17a)@(ap@(c_2Ebool_2E_21@A_27f)@(lam@A_27f@(^ [V18x:$i] : (ap@(c_2Ebool_2E_21@A_27g)@(lam@A_27g@(^ [V19y:$i] : (ap@(c_2Ebool_2E_21@A_27h)@(lam@A_27h@(^ [V20z:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(ap@V12P@V18x)@V19y)@V20z))@(ap@(ap@(c_2Ebool_2EIN@A_27i)@V17a)@(ap@(ap@(ap@V13f@V18x)@V19y)@V20z))))))))))))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__FORALL__IN,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1Q:$i] : ((mem@V1Q@(arr@A_27a@(arr@ty_2Erealax_2Ereal@bool))) => (( ! [V2a:$i] : ((mem@V2a@A_27a) => ((p@(ap@V0P@V2a)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@V1Q@V2a)@V3x)))))))))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(c_2Ebool_2E_21@A_27a)@(lam@A_27a@(^ [V5a:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V0P@V5a))@(ap@(ap@V1Q@V5a)@V4x)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__FORALL,lemma,( ! [A_27a:del]: ( ! [V0Q:$i] : ((mem@V0Q@(arr@A_27a@(arr@ty_2Erealax_2Ereal@bool))) => (( ! [V1a:$i] : ((mem@V1a@A_27a) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@V0Q@V1a)@V2x))))))))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(c_2Ebool_2E_21@A_27a)@(lam@A_27a@(^ [V4a:$i] : (ap@(ap@V0Q@V4a)@V3x))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__DIFF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__DIFF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__BIGINTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V0s)) & ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V0s)) => (p@(ap@c_2Ereal__topology_2EOpen@V1t)))))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__BIGUNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V0s)) & ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V0s)) => (p@(ap@c_2Ereal__topology_2EClosed@V1t)))))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0s))))))).
thf(ax_thm_2Ereal__topology_2Eball,axiom,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2y)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2y)))@(inj__ty_2Erealax_2Ereal @ V1e)))))))))).
thf(ax_thm_2Ereal__topology_2Ecball,axiom,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2y)@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2y)))@(inj__ty_2Erealax_2Ereal @ V1e)))))))))).
thf(ax_thm_2Ereal__topology_2Esphere,axiom,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2y)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2y)))@(inj__ty_2Erealax_2Ereal @ V1e)))))))))).
thf(conj_thm_2Ereal__topology_2EIN__BALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e))))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(inj__ty_2Erealax_2Ereal @ V2e)))))))).
thf(conj_thm_2Ereal__topology_2EIN__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))@(inj__ty_2Erealax_2Ereal @ V2e)))))))).
thf(conj_thm_2Ereal__topology_2EIN__SPHERE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e))))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) = V2e)))))).
thf(conj_thm_2Ereal__topology_2EIN__BALL__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1e))))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1e))))))).
thf(conj_thm_2Ereal__topology_2EIN__CBALL__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1e))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1e))))))).
thf(conj_thm_2Ereal__topology_2EIN__SPHERE__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1e))))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x))) = V1e))))).
thf(conj_thm_2Ereal__topology_2EBALL__TRIVIAL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECBALL__TRIVIAL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ECENTRE__IN__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))))) <=> (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 @ V1e))))))).
thf(conj_thm_2Ereal__topology_2EBALL__SUBSET__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE__SUBSET__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__BALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1d:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V2e))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1d))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1d:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V2e))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1d))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))).
thf(conj_thm_2Ereal__topology_2EBALL__MAX__UNION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ( ! [V2s:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@c_2Ereal_2Emax@(inj__ty_2Erealax_2Ereal @ V1r))@(inj__ty_2Erealax_2Ereal @ V2s)))) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2s))))))))).
thf(conj_thm_2Ereal__topology_2EBALL__MIN__INTER,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ( ! [V2s:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@c_2Ereal_2Emin@(inj__ty_2Erealax_2Ereal @ V1r))@(inj__ty_2Erealax_2Ereal @ V2s)))) = (ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2s))))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__MAX__UNION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ( ! [V2s:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@c_2Ereal_2Emax@(inj__ty_2Erealax_2Ereal @ V1r))@(inj__ty_2Erealax_2Ereal @ V2s)))) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2s))))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__MIN__INTER,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1d:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@c_2Ereal_2Emin@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V2e)))) = (ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1d))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e))))))))).
thf(conj_thm_2Ereal__topology_2EBALL__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V2r))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3y))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r))))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V2r))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3y))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r))))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V2r))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3y))))@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r))))))))).
thf(conj_thm_2Ereal__topology_2EBALL__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (( ! [V3y:tp__ty_2Erealax_2Ereal] : (? [V4x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = V3y))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V5x))))))) => ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V2r))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r))))))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (( ! [V3y:tp__ty_2Erealax_2Ereal] : (? [V4x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = V3y))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V5x))))))) => ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V2r))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r))))))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (( ! [V3y:tp__ty_2Erealax_2Ereal] : (? [V4x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = V3y))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V5x))))))) => ((ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V2r))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r))))))))))).
thf(conj_thm_2Ereal__topology_2EBALL__SCALING,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0c))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V2r)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@V3x))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r)))))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__SCALING,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : ((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 @ V0c))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V2r)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@V3x))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r)))))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__DIFF__BALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))) = (ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))))).
thf(conj_thm_2Ereal__topology_2EBALL__UNION__SPHERE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))) = (ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE__UNION__BALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))) = (ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__DIFF__SPHERE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))) = (ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__BALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2ECENTRE__IN__BALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))))) <=> (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 @ V1e))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__BALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (? [V2e:tp__ty_2Erealax_2Ereal] : ((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@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e))))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__BALL__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> (? [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e))))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2EBALL__EQ__EMPTY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1e))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2EBALL__EMPTY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1e))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (? [V2e:tp__ty_2Erealax_2Ereal] : ((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@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e))))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> (? [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e))))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE__EQ__EMPTY,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE__EMPTY,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ENEGATIONS__BALL,lemma,( ! [V0r:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0r)))) = (ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0r)))))).
thf(conj_thm_2Ereal__topology_2ENEGATIONS__CBALL,lemma,( ! [V0r:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0r)))) = (ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0r)))))).
thf(conj_thm_2Ereal__topology_2ENEGATIONS__SPHERE,lemma,( ! [V0r:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0r)))) = (ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0r)))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1u:$i] : ((mem@V1u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1u))@V0s)) <=> (? [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2t)) & (V0s = (ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1u)@V2t))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__INTER__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__INTER,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V1s)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0u)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__OPEN__IN__TRANS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & ((p@(ap@c_2Ereal__topology_2EOpen@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@c_2Ereal__topology_2EOpen@V0s))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1u:$i] : ((mem@V1u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1u))@V0s)) <=> (? [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EClosed@V2t)) & (V0s = (ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1u)@V2t))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__SUBSET__EQ,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V1s)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) <=> (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1s)@V0u))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__INTER,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V1s)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0u)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@c_2Ereal__topology_2EClosed@V0s))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBSET__TRANS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2u)))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBSET__TRANS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2u)))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2Eopen__in,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) <=> ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1s)@V0u)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x_27))@V0u)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x_27))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V3e)))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x_27))@V1s))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__CONTAINS__BALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)) <=> ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3e))))@V1t))@V0s))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__CONTAINS__CBALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)) <=> ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3e))))@V1t))@V0s))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__TRANS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)) & (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V1t))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__TRANS__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V2u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__TRANS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => (p@(ap@c_2Ereal__topology_2EOpen@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__TRANS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)) & (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V1t))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__TRANS__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V2u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t))))) <=> (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__TRANS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__INTER__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1u:$i] : ((mem@V1u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2v:$i] : ((mem@V2v@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1u))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1u)@V0s))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2v)@V1u))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2v))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2v)@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)) <=> ((p@(ap@c_2Ereal__topology_2EOpen@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)) <=> ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)))))))))).
thf(ax_thm_2Ereal__topology_2Eclosed__segment,axiom,( ! [V0l:tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosed__segment@(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l)) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1u:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@V1u))@(ap@(c_2Epair_2EFST@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(c_2Elist_2EHD@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l)))))@(ap@(ap@c_2Erealax_2Ereal__mul@V1u)@(ap@(c_2Epair_2ESND@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(c_2Elist_2EHD@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l))))))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V1u))@(ap@(ap@c_2Ereal_2Ereal__lte@V1u)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))))).
thf(ax_thm_2Ereal__topology_2Eopen__segment,axiom,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SEGMENT__ALT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = V1b)) => ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2u:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@V2u))@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(ap@c_2Erealax_2Ereal__mul@V2u)@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V2u))@(ap@(ap@c_2Erealax_2Ereal__lt@V2u)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))))))).
thf(conj_thm_2Ereal__topology_2Esegment,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2u:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@V2u))@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(ap@c_2Erealax_2Ereal__mul@V2u)@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V2u))@(ap@(ap@c_2Ereal_2Ereal__lte@V2u)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))) & ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT__REFL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V0a)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) & ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V1a))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2EIN__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> (? [V3u:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3u))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3u))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) & (V2x = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V3u)))@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3u))@(inj__ty_2Erealax_2Ereal @ V1b))))))))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) <=> ((~~ (V0a = V1b)) & (? [V4u:tp__ty_2Erealax_2Ereal] : ((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 @ V4u))) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V4u))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) & (V2x = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V4u)))@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V4u))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT__SYM,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))) = (ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3b))@(inj__ty_2Erealax_2Ereal @ V2a)))))))).
thf(conj_thm_2Ereal__topology_2EENDS__IN__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2EENDS__NOT__IN__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT__CLOSED__OPEN,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT__OPEN__SUBSET__CLOSED,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EMIDPOINT__IN__SEGMENT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) <=> (~~ (V2a = V3b))))))).
thf(conj_thm_2Ereal__topology_2EBETWEEN__IN__SEGMENT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Ebetween@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__CONVEX__BOUND__LE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3u:tp__ty_2Erealax_2Ereal] : ( ! [V4v:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2a))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2a))) & ((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 @ V3u))) & ((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 @ V4v))) & ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V3u))@(inj__ty_2Erealax_2Ereal @ V4v))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3u))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V4v))@(inj__ty_2Erealax_2Ereal @ V1y))))@(inj__ty_2Erealax_2Ereal @ V2a)))))))))).
thf(conj_thm_2Ereal__topology_2EIN__SEGMENT__COMPONENT,lemma,( ! [A_27a:del]: ( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3i:$i] : ((mem@V3i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Ereal_2Emin@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(inj__ty_2Erealax_2Ereal @ V2x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@c_2Ereal_2Emax@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT__TRANSLATION,lemma,(( ! [V0c:tp__ty_2Erealax_2Ereal] : ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1a)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V2b))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0c))@V3x))))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))) & ( ! [V4c:tp__ty_2Erealax_2Ereal] : ( ! [V5a:tp__ty_2Erealax_2Ereal] : ( ! [V6b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V4c))@(inj__ty_2Erealax_2Ereal @ V5a)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V4c))@(inj__ty_2Erealax_2Ereal @ V6b)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V4c))@V7x))))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5a))@(inj__ty_2Erealax_2Ereal @ V6b)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__SEGMENT__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => ((ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1a)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2b))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SEGMENT__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))) => (V3x = V4y))))) => ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1a)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2b)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b))))))))))).
thf(conj_thm_2Ereal__topology_2EIN__OPEN__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) <=> ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) & ((~~ (V2x = V0a)) & (~~ (V2x = V1b))))))))).
thf(conj_thm_2Ereal__topology_2EIN__OPEN__SEGMENT__ALT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) <=> ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) & ((~~ (V2x = V0a)) & ((~~ (V2x = V1b)) & (~~ (V0a = V1b)))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__CLOSED__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__OPEN__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ecollinear@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__IN__OPEN__CLOSED__SEGMENT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))))) & ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V4b))))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V3a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V4b))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V4b))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V4b)))))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__IN__CLOSED__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EDIST__IN__OPEN__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(ax_thm_2Ereal__topology_2Econnected,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (~ (? [V1e1:$i] : ((mem@V1e1@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2e2:$i] : ((mem@V2e2@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V1e1)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2e2)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1e1)@V2e2))) & (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V2e2))@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2e2)@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (~~ (? [V1e1:$i] : ((mem@V1e1@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2e2:$i] : ((mem@V2e2@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1e1)) & ((p@(ap@c_2Ereal__topology_2EClosed@V2e2)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1e1)@V2e2))) & (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V2e2))@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2e2)@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__IN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (~~ (? [V1e1:$i] : ((mem@V1e1@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2e2:$i] : ((mem@V2e2@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1e1)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V2e2)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1e1)@V2e2))) & (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V2e2) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((~~ (V1e1 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V2e2 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__IN__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (~~ (? [V1e1:$i] : ((mem@V1e1@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2e2:$i] : ((mem@V2e2@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1e1)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V2e2)) & (((ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1e1)@V2e2) = V0s) & (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V2e2) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((~~ (V1e1 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V2e2 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__IN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (~~ (? [V1e1:$i] : ((mem@V1e1@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2e2:$i] : ((mem@V2e2@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1e1)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V2e2)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1e1)@V2e2))) & (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V2e2) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((~~ (V1e1 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V2e2 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__IN__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (~~ (? [V1e1:$i] : ((mem@V1e1@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2e2:$i] : ((mem@V2e2@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1e1)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V2e2)) & (((ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1e1)@V2e2) = V0s) & (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V2e2) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((~~ (V1e1 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V2e2 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EEXISTS__DIFF,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@(arr@A_27a@bool)@bool)) => ((? [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) & (p@(ap@V0P@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@(c_2Epred__set_2EUNIV@A_27a))@V1s))))) <=> (? [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) & (p@(ap@V0P@V2s))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)) & (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t))) => ((V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (V1t = V0s))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__SET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (~~ (? [V1e1:$i] : ((mem@V1e1@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2e2:$i] : ((mem@V2e2@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1e1)) & ((p@(ap@c_2Ereal__topology_2EClosed@V2e2)) & ((~~ (V1e1 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V2e2 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (((ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1e1)@V2e2) = V0s) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V2e2) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__SET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (~~ (? [V1e1:$i] : ((mem@V1e1@(arr@ty_2Erealax_2Ereal@bool)) & (? [V2e2:$i] : ((mem@V2e2@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V1e1)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2e2)) & ((~~ (V1e1 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V2e2 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (((ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1e1)@V2e2) = V0s) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1e1)@V2e2) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTABLE__POINTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V0s))) => (? [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2Econnected@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t)@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V3t)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V3t)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EMPTY,lemma,(p@(ap@c_2Ereal__topology_2Econnected@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__REAL__LEMMA,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3e1:$i] : ((mem@V3e1@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4e2:$i] : ((mem@V4e2@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1a)))@V3e1)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2b)))@V4e2)) & (( ! [V5e:tp__ty_2Erealax_2Ereal] : ( ! [V6x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V6x))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V6x))@(inj__ty_2Erealax_2Ereal @ V2b))) & (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 @ V5e))))) => (? [V7d:tp__ty_2Erealax_2Ereal] : ((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 @ V7d))) & ( ! [V8y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V8y))@(inj__ty_2Erealax_2Ereal @ V6x))))@(inj__ty_2Erealax_2Ereal @ V7d))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V8y)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V6x)))))@(inj__ty_2Erealax_2Ereal @ V5e)))))))))) & (( ! [V9y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9y))@V3e1)) => (? [V10e:tp__ty_2Erealax_2Ereal] : ((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 @ V10e))) & ( ! [V11y_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11y_27))@(inj__ty_2Erealax_2Ereal @ V9y))))@(inj__ty_2Erealax_2Ereal @ V10e))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11y_27))@V3e1)))))))) & (( ! [V12y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12y))@V4e2)) => (? [V13e:tp__ty_2Erealax_2Ereal] : ((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 @ V13e))) & ( ! [V14y_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V14y_27))@(inj__ty_2Erealax_2Ereal @ V12y))))@(inj__ty_2Erealax_2Ereal @ V13e))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V14y_27))@V4e2)))))))) & (~~ (? [V15x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V15x))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V15x))@(inj__ty_2Erealax_2Ereal @ V2b))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V15x)))@V3e1)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V15x)))@V4e2))))))))))))) => (? [V16x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V16x))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V16x))@(inj__ty_2Erealax_2Ereal @ V2b))) & ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V16x)))@V3e1))) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V16x)))@V4e2))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__SEGMENT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Econnected@(ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Econnected@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__UNIV,lemma,(p@(ap@c_2Ereal__topology_2Econnected@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECLOPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V0s))) <=> ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (V0s = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__BIGUNION,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0P)) => (p@(ap@c_2Ereal__topology_2Econnected@V1s))))) & (~~ ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0P) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0P))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & ((p@(ap@c_2Ereal__topology_2Econnected@V1t)) & (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__DIFF__OPEN__FROM__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2u)) & ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) & ((p@(ap@c_2Ereal__topology_2Econnected@V2u)) & (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V1t)@V0s)))))))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V2u)@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__DISJOINT__BIGUNION__OPEN__UNIQUE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ( ! [V1f_27:$i] : ((mem@V1f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(ap@(c_2Ereal__topology_2Epairwise@(arr@ty_2Erealax_2Ereal@bool))@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal))@V0f)) & ((p@(ap@(ap@(c_2Ereal__topology_2Epairwise@(arr@ty_2Erealax_2Ereal@bool))@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal))@V1f_27)) & (( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V0f)) => ((p@(ap@c_2Ereal__topology_2EOpen@V2s)) & ((p@(ap@c_2Ereal__topology_2Econnected@V2s)) & (~~ (V2s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) & (( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V3s)@V1f_27)) => ((p@(ap@c_2Ereal__topology_2EOpen@V3s)) & ((p@(ap@c_2Ereal__topology_2Econnected@V3s)) & (~~ (V3s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) & ((ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0f) = (ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1f_27)))))) => (V0f = V1f_27))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__FROM__CLOSED__UNION__AND__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) & ((p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) & (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)))))) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & (p@(ap@c_2Ereal__topology_2Econnected@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__FROM__OPEN__UNION__AND__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & ((p@(ap@c_2Ereal__topology_2EOpen@V1t)) & ((p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) & (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)))))) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & (p@(ap@c_2Ereal__topology_2Econnected@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__INDUCTION,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1Q:$i] : ((mem@V1Q@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V2s)) & (( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V3t)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@V3t))) => (? [V5z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5z))@V3t)) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5z))))))))) & ( ! [V6a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@V2s)) => (? [V7t:$i] : ((mem@V7t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V7t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@V7t)) & ( ! [V8x:tp__ty_2Erealax_2Ereal] : ( ! [V9y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@V7t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9y))@V7t)) & ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V8x))) & ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V9y))) & (p@(ap@V1Q@(inj__ty_2Erealax_2Ereal @ V8x))))))) => (p@(ap@V1Q@(inj__ty_2Erealax_2Ereal @ V9y)))))))))))))) => ( ! [V10a:tp__ty_2Erealax_2Ereal] : ( ! [V11b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10a))@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11b))@V2s)) & ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V10a))) & ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V11b))) & (p@(ap@V1Q@(inj__ty_2Erealax_2Ereal @ V10a))))))) => (p@(ap@V1Q@(inj__ty_2Erealax_2Ereal @ V11b)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION__GEN,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1R:$i] : ((mem@V1R@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V2s)) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@V1R@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y))) => (p@(ap@(ap@V1R@(inj__ty_2Erealax_2Ereal @ V4y))@(inj__ty_2Erealax_2Ereal @ V3x)))))) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6y:tp__ty_2Erealax_2Ereal] : ( ! [V7z:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@V1R@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V6y))) & (p@(ap@(ap@V1R@(inj__ty_2Erealax_2Ereal @ V6y))@(inj__ty_2Erealax_2Ereal @ V7z)))) => (p@(ap@(ap@V1R@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V7z))))))) & (( ! [V8t:$i] : ((mem@V8t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V9a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V8t)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9a))@V8t))) => (? [V10z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10z))@V8t)) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V10z))))))))) & ( ! [V11a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11a))@V2s)) => (? [V12t:$i] : ((mem@V12t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V12t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11a))@V12t)) & ( ! [V13x:tp__ty_2Erealax_2Ereal] : ( ! [V14y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V13x))@V12t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V14y))@V12t)) & ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V13x))) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V14y)))))) => (p@(ap@(ap@V1R@(inj__ty_2Erealax_2Ereal @ V13x))@(inj__ty_2Erealax_2Ereal @ V14y)))))))))))))))) => ( ! [V15a:tp__ty_2Erealax_2Ereal] : ( ! [V16b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V15a))@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V16b))@V2s)) & ((p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V15a))) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V16b)))))) => (p@(ap@(ap@V1R@(inj__ty_2Erealax_2Ereal @ V15a))@(inj__ty_2Erealax_2Ereal @ V16b)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__INDUCTION__SIMPLE,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@V1s)) => (? [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@V3t)) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ( ! [V5y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V3t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V3t)) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V4x))))) => (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5y))))))))))))) => ( ! [V6a:tp__ty_2Erealax_2Ereal] : ( ! [V7b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7b))@V1s)) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V6a))))) => (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V7b)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION,lemma,( ! [V0R:$i] : ((mem@V0R@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))) => (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V2x)))))) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ( ! [V5y:tp__ty_2Erealax_2Ereal] : ( ! [V6z:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5y))) & (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V5y))@(inj__ty_2Erealax_2Ereal @ V6z)))) => (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V6z))))))) & ( ! [V7a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7a))@V1s)) => (? [V8t:$i] : ((mem@V8t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V8t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7a))@V8t)) & ( ! [V9x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9x))@V8t)) => (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V7a))@(inj__ty_2Erealax_2Ereal @ V9x)))))))))))))) => ( ! [V10a:tp__ty_2Erealax_2Ereal] : ( ! [V11b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10a))@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11b))@V1s))) => (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V10a))@(inj__ty_2Erealax_2Ereal @ V11b)))))))))))).
thf(ax_thm_2Ereal__topology_2Elimit__point__of,axiom,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2t)) & (p@(ap@c_2Ereal__topology_2EOpen@V2t))) => (? [V3y:tp__ty_2Erealax_2Ereal] : ((~ (V3y = V0x)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V2t))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__SUBSET,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1s)@V2t))) => (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V2t))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__APPROACHABLE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => (? [V3x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x_27))@V1s)) & ((~~ (V3x_27 = V0x)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x_27))@(inj__ty_2Erealax_2Ereal @ V0x))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))).
thf(conj_thm_2Ereal__topology_2EAPPROACHABLE__LT__LE,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ((? [V2d:tp__ty_2Erealax_2Ereal] : ((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 @ V2d))) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@V1f@V3x))@(inj__ty_2Erealax_2Ereal @ V2d))) => (p@(ap@V0P@V3x))))))) <=> (? [V4d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4d))) & ( ! [V5x:$i] : ((mem@V5x@A_27a) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V1f@V5x))@(inj__ty_2Erealax_2Ereal @ V4d))) => (p@(ap@V0P@V5x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__APPROACHABLE__LE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (? [V3x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x_27))@V1s)) & ((~~ (V3x_27 = V0x)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x_27))@(inj__ty_2Erealax_2Ereal @ V0x))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__CHOOSE__SIZE,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0c))) => (? [V1x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1x))) = V0c))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__UNIV,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__LIMPT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__EMPTY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ENO__LIMIT__POINT__IMP__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((~~ (? [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))) => (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__POSITIVE__ORTHANT,lemma,(p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V0x)@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0x)))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__SET__AVOID,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1s)) => (? [V2d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2d))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & (~~ (V3x = V0a))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V3x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIMIT__POINT__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) => (~~ (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__SING,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2ELIMIT__POINT__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) <=> ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) | (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__INSERT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s))) <=> (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__LIMPTS,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2y)@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@V2y)@V1s))))))) => (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__LIMPTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@V1x)@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2EDISCRETE__IMP__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1e:tp__ty_2Erealax_2Ereal] : (((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 @ V1e))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V1e))))) => (V3y = V2x))))) => (p@(ap@c_2Ereal__topology_2EClosed@V0s))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__UNIV,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__OPEN__IN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)) & ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1t)))) => (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))) => (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__SING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) <=> ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)) & (~~ (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1a))@V0s))))))))).
thf(ax_thm_2Ereal__topology_2Einterior,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@V0s) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(c_2Ebool_2E_3F@(arr@ty_2Erealax_2Ereal@bool))@(lam@(arr@ty_2Erealax_2Ereal@bool)@(^ [V2t:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@c_2Ereal__topology_2EOpen@V2t))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V1x)@V2t))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t)@V0s)))))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Einterior@V0s) = V0s) <=> (p@(ap@c_2Ereal__topology_2EOpen@V0s)))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ((ap@c_2Ereal__topology_2Einterior@V0s) = V0s))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__EMPTY,lemma,((ap@c_2Ereal__topology_2Einterior@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__UNIV,lemma,((ap@c_2Ereal__topology_2Einterior@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@c_2Ereal__topology_2Einterior@V0s)))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Einterior@V0s)) = (ap@c_2Ereal__topology_2Einterior@V0s))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@V0s))@V0s))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__INTERIOR__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Einterior@V0s))) <=> (p@(ap@c_2Ereal__topology_2EOpen@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@V0s))@(ap@c_2Ereal__topology_2Einterior@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__MAXIMAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@(ap@c_2Ereal__topology_2Einterior@V0s))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__MAXIMAL__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Einterior@V1t))) <=> (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__UNIQUE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & ((p@(ap@c_2Ereal__topology_2EOpen@V1t)) & ( ! [V2t_27:$i] : ((mem@V2t_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t_27)@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V2t_27))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t_27)@V1t))))))) => ((ap@c_2Ereal__topology_2Einterior@V0s) = V1t))))))).
thf(conj_thm_2Ereal__topology_2EIN__INTERIOR,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Einterior@V1s))) <=> (? [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e))))@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SUBSET__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Einterior@V1t))) <=> (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)) = (ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@V0s))@(ap@c_2Ereal__topology_2Einterior@V1t)))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__FINITE__BIGINTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V0s)) => ((ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0s)) = (ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@c_2Ereal__topology_2Einterior)@V0s))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__BIGINTER__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f)))@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@c_2Ereal__topology_2Einterior)@V0f))))))).
thf(conj_thm_2Ereal__topology_2EUNION__INTERIOR__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@V0s))@(ap@c_2Ereal__topology_2Einterior@V1t)))@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Einterior@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s))) => (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY__ALT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Einterior@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V1t)) & (~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (~~ ((ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V1t)@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__LIMIT__POINT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Einterior@V0s))) => (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__UNION__EMPTY__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & ((ap@c_2Ereal__topology_2Einterior@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t)) = (ap@c_2Ereal__topology_2Einterior@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__UNION__EQ__EMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) | (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (((ap@c_2Ereal__topology_2Einterior@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((ap@c_2Ereal__topology_2Einterior@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__UNIONS__OPEN__SUBSETS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(lam@(arr@ty_2Erealax_2Ereal@bool)@(^ [V1t:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@V1t)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@c_2Ereal__topology_2EOpen@V1t))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s))))))) = (ap@c_2Ereal__topology_2Einterior@V0s))))).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__INV,lemma,( ! [V0e:tp__ty_2Erealax_2Ereal] : ((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 @ V0e))) <=> (? [V1n:tp__ty_2Enum_2Enum] : ((~~ (V1n = fo__c_2Enum_2E0)) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Erealax_2Einv@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V1n))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Erealax_2Einv@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V1n))))@(inj__ty_2Erealax_2Ereal @ V0e))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__POW__LBOUND,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((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 @ V0x))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V1n)))@(inj__ty_2Erealax_2Ereal @ V0x))))@(ap@(ap@c_2Ereal_2Epow@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Enum_2Enum @ V1n)))))))).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__POW,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V0x))) => (? [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Enum_2Enum @ V2n))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__POW2,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (? [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@c_2Ereal_2Epow@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__POW__INV,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((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 @ V1y))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) => (? [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Enum_2Enum @ V2n)))@(inj__ty_2Erealax_2Ereal @ V1y)))))))).
thf(conj_thm_2Ereal__topology_2EFORALL__POS__MONO,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V1d:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V2e))) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V1d)))) => (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V2e)))))) & ( ! [V3n:tp__ty_2Enum_2Enum] : ((~~ (V3n = fo__c_2Enum_2E0)) => (p@(ap@V0P@(ap@c_2Erealax_2Einv@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V3n)))))))) => ( ! [V4e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4e))) => (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V4e))))))))).
thf(conj_thm_2Ereal__topology_2EFORALL__SUC,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Enum_2Enum@bool)) => (( ! [V1n:tp__ty_2Enum_2Enum] : ((~~ (V1n = fo__c_2Enum_2E0)) => (p@(ap@V0P@(inj__ty_2Enum_2Enum @ V1n))))) <=> ( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@V0P@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2n))))))))).
thf(conj_thm_2Ereal__topology_2ELT__NZ,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V0n))) <=> (~~ (V0n = fo__c_2Enum_2E0))))).
thf(conj_thm_2Ereal__topology_2EREAL__ARCH__RDIV__EQ__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) & ((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))) & ( ! [V2m:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V2m))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V2m)))@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1c))))))) => (V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(ax_thm_2Ereal__topology_2Eclosure,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@V0s) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@V1x)@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__APPROACHABLE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (? [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V0x))))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__NONEMPTY__OPEN__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Eclosure@V0s))) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V2t)) & (p@(ap@c_2Ereal__topology_2EOpen@V2t))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V2t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@V0s) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@V0s) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@c_2Ereal__topology_2Eclosure@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HULL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@V0s) = (ap@(ap@(c_2Etopology_2Ehull@ty_2Erealax_2Ereal)@c_2Ereal__topology_2EClosed)@V0s))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Eclosure@V0s) = V0s) <=> (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => ((ap@c_2Ereal__topology_2Eclosure@V0s) = V0s))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Eclosure@V0s)) = (ap@c_2Ereal__topology_2Eclosure@V0s))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eclosure@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t)) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTER__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__BIGINTER__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f)))@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@c_2Ereal__topology_2Eclosure)@V0f))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__MINIMAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__MINIMAL__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@V1t)) <=> (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__UNIQUE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) & ( ! [V2t_27:$i] : ((mem@V2t_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V2t_27)) & (p@(ap@c_2Ereal__topology_2EClosed@V2t_27))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2t_27))))))) => ((ap@c_2Ereal__topology_2Eclosure@V0s) = V1t))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__EMPTY,lemma,((ap@c_2Ereal__topology_2Eclosure@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__UNIV,lemma,((ap@c_2Ereal__topology_2Eclosure@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__BIGUNION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V0f)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0f)) = (ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(lam@(arr@ty_2Erealax_2Ereal@bool)@(^ [V1s:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@c_2Ereal__topology_2Eclosure@V1s))@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__EQ__EMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Eclosure@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SUBSET__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@V0s)) <=> (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__EQ__EMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eclosure@V1t)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__IN__INTER__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2u))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eclosure@V1t))) = (ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eclosure@V1t))) = (ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eclosure@V1t)))@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__SUPERSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eclosure@V1t)))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)) = (ap@c_2Ereal__topology_2Eclosure@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__COMPLEMENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s)) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(ap@c_2Ereal__topology_2Einterior@V0s)))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__COMPLEMENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s)) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(ap@c_2Ereal__topology_2Eclosure@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__INTERMEDIATE__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@(ap@c_2Ereal__topology_2Eclosure@V0s))))) => (p@(ap@c_2Ereal__topology_2Econnected@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@c_2Ereal__topology_2Eclosure@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__UNION__STRONG,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & ((p@(ap@c_2Ereal__topology_2Econnected@V1t)) & (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__DIFF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@V1t)) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__CLOSURE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) <=> (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__LIMPT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)) <=> ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1t))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)) <=> ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eclosure@V1t)) = V1t))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE__IDEMP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V0s)))) = (ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR__IDEMP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@V0s)))) = (ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@V0s)))))).
thf(conj_thm_2Ereal__topology_2ENOWHERE__DENSE__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V0s)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V1t)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ENOWHERE__DENSE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V0s)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V1t)) & (~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2u)) & ((~~ (V2u = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2u)@V1t)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2u)@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE__INTER__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))) = (ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V0s)))@(ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR__UNION__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@V0s)))@(ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2EREGULAR__OPEN__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V0s)) = V0s) & ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V1t)) = V1t)) => ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))) = (ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)))))))).
thf(conj_thm_2Ereal__topology_2EREGULAR__CLOSED__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@V0s)) = V0s) & ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@V1t)) = V1t)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t)))))))).
thf(conj_thm_2Ereal__topology_2EREGULAR__CLOSED__BIGUNION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V0f)) & ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V0f)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@V1t)) = V1t))))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0f))) = (ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0f)))))).
thf(conj_thm_2Ereal__topology_2EDIFF__CLOSURE__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t)))@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EDENSE__OPEN__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => ((((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2u))) | ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V2u)))) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2u)@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)))) <=> ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2u)@(ap@c_2Ereal__topology_2Eclosure@V0s))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2u)@(ap@c_2Ereal__topology_2Eclosure@V1t))))))))))))).
thf(ax_thm_2Ereal__topology_2Efrontier,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Efrontier@V0s) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Einterior@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@c_2Ereal__topology_2Efrontier@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CLOSURES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Efrontier@V0s) = (ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__STRADDLE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Efrontier@V1s))) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => ((? [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V3x))))@(inj__ty_2Erealax_2Ereal @ V2e))))) & (? [V4x:tp__ty_2Erealax_2Ereal] : ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V4x))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__EMPTY,lemma,((ap@c_2Ereal__topology_2Efrontier@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__UNIV,lemma,((ap@c_2Ereal__topology_2Efrontier@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@V0s)) <=> (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__COMPLEMENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Efrontier@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s)) = (ap@c_2Ereal__topology_2Efrontier@V0s))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__DISJOINT__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (p@(ap@c_2Ereal__topology_2EOpen@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@(ap@c_2Ereal__topology_2Efrontier@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__UNION__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t)))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@(ap@c_2Ereal__topology_2Efrontier@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__INTERIORS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Efrontier@V0s) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(ap@c_2Ereal__topology_2Einterior@V0s)))@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Efrontier@V0s)))@(ap@c_2Ereal__topology_2Efrontier@V0s)))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Efrontier@V0s)) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eclosure@V0s)))@(ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Einterior@V0s))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__FRONTIER__EMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) | (p@(ap@c_2Ereal__topology_2EClosed@V0s))) => ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Efrontier@V0s)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) | (p@(ap@c_2Ereal__topology_2EClosed@V0s))) => ((ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Efrontier@V0s)) = (ap@c_2Ereal__topology_2Efrontier@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Efrontier@V0s))) = (ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Efrontier@V0s)))))).
thf(conj_thm_2Ereal__topology_2EUNION__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@(ap@c_2Ereal__topology_2Efrontier@V1t)) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t)))@(ap@c_2Ereal__topology_2Efrontier@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@(ap@c_2Ereal__topology_2Efrontier@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__INTER__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & ((~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ ((ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Efrontier@V1t)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__EQ__EMPTY__AS__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & ((ap@c_2Ereal__topology_2Einterior@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) <=> (? [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V1t)) & (V0s = (ap@c_2Ereal__topology_2Efrontier@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) => ((ap@c_2Ereal__topology_2Efrontier@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t)) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@(ap@c_2Ereal__topology_2Efrontier@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__UNION__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@V0s) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Efrontier@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__INTERIOR__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Einterior@V0s)))@(ap@c_2Ereal__topology_2Efrontier@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CLOSURE__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Eclosure@V0s)))@(ap@c_2Ereal__topology_2Efrontier@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESET__DIFF__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Efrontier@V0s)) = (ap@c_2Ereal__topology_2Einterior@V0s))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t)))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Efrontier@V1t)))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t)))))))))).
thf(ax_thm_2Ereal__topology_2Eisnet,axiom,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0g:$i] : ((mem@V0g@(arr@A_27a@(arr@A_27b@bool))) => ((p@(ap@(c_2Ereal__topology_2Eisnet@A_27a@A_27b)@V0g)) <=> ( ! [V1x:$i] : ((mem@V1x@A_27b) => ( ! [V2y:$i] : ((mem@V2y@A_27b) => (( ! [V3z:$i] : ((mem@V3z@A_27a) => ((p@(ap@(ap@V0g@V3z)@V1x)) => (p@(ap@(ap@V0g@V3z)@V2y))))) | ( ! [V4z:$i] : ((mem@V4z@A_27a) => ((p@(ap@(ap@V0g@V4z)@V2y)) => (p@(ap@(ap@V0g@V4z)@V1x)))))))))))))))).
thf(ax_thm_2Ereal__topology_2Enet__TY__DEF,axiom,( ! [A_27a:del]: (? [V0rep:$i] : ((mem@V0rep@(arr@(ty_2Ereal__topology_2Enet@A_27a)@(arr@A_27a@(arr@A_27a@bool)))) & (p@(ap@(ap@(c_2Ebool_2ETYPE__DEFINITION@(arr@A_27a@(arr@A_27a@bool))@(ty_2Ereal__topology_2Enet@A_27a))@(c_2Ereal__topology_2Eisnet@A_27a@A_27a))@V0rep)))))).
thf(conj_thm_2Ereal__topology_2Enet__tybij,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: (( ! [V0a:$i] : ((mem@V0a@(ty_2Ereal__topology_2Enet@A_27a)) => ((ap@(c_2Ereal__topology_2Emk__net@A_27a)@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0a)) = V0a))) & ( ! [V1r:$i] : ((mem@V1r@(arr@A_27b@(arr@A_27b@bool))) => (( ! [V2x:$i] : ((mem@V2x@A_27b) => ( ! [V3y:$i] : ((mem@V3y@A_27b) => (( ! [V4z:$i] : ((mem@V4z@A_27b) => ((p@(ap@(ap@V1r@V4z)@V2x)) => (p@(ap@(ap@V1r@V4z)@V3y))))) | ( ! [V5z:$i] : ((mem@V5z@A_27b) => ((p@(ap@(ap@V1r@V5z)@V3y)) => (p@(ap@(ap@V1r@V5z)@V2x)))))))))) <=> ((ap@(c_2Ereal__topology_2Enetord@A_27b)@(ap@(c_2Ereal__topology_2Emk__net@A_27b)@V1r)) = V1r)))))))).
thf(conj_thm_2Ereal__topology_2ENET,lemma,( ! [A_27a:del]: ( ! [V0n:$i] : ((mem@V0n@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1x:$i] : ((mem@V1x@A_27a) => ( ! [V2y:$i] : ((mem@V2y@A_27a) => (( ! [V3z:$i] : ((mem@V3z@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V3z)@V1x)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V3z)@V2y))))) | ( ! [V4z:$i] : ((mem@V4z@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V4z)@V2y)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V4z)@V1x)))))))))))))).
thf(conj_thm_2Ereal__topology_2EOLDNET,lemma,( ! [A_27a:del]: ( ! [V0n:$i] : ((mem@V0n@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1x:$i] : ((mem@V1x@A_27a) => ( ! [V2y:$i] : ((mem@V2y@A_27a) => (((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V1x)@V1x)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V2y)@V2y))) => (? [V3z:$i] : ((mem@V3z@A_27a) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V3z)@V3z)) & ( ! [V4w:$i] : ((mem@V4w@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V4w)@V3z)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V4w)@V1x)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V4w)@V2y)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ENET__DILEMMA,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1Q:$i] : ((mem@V1Q@(arr@A_27a@bool)) => ( ! [V2net:$i] : ((mem@V2net@(ty_2Ereal__topology_2Enet@A_27a)) => (((? [V3a:$i] : ((mem@V3a@A_27a) & ((? [V4x:$i] : ((mem@V4x@A_27a) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V2net)@V4x)@V3a)))) & ( ! [V5x:$i] : ((mem@V5x@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V2net)@V5x)@V3a)) => (p@(ap@V0P@V5x)))))))) & (? [V6b:$i] : ((mem@V6b@A_27a) & ((? [V7x:$i] : ((mem@V7x@A_27a) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V2net)@V7x)@V6b)))) & ( ! [V8x:$i] : ((mem@V8x@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V2net)@V8x)@V6b)) => (p@(ap@V1Q@V8x))))))))) => (? [V9c:$i] : ((mem@V9c@A_27a) & ((? [V10x:$i] : ((mem@V10x@A_27a) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V2net)@V10x)@V9c)))) & ( ! [V11x:$i] : ((mem@V11x@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V2net)@V11x)@V9c)) => ((p@(ap@V0P@V11x)) & (p@(ap@V1Q@V11x)))))))))))))))))).
thf(ax_thm_2Ereal__topology_2Eat,axiom,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a))) = (surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@(c_2Ereal__topology_2Emk__net@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1x)@(inj__ty_2Erealax_2Ereal @ V0a)))))@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2y)@(inj__ty_2Erealax_2Ereal @ V0a)))))))))))))).
thf(ax_thm_2Ereal__topology_2Eat__infinity,axiom,((surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__infinity) = (surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@(c_2Ereal__topology_2Emk__net@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V1y:$i] : (ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@V0x))@(ap@c_2Ereal_2Eabs@V1y)))))))))).
thf(ax_thm_2Ereal__topology_2Eat__posinfinity,axiom,((surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__posinfinity) = (surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@(c_2Ereal__topology_2Emk__net@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V1y:$i] : (ap@(ap@c_2Ereal_2Ereal__ge@V0x)@V1y))))))))).
thf(ax_thm_2Ereal__topology_2Eat__neginfinity,axiom,((surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ c_2Ereal__topology_2Eat__neginfinity) = (surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@(c_2Ereal__topology_2Emk__net@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V1y:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@V0x)@V1y))))))))).
thf(ax_thm_2Ereal__topology_2Esequentially,axiom,((surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ c_2Ereal__topology_2Esequentially) = (surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum @ (ap@(c_2Ereal__topology_2Emk__net@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V0m:$i] : (lam@ty_2Enum_2Enum@(^ [V1n:$i] : (ap@(ap@c_2Earithmetic_2E_3E_3D@V0m)@V1n))))))))).
thf(ax_thm_2Ereal__topology_2Ewithin,axiom,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((ap@(ap@(c_2Ereal__topology_2Ewithin@A_27a)@V0net)@V1s) = (ap@(c_2Ereal__topology_2Emk__net@A_27a)@(lam@A_27a@(^ [V2x:$i] : (lam@A_27a@(^ [V3y:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0net)@V2x)@V3y))@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)))))))))))))).
thf(ax_thm_2Ereal__topology_2Ein__direction,axiom,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1v:tp__ty_2Erealax_2Ereal] : ((surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Ein__direction@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1v))) = (surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2b:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2b)@(ap@(c_2Ebool_2E_3F@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3c:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V3c))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@V2b)@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(ap@c_2Erealax_2Ereal__mul@V3c)@(inj__ty_2Erealax_2Ereal @ V1v))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EAT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V0a))))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V0a))))))))))).
thf(conj_thm_2Ereal__topology_2EAT__INFINITY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eat__infinity)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1y)))))))).
thf(conj_thm_2Ereal__topology_2EAT__POSINFINITY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eat__posinfinity)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__ge@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))))).
thf(conj_thm_2Ereal__topology_2EAT__NEGINFINITY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eat__neginfinity)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))))).
thf(conj_thm_2Ereal__topology_2ESEQUENTIALLY,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@ty_2Enum_2Enum)@c_2Ereal__topology_2Esequentially)@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))) <=> (p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Ereal__topology_2EWITHIN,lemma,( ! [A_27a:del]: ( ! [V0n:$i] : ((mem@V0n@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2x:$i] : ((mem@V2x@A_27a) => ( ! [V3y:$i] : ((mem@V3y@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@(ap@(ap@(c_2Ereal__topology_2Ewithin@A_27a)@V0n)@V1s))@V2x)@V3y)) <=> ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0n)@V2x)@V3y)) & (p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)))))))))))))).
thf(conj_thm_2Ereal__topology_2EIN__DIRECTION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1v:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Ein__direction@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1v)))@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V0a))))) & (? [V4c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4c))) & ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0a))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V4c))@(inj__ty_2Erealax_2Ereal @ V1v)))))))))))))).
thf(conj_thm_2Ereal__topology_2EWITHIN__UNIV,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0x)))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))) = (surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0x)))))).
thf(conj_thm_2Ereal__topology_2EWITHIN__WITHIN,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => ((ap@(ap@(c_2Ereal__topology_2Ewithin@A_27a)@(ap@(ap@(c_2Ereal__topology_2Ewithin@A_27a)@V0net)@V1s))@V2t) = (ap@(ap@(c_2Ereal__topology_2Ewithin@A_27a)@V0net)@(ap@(ap@(c_2Epred__set_2EINTER@A_27a)@V1s)@V2t))))))))))).
thf(ax_thm_2Ereal__topology_2Etrivial__limit,axiom,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net)) <=> (( ! [V1a:$i] : ((mem@V1a@A_27a) => ( ! [V2b:$i] : ((mem@V2b@A_27a) => (V1a = V2b))))) | (? [V3a:$i] : ((mem@V3a@A_27a) & (? [V4b:$i] : ((mem@V4b@A_27a) & ((~ (V3a = V4b)) & ( ! [V5x:$i] : ((mem@V5x@A_27a) => ((~ (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0net)@V5x)@V3a))) & (~ (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0net)@V5x)@V4b))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__WITHIN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Ereal__topology_2Etrivial__limit@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))@V0s))) <=> (~~ (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a))))))).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__INFINITY,lemma,(~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eat__infinity)))).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__POSINFINITY,lemma,(~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eat__posinfinity)))).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__NEGINFINITY,lemma,(~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eat__neginfinity)))).
thf(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__SEQUENTIALLY,lemma,(~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@ty_2Enum_2Enum)@c_2Ereal__topology_2Esequentially)))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__CLOSED__TRIVIAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V1s)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V1s)))) => (p@(ap@(c_2Ereal__topology_2Etrivial__limit@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ENONTRIVIAL__LIMIT__WITHIN,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net)) => (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@(ap@(ap@(c_2Ereal__topology_2Ewithin@A_27a)@V0net)@V1s)))))))))).
thf(ax_thm_2Ereal__topology_2Eeventually,axiom,( ! [A_27a:del]: ( ! [V0p:$i] : ((mem@V0p@(arr@A_27a@bool)) => ( ! [V1net:$i] : ((mem@V1net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V0p)@V1net)) <=> ((p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V1net)) | (? [V2y:$i] : ((mem@V2y@A_27a) & ((? [V3x:$i] : ((mem@V3x@A_27a) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V1net)@V3x)@V2y)))) & ( ! [V4x:$i] : ((mem@V4x@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V1net)@V4x)@V2y)) => (p@(ap@V0p@V4x)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__HAPPENS,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V1p)@V0net)) => ((p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net)) | (? [V2x:$i] : ((mem@V2x@A_27a) & (p@(ap@V1p@V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2p:$i] : ((mem@V2p@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V2p)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))@V0s))) <=> (? [V3d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3d))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V1a))))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V1a))))@(inj__ty_2Erealax_2Ereal @ V3d))))) => (p@(ap@V2p@(inj__ty_2Erealax_2Ereal @ V4x)))))))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2p:$i] : ((mem@V2p@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V2p)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))@V0s))) <=> (? [V3d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3d))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V1a))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V1a))))@(inj__ty_2Erealax_2Ereal @ V3d))))) => (p@(ap@V2p@(inj__ty_2Erealax_2Ereal @ V4x)))))))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1p:$i] : ((mem@V1p@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V1p)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))) <=> (? [V2d:tp__ty_2Erealax_2Ereal] : ((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 @ V2d))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V0a))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V0a))))@(inj__ty_2Erealax_2Ereal @ V2d)))) => (p@(ap@V1p@(inj__ty_2Erealax_2Ereal @ V3x)))))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__SEQUENTIALLY,lemma,( ! [V0p:$i] : ((mem@V0p@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Enum_2Enum)@V0p)@c_2Ereal__topology_2Esequentially)) <=> (? [V1N:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1N))@(inj__ty_2Enum_2Enum @ V2n))) => (p@(ap@V0p@(inj__ty_2Enum_2Enum @ V2n)))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY,lemma,( ! [V0p:$i] : ((mem@V0p@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V0p)@c_2Ereal__topology_2Eat__infinity)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1b))) => (p@(ap@V0p@(inj__ty_2Erealax_2Ereal @ V2x)))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__POSINFINITY,lemma,( ! [V0p:$i] : ((mem@V0p@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V0p)@c_2Ereal__topology_2Eat__posinfinity)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__ge@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))) => (p@(ap@V0p@(inj__ty_2Erealax_2Ereal @ V2x)))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__NEGINFINITY,lemma,( ! [V0p:$i] : ((mem@V0p@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V0p)@c_2Ereal__topology_2Eat__neginfinity)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b))) => (p@(ap@V0p@(inj__ty_2Erealax_2Ereal @ V2x)))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY__POS,lemma,( ! [V0p:$i] : ((mem@V0p@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V0p)@c_2Ereal__topology_2Eat__infinity)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1b))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1b))) => (p@(ap@V0p@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2EALWAYS__EVENTUALLY,lemma,( ! [A_27a:del]: ( ! [V0p:$i] : ((mem@V0p@(arr@A_27a@bool)) => ( ! [V1net:$i] : ((mem@V1net@(ty_2Ereal__topology_2Enet@A_27a)) => (( ! [V2x:$i] : ((mem@V2x@A_27a) => (p@(ap@V0p@V2x)))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V0p)@V1net))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__AND,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27a@bool)) => ( ! [V2q:$i] : ((mem@V2q@(arr@A_27a@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@V1p@V3x))@(ap@V2q@V3x)))))@V0net)) <=> ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V1p)@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V2q)@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__MONO,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27a@bool)) => ( ! [V2q:$i] : ((mem@V2q@(arr@A_27a@bool)) => ((( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@V1p@V3x)) => (p@(ap@V2q@V3x))))) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V1p)@V0net))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V2q)@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__MP,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27a@bool)) => ( ! [V2q:$i] : ((mem@V2q@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1p@V3x))@(ap@V2q@V3x)))))@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V1p)@V0net))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V2q)@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__FALSE,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V1x:$i] : c_2Ebool_2EF)))@V0net)) <=> (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__TRUE,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V1x:$i] : c_2Ebool_2ET)))@V0net)) <=> $true))))).
thf(conj_thm_2Ereal__topology_2ENOT__EVENTUALLY,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27a@bool)) => ((( ! [V2x:$i] : ((mem@V2x@A_27a) => (~~ (p@(ap@V1p@V2x))))) & (~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net)))) => (~~ (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@V1p)@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__FORALL,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27b@(arr@A_27a@bool))) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2s)) & (~~ (V2s = (c_2Epred__set_2EEMPTY@A_27b)))) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V4a:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V4a)@V2s))@(ap@(ap@V1p@V4a)@V3x))))))))@V0net)) <=> ( ! [V5a:$i] : ((mem@V5a@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V5a)@V2s)) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(ap@V1p@V5a))@V0net)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EFORALL__EVENTUALLY,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1p:$i] : ((mem@V1p@(arr@A_27b@(arr@A_27a@bool))) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2s)) & (~~ (V2s = (c_2Epred__set_2EEMPTY@A_27b)))) => (( ! [V3a:$i] : ((mem@V3a@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3a)@V2s)) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(ap@V1p@V3a))@V0net))))) <=> (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V5a:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V5a)@V2s))@(ap@(ap@V1p@V5a)@V4x))))))))@V0net))))))))))))).
thf(ax_thm_2Ereal__topology_2Etendsto,axiom,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2net:$i] : ((mem@V2net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@V2net)) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V3e)))))@V2net)))))))))))).
thf(ax_thm_2Ereal__topology_2Elim__def,axiom,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Ereal__topology_2Elim__def@A_27a)@V0net)@V1f)) = (surj__ty_2Erealax_2Ereal @ (ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2l:$i] : (ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@V2l)@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2net:$i] : ((mem@V2net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@V2net)) <=> ((p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V2net)) | ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) => (? [V4y:$i] : ((mem@V4y@A_27a) & ((? [V5x:$i] : ((mem@V5x@A_27a) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V2net)@V5x)@V4y)))) & ( ! [V6x:$i] : ((mem@V6x@A_27a) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V2net)@V6x)@V4y)) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V3e)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__LE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V3s))) <=> ( ! [V4e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4e))) => (? [V5d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V5d))) & ( ! [V6x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@V3s)) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@(inj__ty_2Erealax_2Ereal @ V2a))))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@(inj__ty_2Erealax_2Ereal @ V2a))))@(inj__ty_2Erealax_2Ereal @ V5d))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V6x)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V4e))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V3s))) <=> ( ! [V4e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4e))) => (? [V5d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V5d))) & ( ! [V6x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@V3s)) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@(inj__ty_2Erealax_2Ereal @ V2a))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@(inj__ty_2Erealax_2Ereal @ V2a))))@(inj__ty_2Erealax_2Ereal @ V5d))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V6x)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V4e))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT__LE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : ((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 @ V4d))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V2a))))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V2a))))@(inj__ty_2Erealax_2Ereal @ V4d)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4d))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V2a))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V2a))))@(inj__ty_2Erealax_2Ereal @ V4d)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT__INFINITY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Eat__infinity)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => (? [V3b:tp__ty_2Erealax_2Ereal] : ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V4x)))@(inj__ty_2Erealax_2Ereal @ V3b))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT__INFINITY__POS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Eat__infinity)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (? [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3b))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V4x)))@(inj__ty_2Erealax_2Ereal @ V3b))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT__POSINFINITY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Eat__posinfinity)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => (? [V3b:tp__ty_2Erealax_2Ereal] : ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__ge@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V3b))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT__NEGINFINITY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Eat__neginfinity)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (? [V3b:tp__ty_2Erealax_2Ereal] : ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V3b))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__SEQUENTIALLY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0s)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => (? [V3N:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3N))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V4n)))@(inj__ty_2Erealax_2Ereal @ V1l))))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__EVENTUALLY,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V1f@V3x))@(inj__ty_2Erealax_2Ereal @ V2l)))))@V0net)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__POSINFINITY__SEQUENTIALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Eat__posinfinity)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@V0f@(ap@c_2Ereal_2Ereal__of__num@V2n)))))@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially))))))).
thf(conj_thm_2Ereal__topology_2ELIM__INFINITY__POSINFINITY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Eat__infinity)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Eat__posinfinity))))))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__EMPTY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__SUBSET,lemma,( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4s:$i] : ((mem@V4s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a)))@V4s))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0t)@V4s))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a)))@V0t))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__UNION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V3s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V4t)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__UNION__UNIV,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V3s))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V4t))) & ((ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__COMPOSE__WITHIN,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4y:tp__ty_2Erealax_2Ereal] : ( ! [V5z:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V4y))@V0net)) & ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V6w:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1f@V6w))@V3s))@(ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V1f@V6w))@(inj__ty_2Erealax_2Ereal @ V4y)))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V2g@(inj__ty_2Erealax_2Ereal @ V4y)))@(inj__ty_2Erealax_2Ereal @ V5z)))))))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2g)@(inj__ty_2Erealax_2Ereal @ V5z))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V4y)))@V3s))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2g)@V1f))@(inj__ty_2Erealax_2Ereal @ V5z))@V0net))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__COMPOSE__AT,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3y:tp__ty_2Erealax_2Ereal] : ( ! [V4z:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3y))@V0net)) & ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V5w:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V1f@V5w))@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V2g@(inj__ty_2Erealax_2Ereal @ V3y)))@(inj__ty_2Erealax_2Ereal @ V4z))))))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2g)@(inj__ty_2Erealax_2Ereal @ V4z))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3y)))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2g)@V1f))@(inj__ty_2Erealax_2Ereal @ V4z))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT__WITHIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V3s))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@V3s)) & (p@(ap@c_2Ereal__topology_2EOpen@V3s))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V3s))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a))))))))))))).
thf(ax_thm_2Ereal__topology_2Efrom,axiom,( ! [V0n:tp__ty_2Enum_2Enum] : ((ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V0n)) = (ap@(c_2Epred__set_2EGSPEC@ty_2Enum_2Enum@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V1m:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Enum_2Enum@bool)@V1m)@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V0n))@V1m)))))))).
thf(conj_thm_2Ereal__topology_2EFROM__0,lemma,((ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)) = (c_2Epred__set_2EUNIV@ty_2Enum_2Enum))).
thf(conj_thm_2Ereal__topology_2EIN__FROM,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V0m))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V1n)))) <=> (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V0m))))))).
thf(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG__GEN,lemma,( ! [V0k:tp__ty_2Enum_2Enum] : ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V0k)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Enum_2Enum@bool))@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V0k)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0k))@(inj__ty_2Enum_2Enum @ V2n)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))))))).
thf(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG__MAX,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2p:tp__ty_2Enum_2Enum] : ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V2p)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))) = (ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2EMAX@(inj__ty_2Enum_2Enum @ V2p))@(inj__ty_2Enum_2Enum @ V0m)))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG,lemma,( ! [V0k:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V0k)))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V1n))) = (ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V0k))@(inj__ty_2Enum_2Enum @ V1n)))))).
thf(conj_thm_2Ereal__topology_2EINFINITE__FROM,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V0n))))))).
thf(conj_thm_2Ereal__topology_2EWLOG__LT,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@bool))) => ((( ! [V1m:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V1m)))) & (( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))) <=> (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V3n))@(inj__ty_2Enum_2Enum @ V2m)))))) & ( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V4m))@(inj__ty_2Enum_2Enum @ V5n))) => (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V4m))@(inj__ty_2Enum_2Enum @ V5n)))))))) => ( ! [V6m:tp__ty_2Enum_2Enum] : ( ! [V7y:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0P@(inj__ty_2Enum_2Enum @ V6m))@(inj__ty_2Enum_2Enum @ V7y))))))))).
thf(conj_thm_2Ereal__topology_2ELT__EXISTS,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V1n))) <=> (? [V2d:tp__ty_2Enum_2Enum] : (V1n = (surj__ty_2Enum_2Enum @ (ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V0m))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2d)))))))))).
thf(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT__EQ,lemma,( ! [V0R:$i] : ((mem@V0R@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@bool))) => (( ! [V1x:tp__ty_2Enum_2Enum] : ( ! [V2y:tp__ty_2Enum_2Enum] : ( ! [V3z:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V1x))@(inj__ty_2Enum_2Enum @ V2y))) & (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V2y))@(inj__ty_2Enum_2Enum @ V3z)))) => (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V1x))@(inj__ty_2Enum_2Enum @ V3z))))))) => (( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V4m))@(inj__ty_2Enum_2Enum @ V5n))) => (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V4m))@(inj__ty_2Enum_2Enum @ V5n)))))) <=> ( ! [V6n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V6n))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V6n)))))))))).
thf(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT,lemma,( ! [V0R:$i] : ((mem@V0R@(arr@ty_2Enum_2Enum@(arr@ty_2Enum_2Enum@bool))) => ((( ! [V1x:tp__ty_2Enum_2Enum] : ( ! [V2y:tp__ty_2Enum_2Enum] : ( ! [V3z:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V1x))@(inj__ty_2Enum_2Enum @ V2y))) & (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V2y))@(inj__ty_2Enum_2Enum @ V3z)))) => (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V1x))@(inj__ty_2Enum_2Enum @ V3z))))))) & ( ! [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V4n))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V4n)))))) => ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))) => (p@(ap@(ap@V0R@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n)))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__SEQUENTIAL__INJ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) <=> (? [V2f:$i] : ((mem@V2f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) & (( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2f@(inj__ty_2Enum_2Enum @ V3n)))@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V1s)@(inj__ty_2Erealax_2Ereal @ V0x))))) & (( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : (((surj__ty_2Erealax_2Ereal @ (ap@V2f@(inj__ty_2Enum_2Enum @ V4m))) = (surj__ty_2Erealax_2Ereal @ (ap@V2f@(inj__ty_2Enum_2Enum @ V5n)))) <=> (V4m = V5n)))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V2f)@(inj__ty_2Erealax_2Ereal @ V0x))@c_2Ereal__topology_2Esequentially))))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__SEQUENTIAL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) <=> (? [V2f:$i] : ((mem@V2f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) & (( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2f@(inj__ty_2Enum_2Enum @ V3n)))@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V1s)@(inj__ty_2Erealax_2Ereal @ V0x))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V2f)@(inj__ty_2Erealax_2Ereal @ V0x))@c_2Ereal__topology_2Esequentially)))))))))).
thf(conj_thm_2Ereal__topology_2EINFINITE__SUPERSET,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => (((~~ (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@A_27a)@V0s)@V1t))) => (~~ (p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN__BALL__CBALL,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V2t)) & (p@(ap@c_2Ereal__topology_2EOpen@V2t))) => (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V2t))))))))))) & (( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V4x))@V3s)) <=> ( ! [V5e:tp__ty_2Erealax_2Ereal] : ((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 @ V5e))) => (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V3s)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5e))))))))))))) & ( ! [V6s:$i] : ((mem@V6s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V7x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V7x))@V6s)) <=> ( ! [V8e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V8e))) => (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V6s)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@(inj__ty_2Erealax_2Ereal @ V8e)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V2t)) & (p@(ap@c_2Ereal__topology_2EOpen@V2t))) => (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V2t)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__BALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__CBALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))).
thf(conj_thm_2Ereal__topology_2EINFINITE__OPEN__IN,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) & (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) & (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V0u))))) => (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1s))))))))).
thf(ax_thm_2Ereal__topology_2Econdensation__point__of,axiom,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econdensation__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2t)) & (p@(ap@c_2Ereal__topology_2EOpen@V2t))) => (~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1s)@V2t)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__OF__SUBSET,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econdensation__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1s)@V2t))) => (p@(ap@(ap@c_2Ereal__topology_2Econdensation__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V2t))))))))).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__IMP__LIMPT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econdensation__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))))))).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL__CBALL,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Econdensation__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))) & ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Econdensation__point__of@(inj__ty_2Erealax_2Ereal @ V4x))@V3s)) <=> ( ! [V5e:tp__ty_2Erealax_2Ereal] : ((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 @ V5e))) => (~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V3s)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5e))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Econdensation__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__CBALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Econdensation__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__LINEAR,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1h:$i] : ((mem@V1h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) & (p@(ap@c_2Ereal__topology_2Elinear@V1h))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@V1h@(ap@V2f@V4x)))))@(ap@V1h@(inj__ty_2Erealax_2Ereal @ V3l)))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CONST,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V2x:$i] : (inj__ty_2Erealax_2Ereal @ V1a))))@(inj__ty_2Erealax_2Ereal @ V1a))@V0net))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CMUL,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3c))@(ap@V1f@V4x)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3c))@(inj__ty_2Erealax_2Ereal @ V2l)))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CMUL__EQ,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3c:tp__ty_2Erealax_2Ereal] : ((~~ (V3c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3c))@(ap@V1f@V4x)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3c))@(inj__ty_2Erealax_2Ereal @ V2l)))@V0net)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NEG,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V1f@V3x)))))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V2l)))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NEG__EQ,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V1f@V3x)))))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V2l)))@V0net)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__ADD,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V4m))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V1f@V5x))@(ap@V2g@V5x)))))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V3l))@(inj__ty_2Erealax_2Ereal @ V4m)))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__ABS,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@c_2Ereal_2Eabs@(ap@V1f@V3x)))))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2l)))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__SUB,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V4m))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1f@V5x))@(ap@V2g@V5x)))))@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3l))@(inj__ty_2Erealax_2Ereal @ V4m)))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__MAX,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V4m))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Ereal_2Emax@(ap@V1f@V5x))@(ap@V2g@V5x)))))@(ap@(ap@c_2Ereal_2Emax@(inj__ty_2Erealax_2Ereal @ V3l))@(inj__ty_2Erealax_2Ereal @ V4m)))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__MIN,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V4m))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Ereal_2Emin@(ap@V1f@V5x))@(ap@V2g@V5x)))))@(ap@(ap@c_2Ereal_2Emin@(inj__ty_2Erealax_2Ereal @ V3l))@(inj__ty_2Erealax_2Ereal @ V4m)))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1f@V3x))@(inj__ty_2Erealax_2Ereal @ V2l)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__ABS,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@c_2Ereal_2Eabs@(ap@V1f@V2x)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL__EQ,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((~~ (V2c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@V1f@V3x)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@V1f@V3x)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__ADD,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V1f@V3x))@(ap@V2g@V3x)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__SUB,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1f@V3x))@(ap@V2g@V3x)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__COMPARISON,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V1f@V3x)))@(ap@V2g@V3x)))))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@V2g@V4x))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2i:$i] : ((mem@V2i@A_27b) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4a:$i] : (ap@V1f@V4a))))@(inj__ty_2Erealax_2Ereal @ V3l))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__BOUND,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V3n:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V1f@V3n)))@(ap@c_2Ereal_2Eabs@(ap@V2g@V3n))))))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL__BOUNDED,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3B:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4a:$i] : (ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V2g@V4a))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V1f@V4a)))@(inj__ty_2Erealax_2Ereal @ V3B))))))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V5n:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V1f@V5n))@(ap@V2g@V5n)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__SUM,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27b)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@A_27b@ty_2Erealax_2Ereal))) => ( ! [V2l:$i] : ((mem@V2l@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3s)) & ( ! [V4i:$i] : ((mem@V4i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4i)@V3s)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27b)@(ap@V1f@V4i))@(ap@V2l@V4i))@V0net)))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27b)@(lam@A_27b@(^ [V5x:$i] : (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V3s)@(lam@A_27a@(^ [V6i:$i] : (ap@(ap@V1f@V6i)@V5x)))))))@(ap@(ap@(c_2Eiterate_2ESum@A_27a)@V3s)@V2l))@V0net)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__NULL__SUM,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@A_27b@ty_2Erealax_2Ereal))) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2s)) & ( ! [V3a:$i] : ((mem@V3a@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3a)@V2s)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@V1f@V4x)@V3a))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@(c_2Eiterate_2ESum@A_27b)@V2s)@(ap@V1f@V5x)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__IN__CLOSED__SET,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V2s)) & ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1f@V4x))@V2s))))@V0net)) & ((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3l))@V2s)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__ABS__UBOUND,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V1f@V4x)))@(inj__ty_2Erealax_2Ereal @ V3b)))))@V0net)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2l)))@(inj__ty_2Erealax_2Ereal @ V3b)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__ABS__LBOUND,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(ap@c_2Ereal_2Eabs@(ap@V1f@V4x))))))@V0net)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2l))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__UNIQUE,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3l_27:tp__ty_2Erealax_2Ereal] : (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l_27))@V0net)))) => (V2l = V3l_27)))))))))).
thf(conj_thm_2Ereal__topology_2ETENDSTO__LIM,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Ereal__topology_2Elim__def@A_27a)@V0net)@V1f)) = V2l))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CONST__EQ,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V3x:$i] : (inj__ty_2Erealax_2Ereal @ V1c))))@(inj__ty_2Erealax_2Ereal @ V2d))@V0net)) <=> ((p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net)) | (V1c = V2d))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORM__LIM__ADD,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27b@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27b@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V3g:$i] : ((mem@V3g@(arr@A_27b@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V4l:$i] : ((mem@V4l@(arr@A_27b@ty_2Erealax_2Ereal)) => ( ! [V5m:$i] : ((mem@V5m@(arr@A_27b@ty_2Erealax_2Ereal)) => ((( ! [V6e:tp__ty_2Erealax_2Ereal] : ((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 @ V6e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V7x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V8n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V8n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V2f@V8n)@V7x))@(ap@V4l@V8n))))@(inj__ty_2Erealax_2Ereal @ V6e)))))))))@V0net)))) & ( ! [V9e:tp__ty_2Erealax_2Ereal] : ((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 @ V9e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V10x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V11n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V11n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V3g@V11n)@V10x))@(ap@V5m@V11n))))@(inj__ty_2Erealax_2Ereal @ V9e)))))))))@V0net))))) => ( ! [V12e:tp__ty_2Erealax_2Ereal] : ((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 @ V12e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V13x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V14n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V14n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@V2f@V14n)@V13x))@(ap@(ap@V3g@V14n)@V13x)))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@V4l@V14n))@(ap@V5m@V14n)))))@(inj__ty_2Erealax_2Ereal @ V12e)))))))))@V0net)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORM__LIM__SUB,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27b@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27b@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V3g:$i] : ((mem@V3g@(arr@A_27b@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V4l:$i] : ((mem@V4l@(arr@A_27b@ty_2Erealax_2Ereal)) => ( ! [V5m:$i] : ((mem@V5m@(arr@A_27b@ty_2Erealax_2Ereal)) => ((( ! [V6e:tp__ty_2Erealax_2Ereal] : ((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 @ V6e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V7x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V8n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V8n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V2f@V8n)@V7x))@(ap@V4l@V8n))))@(inj__ty_2Erealax_2Ereal @ V6e)))))))))@V0net)))) & ( ! [V9e:tp__ty_2Erealax_2Ereal] : ((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 @ V9e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V10x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V11n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V11n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V3g@V11n)@V10x))@(ap@V5m@V11n))))@(inj__ty_2Erealax_2Ereal @ V9e)))))))))@V0net))))) => ( ! [V12e:tp__ty_2Erealax_2Ereal] : ((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 @ V12e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V13x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V14n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V14n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V2f@V14n)@V13x))@(ap@(ap@V3g@V14n)@V13x)))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V4l@V14n))@(ap@V5m@V14n)))))@(inj__ty_2Erealax_2Ereal @ V12e)))))))))@V0net)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORM__LIM__BILINEAR,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27b@bool)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V3f:$i] : ((mem@V3f@(arr@A_27b@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V4g:$i] : ((mem@V4g@(arr@A_27b@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V5l:$i] : ((mem@V5l@(arr@A_27b@ty_2Erealax_2Ereal)) => ( ! [V6m:$i] : ((mem@V6m@(arr@A_27b@ty_2Erealax_2Ereal)) => ( ! [V7b1:tp__ty_2Erealax_2Ereal] : ( ! [V8b2:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ebilinear@V2h)) & ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V9x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V10n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V10n))@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V5l@V10n)))@(inj__ty_2Erealax_2Ereal @ V7b1)))))))))@V0net)) & ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V11x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V12n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V12n))@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V6m@V12n)))@(inj__ty_2Erealax_2Ereal @ V8b2)))))))))@V0net)) & (( ! [V13e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V13e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V14x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V15n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V15n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V3f@V15n)@V14x))@(ap@V5l@V15n))))@(inj__ty_2Erealax_2Ereal @ V13e)))))))))@V0net)))) & ( ! [V16e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V16e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V17x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V18n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V18n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V4g@V18n)@V17x))@(ap@V6m@V18n))))@(inj__ty_2Erealax_2Ereal @ V16e)))))))))@V0net)))))))) => ( ! [V19e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V19e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V20x:$i] : (ap@(c_2Ebool_2E_21@A_27b)@(lam@A_27b@(^ [V21n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@V1P@V21n))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V2h@(ap@(ap@V3f@V21n)@V20x))@(ap@(ap@V4g@V21n)@V20x)))@(ap@(ap@V2h@(ap@V5l@V21n))@(ap@V6m@V21n)))))@(inj__ty_2Erealax_2Ereal @ V19e)))))))))@V0net)))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__BILINEAR,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1h:$i] : ((mem@V1h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3g:$i] : ((mem@V3g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V4l:tp__ty_2Erealax_2Ereal] : ( ! [V5m:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2f)@(inj__ty_2Erealax_2Ereal @ V4l))@V0net)) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V3g)@(inj__ty_2Erealax_2Ereal @ V5m))@V0net)) & (p@(ap@c_2Ereal__topology_2Ebilinear@V1h)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V6x:$i] : (ap@(ap@V1h@(ap@V2f@V6x))@(ap@V3g@V6x)))))@(ap@(ap@V1h@(inj__ty_2Erealax_2Ereal @ V4l))@(inj__ty_2Erealax_2Ereal @ V5m)))@V0net))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__ID,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : V2x)))@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@V1s))))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT__ID,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : V1x)))@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))))).
thf(conj_thm_2Ereal__topology_2ELIM__AT__ZERO,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@V0f@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V2a))@V3x)))))@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Eat@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))).
thf(ax_thm_2Ereal__topology_2Enetlimit,axiom,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ((ap@(c_2Ereal__topology_2Enetlimit@A_27a)@V0net) = (ap@(c_2Emin_2E_40@A_27a)@(lam@A_27a@(^ [V1a:$i] : (ap@(c_2Ebool_2E_21@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@c_2Ebool_2E_7E@(ap@(ap@(ap@(c_2Ereal__topology_2Enetord@A_27a)@V0net)@V2x)@V1a))))))))))))).
thf(conj_thm_2Ereal__topology_2ENETLIMIT__WITHIN,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@V1s)))) => ((surj__ty_2Erealax_2Ereal @ (ap@(c_2Ereal__topology_2Enetlimit@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@V1s))) = V0a)))))).
thf(conj_thm_2Ereal__topology_2ENETLIMIT__AT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@(c_2Ereal__topology_2Enetlimit@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))) = V0a))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1f@V4x))@(ap@V2g@V4x)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__EVENTUALLY,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V1f@V4x))@(ap@V2g@V4x)))))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN,lemma,( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4s:$i] : ((mem@V4s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V5d:tp__ty_2Erealax_2Ereal] : (((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 @ V5d))) & (( ! [V6x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x_27))@V4s)) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x_27))@(inj__ty_2Erealax_2Ereal @ V3x))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x_27))@(inj__ty_2Erealax_2Ereal @ V3x))))@(inj__ty_2Erealax_2Ereal @ V5d))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V6x_27))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V6x_27)))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x)))@V4s))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2g)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x)))@V4s)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AT,lemma,( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4d:tp__ty_2Erealax_2Ereal] : (((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 @ V4d))) & (( ! [V5x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@(inj__ty_2Erealax_2Ereal @ V3x))))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@(inj__ty_2Erealax_2Ereal @ V3x))))@(inj__ty_2Erealax_2Ereal @ V4d)))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x_27))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V5x_27)))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x)))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2g)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__EQ,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1f@V4x))@(ap@V2g@V4x)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0net)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET,lemma,( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Emin_2E_3D@bool)@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V3s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V4t)))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V3s))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V4t)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET__IMP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V4t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V3s)))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V3s)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V4t))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__WITHIN,lemma,( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : ( ! [V5s:$i] : ((mem@V5s@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V3a = V4b)) & (( ! [V6x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@V5s)) & ((~~ (V6x = V3a)) & (~~ (V6x = V4b)))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V6x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V6x)))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a)))@V5s))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2g)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a)))@V5s)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__AT,lemma,( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : (((~~ (V3a = V4b)) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : (((~~ (V5x = V3a)) & (~~ (V5x = V4b))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V5x)))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a)))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2g)@(inj__ty_2Erealax_2Ereal @ V0l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4l:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EOpen@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@V2s)) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V2s)) & (~~ (V5x = V3a))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V5x)))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V4l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a))))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1g)@(inj__ty_2Erealax_2Ereal @ V4l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN__IN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@V2s)) & (( ! [V6x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@V2s)) & (~~ (V6x = V4a))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V6x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V6x)))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V5l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V4a)))@V3t)))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1g)@(inj__ty_2Erealax_2Ereal @ V5l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V4a)))@V3t))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CASES__FINITE__SEQUENTIALLY,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@(ap@(c_2Epred__set_2EGSPEC@ty_2Enum_2Enum@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Enum_2Enum@bool)@V4n)@(ap@V0P@V4n))))))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@V0P@V5n))@(ap@V1f@V5n))@(ap@V2g@V5n)))))@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V2g)@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CASES__COFINITE__SEQUENTIALLY,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@(ap@(c_2Epred__set_2EGSPEC@ty_2Enum_2Enum@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Enum_2Enum@bool)@V4n)@(ap@c_2Ebool_2E_7E@(ap@V0P@V4n)))))))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@V0P@V5n))@(ap@V1f@V5n))@(ap@V2g@V5n)))))@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CASES__SEQUENTIALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3m:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@V4n))@(ap@V0f@V4n))@(ap@V1g@V4n)))))@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially))) & (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V3m))@V5n))@(ap@V0f@V5n))@(ap@V1g@V5n)))))@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially))) & (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V6n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Earithmetic_2E_3C_3D@V6n)@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V0f@V6n))@(ap@V1g@V6n)))))@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V1g)@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V7n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Eprim__rec_2E_3C@V7n)@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V0f@V7n))@(ap@V1g@V7n)))))@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V1g)@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CONG__WITHIN,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4s:$i] : ((mem@V4s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V5x:tp__ty_2Erealax_2Ereal] : ((~~ (V5x = V0a)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V5x)))))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@V1f@V6x))))@(inj__ty_2Erealax_2Ereal @ V3l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@V4s))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2g)@(inj__ty_2Erealax_2Ereal @ V3l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@V4s)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CONG__AT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((~~ (V4x = V0a)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V4x)))))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@V1f@V5x))))@(inj__ty_2Erealax_2Ereal @ V3l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2g)@(inj__ty_2Erealax_2Ereal @ V3l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SEQUENTIAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Eclosure@V0s))) <=> (? [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) & (( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V3n)))@V0s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V2x)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__CONTAINS__SEQUENTIAL__LIMIT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:$i] : ((mem@V1x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1x@(inj__ty_2Enum_2Enum @ V3n)))@V0s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V1x)@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially)))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2l))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__SEQUENTIAL__LIMITS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) <=> ( ! [V1x:$i] : ((mem@V1x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1x@(inj__ty_2Enum_2Enum @ V3n)))@V0s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V1x)@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2l))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__APPROACHABLE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V1s)) => (( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => (? [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V0x))))@(inj__ty_2Erealax_2Ereal @ V2e))))))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EIN__CLOSURE__DELETE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))))) <=> (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EDENSE__IMP__PERFECT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Eclosure@V0s) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EDENSE__LIMIT__POINTS,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:$i] : ((mem@V1x@A_27a) => (((ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@V2x)@V0s))))) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) <=> ((ap@c_2Ereal__topology_2Eclosure@V0s) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ESEQ__OFFSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2k:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@V3i)@(inj__ty_2Enum_2Enum @ V2k))))))@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)))))))).
thf(conj_thm_2Ereal__topology_2ESEQ__OFFSET__NEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2k:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@V0f@(ap@(ap@c_2Earithmetic_2E_2D@V3i)@(inj__ty_2Enum_2Enum @ V2k))))))@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)))))))).
thf(conj_thm_2Ereal__topology_2ESEQ__OFFSET__REV,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2k:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@V3i)@(inj__ty_2Enum_2Enum @ V2k))))))@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)))))))).
thf(conj_thm_2Ereal__topology_2ESEQ__HARMONIC__OFFSET,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V1n:$i] : (ap@c_2Erealax_2Einv@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal_2Ereal__of__num@V1n))@(inj__ty_2Erealax_2Ereal @ V0a))))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially)))).
thf(conj_thm_2Ereal__topology_2ESEQ__HARMONIC,lemma,(p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V0n:$i] : (ap@c_2Erealax_2Einv@(ap@c_2Ereal_2Ereal__of__num@V0n)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially))).
thf(conj_thm_2Ereal__topology_2ECLOSED__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2EIN__INTERIOR__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2Einterior@V1s))) <=> (? [V2e:tp__ty_2Erealax_2Ereal] : ((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@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e))))@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__BALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e))))) <=> ((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_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__BALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((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 @ V1e))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))) = (ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__BALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))) = (ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))) = (ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__BALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((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 @ V1e))) => ((ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1e)))) = (ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CBALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1e)))) = (ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1e))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__EQ__EMPTY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1e))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__EMPTY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1e))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECBALL__EQ__SING,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) <=> (V1e = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__SING,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((V1e = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE__SING,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((V1e = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE__EQ__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) <=> ((V2x = V0a) & (V1r = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))).
thf(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN__INTERIOR,lemma,( ! [V0p:$i] : ((mem@V0p@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Einterior@V1s))) => ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V0p)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V1s))) <=> (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@V0p)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__INTERIOR,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal__topology_2Einterior@V2s))) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x)))@V2s))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x))))))))))))).
thf(conj_thm_2Ereal__topology_2ENETLIMIT__WITHIN__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Einterior@V0s))) => ((surj__ty_2Erealax_2Ereal @ (ap@(c_2Ereal__topology_2Enetlimit@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V0s))) = V1x)))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & ((~~ (? [V2a:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))) => (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (? [V2a:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))))))).
thf(ax_thm_2Ereal__topology_2Ebounded__def,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) <=> (? [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1a)))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__EMPTY,lemma,(p@(ap@c_2Ereal__topology_2Ebounded__def@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2Einterior@V0s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2Eclosure@V0s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSURE__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2Eclosure@V0s))) <=> (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__BALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))) <=> ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__BIGUNION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V0f)) & ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@V1s)))))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0f))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__POS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1b))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__POS__LT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : ((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 @ V1b))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) | (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__DIFF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INSERT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))) <=> (p@(ap@c_2Ereal__topology_2Ebounded__def@V1s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__BIGINTER,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((? [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1s))))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f))))))).
thf(conj_thm_2Ereal__topology_2ENOT__BOUNDED__UNIV,lemma,(~~ (p@(ap@c_2Ereal__topology_2Ebounded__def@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ECOBOUNDED__IMP__UNBOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s))) => (~~ (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V1s)) & (p@(ap@c_2Ereal__topology_2Elinear@V0f))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SCALING,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@V2x))))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__NEGATIONS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__TRANSLATION__EQ,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@V1s))) <=> (p@(ap@c_2Ereal__topology_2Ebounded__def@V1s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__DIFFS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Ereal_2Ereal__sub@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUMS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGE,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@V0f@V3x))@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V2t))))))) & (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@V1g@V4x))@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V2t)))))))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@V5x))@(ap@V1g@V5x)))@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGES,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@A_27b@ty_2Erealax_2Ereal))) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2s)) & ( ! [V3a:$i] : ((mem@V3a@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3a)@V2s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@V0f@V4x)@V3a))@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4x)@V1t))))))))))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@(c_2Eiterate_2ESum@A_27b)@V2s)@(ap@V0f@V5x)))@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5x)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__BALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (? [V2r:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2r))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r)))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CBALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (? [V2r:tp__ty_2Erealax_2Ereal] : ((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 @ V2r))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2r)))))))))))).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__INTER__COBOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s))) & (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V1t)))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECOBOUNDED__INTER__UNBOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V0s))) & (~~ (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__BOUNDED__EQ__TRIVIAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Esubspace@V0s)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) <=> (V0s = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__COMPONENTWISE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) <=> (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : V1x)))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__HAS__SUP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal_2Esup@V0s))))) & ( ! [V2b:tp__ty_2Erealax_2Ereal] : (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Esup@V0s))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))).
thf(conj_thm_2Ereal__topology_2ESUP__INSERT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@V1s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@c_2Ereal_2Emax@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Esup@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__HAS__INF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V1x))))) & ( ! [V2b:tp__ty_2Erealax_2Ereal] : (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2b))@(inj__ty_2Erealax_2Ereal @ V3x))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2b))@(ap@c_2Eiterate_2Einf@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2EINF__INSERT,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Eiterate_2Einf@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V1s))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@V1s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@c_2Ereal_2Emin@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Eiterate_2Einf@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__BALLS,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1a_27:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ( ! [V3r_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a_27))@(inj__ty_2Erealax_2Ereal @ V3r_27))))) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1a_27))))@(inj__ty_2Erealax_2Ereal @ V2r)))@(inj__ty_2Erealax_2Ereal @ V3r_27))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))) & (( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5a_27:tp__ty_2Erealax_2Ereal] : ( ! [V6r:tp__ty_2Erealax_2Ereal] : ( ! [V7r_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V6r))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5a_27))@(inj__ty_2Erealax_2Ereal @ V7r_27))))) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V5a_27))))@(inj__ty_2Erealax_2Ereal @ V6r)))@(inj__ty_2Erealax_2Ereal @ V7r_27))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V6r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))) & (( ! [V8a:tp__ty_2Erealax_2Ereal] : ( ! [V9a_27:tp__ty_2Erealax_2Ereal] : ( ! [V10r:tp__ty_2Erealax_2Ereal] : ( ! [V11r_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V10r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9a_27))@(inj__ty_2Erealax_2Ereal @ V11r_27))))) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V9a_27))))@(inj__ty_2Erealax_2Ereal @ V10r)))@(inj__ty_2Erealax_2Ereal @ V11r_27))) | (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V10r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))) & ( ! [V12a:tp__ty_2Erealax_2Ereal] : ( ! [V13a_27:tp__ty_2Erealax_2Ereal] : ( ! [V14r:tp__ty_2Erealax_2Ereal] : ( ! [V15r_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V14r))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V13a_27))@(inj__ty_2Erealax_2Ereal @ V15r_27))))) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V13a_27))))@(inj__ty_2Erealax_2Ereal @ V14r)))@(inj__ty_2Erealax_2Ereal @ V15r_27))) | (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V14r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))))))).
thf(conj_thm_2Ereal__topology_2EINTER__BALLS__EQ__EMPTY,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ( ! [V3s:tp__ty_2Erealax_2Ereal] : (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3s)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V2r))@(inj__ty_2Erealax_2Ereal @ V3s)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))))))) & (( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5b:tp__ty_2Erealax_2Ereal] : ( ! [V6r:tp__ty_2Erealax_2Ereal] : ( ! [V7s:tp__ty_2Erealax_2Ereal] : (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V6r))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5b))@(inj__ty_2Erealax_2Ereal @ V7s)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V6r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V7s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V6r))@(inj__ty_2Erealax_2Ereal @ V7s)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V5b)))))))))))) & (( ! [V8a:tp__ty_2Erealax_2Ereal] : ( ! [V9b:tp__ty_2Erealax_2Ereal] : ( ! [V10r:tp__ty_2Erealax_2Ereal] : ( ! [V11s:tp__ty_2Erealax_2Ereal] : (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V10r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9b))@(inj__ty_2Erealax_2Ereal @ V11s)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V10r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V11s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V10r))@(inj__ty_2Erealax_2Ereal @ V11s)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V9b)))))))))))) & ( ! [V12a:tp__ty_2Erealax_2Ereal] : ( ! [V13b:tp__ty_2Erealax_2Ereal] : ( ! [V14r:tp__ty_2Erealax_2Ereal] : ( ! [V15s:tp__ty_2Erealax_2Ereal] : (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V14r))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V13b))@(inj__ty_2Erealax_2Ereal @ V15s)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V14r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V15s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V14r))@(inj__ty_2Erealax_2Ereal @ V15s)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V13b)))))))))))))))).
thf(ax_thm_2Ereal__topology_2Ecompact,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) <=> ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1f@(inj__ty_2Enum_2Enum @ V2n)))@V0s))) => (? [V3l:tp__ty_2Erealax_2Ereal] : (? [V4r:$i] : ((mem@V4r@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3l))@V0s)) & (( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V4r@(inj__ty_2Enum_2Enum @ V5m)))@(ap@V4r@(inj__ty_2Enum_2Enum @ V6n))))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@V1f)@V4r))@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially)))))))))))))).
thf(conj_thm_2Ereal__topology_2EMONOTONE__BIGGER,lemma,( ! [V0r:$i] : ((mem@V0r@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => (( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V0r@(inj__ty_2Enum_2Enum @ V1m)))@(ap@V0r@(inj__ty_2Enum_2Enum @ V2n))))))) => ( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3n))@(ap@V0r@(inj__ty_2Enum_2Enum @ V3n))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__SUBSEQUENCE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1r:$i] : ((mem@V1r@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V1r@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V1r@(inj__ty_2Enum_2Enum @ V4n))))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0s)@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@V0s)@V1r))@(inj__ty_2Erealax_2Ereal @ V2l))@c_2Ereal__topology_2Esequentially))))))))).
thf(conj_thm_2Ereal__topology_2EMONOTONE__SUBSEQUENCE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (? [V1r:$i] : ((mem@V1r@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) & (( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V1r@(inj__ty_2Enum_2Enum @ V2m)))@(ap@V1r@(inj__ty_2Enum_2Enum @ V3n))))))) & (( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V4m))@(inj__ty_2Enum_2Enum @ V5n))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0s@(ap@V1r@(inj__ty_2Enum_2Enum @ V4m))))@(ap@V0s@(ap@V1r@(inj__ty_2Enum_2Enum @ V5n)))))))) | ( ! [V6m:tp__ty_2Enum_2Enum] : ( ! [V7n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V6m))@(inj__ty_2Enum_2Enum @ V7n))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0s@(ap@V1r@(inj__ty_2Enum_2Enum @ V7n))))@(ap@V0s@(ap@V1r@(inj__ty_2Enum_2Enum @ V6m))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__BOUNDED__INCREASING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0s@(inj__ty_2Enum_2Enum @ V2m)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V3n))))))) & ( ! [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0s@(inj__ty_2Enum_2Enum @ V4n))))@(inj__ty_2Erealax_2Ereal @ V1b))))) => (? [V5l:tp__ty_2Erealax_2Ereal] : ( ! [V6e:tp__ty_2Erealax_2Ereal] : ((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 @ V6e))) => (? [V7N:tp__ty_2Enum_2Enum] : ( ! [V8n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V7N))@(inj__ty_2Enum_2Enum @ V8n))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0s@(inj__ty_2Enum_2Enum @ V8n)))@(inj__ty_2Erealax_2Ereal @ V5l))))@(inj__ty_2Erealax_2Ereal @ V6e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__BOUNDED__MONOTONE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0s@(inj__ty_2Enum_2Enum @ V2n))))@(inj__ty_2Erealax_2Ereal @ V1b)))) & (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0s@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V4n))))))) | ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0s@(inj__ty_2Enum_2Enum @ V6n)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V5m))))))))) => (? [V7l:tp__ty_2Erealax_2Ereal] : ( ! [V8e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V8e))) => (? [V9N:tp__ty_2Enum_2Enum] : ( ! [V10n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V9N))@(inj__ty_2Enum_2Enum @ V10n))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0s@(inj__ty_2Enum_2Enum @ V10n)))@(inj__ty_2Erealax_2Ereal @ V7l))))@(inj__ty_2Erealax_2Ereal @ V8e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__REAL__LEMMA,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : (( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0s@(inj__ty_2Enum_2Enum @ V2n))))@(inj__ty_2Erealax_2Ereal @ V1b)))) => (? [V3l:tp__ty_2Erealax_2Ereal] : (? [V4r:$i] : ((mem@V4r@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) & (( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V4r@(inj__ty_2Enum_2Enum @ V5m)))@(ap@V4r@(inj__ty_2Enum_2Enum @ V6n))))))) & ( ! [V7e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V7e))) => (? [V8N:tp__ty_2Enum_2Enum] : ( ! [V9n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V8N))@(inj__ty_2Enum_2Enum @ V9n))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0s@(ap@V4r@(inj__ty_2Enum_2Enum @ V9n))))@(inj__ty_2Erealax_2Ereal @ V3l))))@(inj__ty_2Erealax_2Ereal @ V7e))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__LEMMA,lemma,( ! [A_27a:del]: ( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V1s)) & ( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0x@(inj__ty_2Enum_2Enum @ V2n)))@V1s)))) => (? [V3l:tp__ty_2Erealax_2Ereal] : (? [V4r:$i] : ((mem@V4r@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) & (( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V4r@(inj__ty_2Enum_2Enum @ V5m)))@(ap@V4r@(inj__ty_2Enum_2Enum @ V6n))))))) & ( ! [V7e:tp__ty_2Erealax_2Ereal] : ((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 @ V7e))) => (? [V8N:tp__ty_2Enum_2Enum] : ( ! [V9n:tp__ty_2Enum_2Enum] : ( ! [V10i:$i] : ((mem@V10i@A_27a) => ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V8N))@(inj__ty_2Enum_2Enum @ V9n))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0x@(ap@V4r@(inj__ty_2Enum_2Enum @ V9n))))@(inj__ty_2Erealax_2Ereal @ V3l))))@(inj__ty_2Erealax_2Ereal @ V7e))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__IMP__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V0s))) => (p@(ap@c_2Ereal__topology_2Ecompact@V0s)))))).
thf(ax_thm_2Ereal__topology_2Ecauchy,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Ecauchy@V0s)) <=> ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1e))) => (? [V2N:tp__ty_2Enum_2Enum] : ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V2N))) & (p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V4n))@(inj__ty_2Enum_2Enum @ V2N)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V4n)))))@(inj__ty_2Erealax_2Ereal @ V1e))))))))))))).
thf(ax_thm_2Ereal__topology_2Ecomplete,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecomplete@V0s)) <=> ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1f@(inj__ty_2Enum_2Enum @ V2n)))@V0s))) & (p@(ap@c_2Ereal__topology_2Ecauchy@V1f))) => (? [V3l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3l))@V0s)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially))))))))))).
thf(conj_thm_2Ereal__topology_2ECAUCHY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Ecauchy@V0s)) <=> ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((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 @ V1e))) => (? [V2N:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V3n))@(inj__ty_2Enum_2Enum @ V2N))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V3n)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V2N)))))@(inj__ty_2Erealax_2Ereal @ V1e)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__IMP__CAUCHY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0s)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)) => (p@(ap@c_2Ereal__topology_2Ecauchy@V0s))))))).
thf(conj_thm_2Ereal__topology_2EGREATER__EQ__REFL,lemma,( ! [V0m:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V0m))@(inj__ty_2Enum_2Enum @ V0m))))).
thf(conj_thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET__REAL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) => (? [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@V3x))@(inj__ty_2Erealax_2Ereal @ V2a)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECAUCHY__IMP__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Ecauchy@V0s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1y)@(ap@(c_2Ebool_2E_3F@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V1y)@(ap@V0s@V2n))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__COMPLETE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@c_2Ereal__topology_2Ecomplete@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__UNIV,lemma,(p@(ap@c_2Ereal__topology_2Ecomplete@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__EQ__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecomplete@V0s)) <=> (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__EQ__CAUCHY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((? [V1l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0s)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially))) <=> (p@(ap@c_2Ereal__topology_2Ecauchy@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECONVERGENT__IMP__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0s)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@V0s)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__TOTALLY__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((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 @ V1e))) => (? [V2k:$i] : ((mem@V2k@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V2k)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2k)@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V3x)@(inj__ty_2Erealax_2Ereal @ V1e))))))@V2k)))))))))))))).
thf(conj_thm_2Ereal__topology_2EHEINE__BOREL__LEMMA,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => ( ! [V1t:$i] : ((mem@V1t@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1t))) & ( ! [V2b:$i] : ((mem@V2b@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2b)@V1t)) => (p@(ap@c_2Ereal__topology_2EOpen@V2b)))))) => (? [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3e))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) => (? [V5b:$i] : ((mem@V5b@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V5b)@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V3e))))@V5b)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__HEINE__BOREL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => ( ! [V1f:$i] : ((mem@V1f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2t)@V1f)) => (p@(ap@c_2Ereal__topology_2EOpen@V2t))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1f)))) => (? [V3f_27:$i] : ((mem@V3f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)@V1f)) & ((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V3f_27)))))))))))))).
thf(conj_thm_2Ereal__topology_2EHEINE__BOREL__IMP__BOLZANO__WEIERSTRASS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V1f:$i] : ((mem@V1f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2t)@V1f)) => (p@(ap@c_2Ereal__topology_2EOpen@V2t))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1f)))) => (? [V3f_27:$i] : ((mem@V3f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)@V1f)) & ((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V3f_27)))))))))) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V4t))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4t)@V0s))) => (? [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V0s)) & (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V5x))@V4t))))))))))).
thf(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s))) => (? [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V1t)))))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)))))).
thf(conj_thm_2Ereal__topology_2EINF__FINITE__LEMMA,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@V0s)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2EINF__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Eiterate_2Einf@V0s))@V0s)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V1x)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__INF__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Eiterate_2Einf@V0s))) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__INF__LE__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V1a))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1a))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LT__INF__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@c_2Eiterate_2Einf@V0s))) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__INF__LT__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Eiterate_2Einf@V0s))@(inj__ty_2Erealax_2Ereal @ V1a))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1a))))))))))).
thf(conj_thm_2Ereal__topology_2ESEQUENCE__INFINITE__LEMMA,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((( ! [V2n:tp__ty_2Enum_2Enum] : (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V2n))) = V1l))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially))) => (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3y)@(ap@(c_2Ebool_2E_3F@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V3y)@(ap@V0f@V4n))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELE__1,lemma,(( ! [V0n:tp__ty_2Enum_2Enum] : ((~~ (V0n = fo__c_2Enum_2E0)) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V0n))))) & (( ! [V1n:tp__ty_2Enum_2Enum] : ((~~ (V1n = fo__c_2Enum_2E0)) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V1n))))) & (( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V2n))) => (~~ (V2n = fo__c_2Enum_2E0)))) & (( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V3n))) => (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V3n))))) & (( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V4n))))) & ( ! [V5n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V5n))) => (~~ (V5n = fo__c_2Enum_2E0)))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__OF__SEQUENCE__SUBSEQUENCE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@V0f)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))) => (? [V2r:$i] : ((mem@V2r@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) & (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@c_2Eprim__rec_2E_3C@(ap@V2r@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V2r@(inj__ty_2Enum_2Enum @ V4n))))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@V0f)@V2r))@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)))))))))).
thf(conj_thm_2Ereal__topology_2ESEQUENCE__UNIQUE__LIMPT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2l_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)) & (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2l_27))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3y)@(ap@(c_2Ebool_2E_3F@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V3y)@(ap@V0f@V4n)))))))))))) => (V2l_27 = V1l))))))).
thf(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s))) => (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V1t))))))) => (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) <=> ( ! [V1f:$i] : ((mem@V1f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2t)@V1f)) => (p@(ap@c_2Ereal__topology_2EOpen@V2t))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1f)))) => (? [V3f_27:$i] : ((mem@V3f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)@V1f)) & ((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V3f_27)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EQ__BOLZANO__WEIERSTRASS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) <=> ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s))) => (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V1t))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EQ__BOUNDED__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) <=> ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__SEQUENCE__WITH__LIMIT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@V0f)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t))) => (p@(ap@c_2Ereal__topology_2Ecompact@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPACT__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1t)) <=> ((p@(ap@c_2Ereal__topology_2Ecompact@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL__SUBTOPOLOGY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) <=> ( ! [V1f:$i] : ((mem@V1f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2t)@V1f)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V2t))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1f)))) => (? [V3f_27:$i] : ((mem@V3f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)@V1f)) & ((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V3f_27)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2Eclosure@V0s))) <=> (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)))))).
thf(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__CONTRAPOS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (~~ (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V1t))))))) => (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t)))))))).
thf(conj_thm_2Ereal__topology_2EDISCRETE__BOUNDED__IMP__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1e:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1e))) & (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V1e))))) => (V3y = V2x)))) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)))) => (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s))))))).
thf(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)))) => (? [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__EQ__BOLZANO__WEIERSTRASS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) <=> ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t)))) => (? [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__EMPTY,lemma,(p@(ap@c_2Ereal__topology_2Ecompact@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__UNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V1t))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V1t))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INTER__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTER__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V1t))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__BIGINTER,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) => (p@(ap@c_2Ereal__topology_2Ecompact@V1s))))) & (~~ (V0f = (c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool))))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) => (p@(ap@c_2Ereal__topology_2EClosed@V0s)))))).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__CLOSED__IN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) => (p@(ap@c_2Ereal__topology_2Ecompact@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INSERT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V1s)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__SING,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0u))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SING,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INSERT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V1s)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__FRONTIER__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2Efrontier@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2Efrontier@V0s))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2Efrontier@V0s))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))@V0s)))))).
thf(conj_thm_2Ereal__topology_2EOPEN__DELETE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__DELETE,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V1s)@(inj__ty_2Erealax_2Ereal @ V2a))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__BIGINTER__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) <=> ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1e))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__BIGUNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V0s)) & ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V0s)) => (p@(ap@c_2Ereal__topology_2Ecompact@V1t)))))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__DIFF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__SPHERE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SPHERE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__SPHERE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Efrontier@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__IMP__FIP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2t)@V1f)) => (p@(ap@c_2Ereal__topology_2EClosed@V2t))))) & ( ! [V3f_27:$i] : ((mem@V3f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)@V1f))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V3f_27)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V1f)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IMP__FIP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2t)@V1f)) => (p@(ap@c_2Ereal__topology_2EClosed@V2t))))) & ((? [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V3t)@V1f)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V3t))))) & ( ! [V4f_27:$i] : ((mem@V4f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V4f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V4f_27)@V1f))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V4f_27)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V1f)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IMP__FIP__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2t)@V1f)) => (p@(ap@c_2Ereal__topology_2Ecompact@V2t))))) & ( ! [V3f_27:$i] : ((mem@V3f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)@V1f))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V3f_27)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) => (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V1f)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__FIP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V0f)) => (p@(ap@c_2Ereal__topology_2EClosed@V1t))))) & ((? [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2t)@V0f)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V2t))))) & ( ! [V3f_27:$i] : ((mem@V3f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V3f_27)@V0f))) => (~~ ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V3f_27) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) => (~~ ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__FIP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V0f)) => (p@(ap@c_2Ereal__topology_2Ecompact@V1t))))) & ( ! [V2f_27:$i] : ((mem@V2f_27@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@V2f_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V2f_27)@V0f))) => (~~ ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V2f_27) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) => (~~ ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__NEST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@V0s@(inj__ty_2Enum_2Enum @ V1n))))) & (( ! [V2n:tp__ty_2Enum_2Enum] : (~~ ((ap@V0s@(inj__ty_2Enum_2Enum @ V2n)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) & (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V4n)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V3m))))))) & (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@V0s@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))) => (? [V5a:tp__ty_2Erealax_2Ereal] : ( ! [V6n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5a))@(ap@V0s@(inj__ty_2Enum_2Enum @ V6n)))))))))).
thf(conj_thm_2Ereal__topology_2EDECREASING__CLOSED__NEST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@V0s@(inj__ty_2Enum_2Enum @ V1n))))) & (( ! [V2n:tp__ty_2Enum_2Enum] : (~~ ((ap@V0s@(inj__ty_2Enum_2Enum @ V2n)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) & (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V4n)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V3m))))))) & ( ! [V5e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V5e))) => (? [V6n:tp__ty_2Enum_2Enum] : ( ! [V7x:tp__ty_2Erealax_2Ereal] : ( ! [V8y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@(ap@V0s@(inj__ty_2Enum_2Enum @ V6n)))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8y))@(ap@V0s@(inj__ty_2Enum_2Enum @ V6n))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@(inj__ty_2Erealax_2Ereal @ V8y))))@(inj__ty_2Erealax_2Ereal @ V5e)))))))))))) => (? [V9a:tp__ty_2Erealax_2Ereal] : ( ! [V10n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9a))@(ap@V0s@(inj__ty_2Enum_2Enum @ V10n)))))))))).
thf(conj_thm_2Ereal__topology_2EDECREASING__CLOSED__NEST__SING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@V0s@(inj__ty_2Enum_2Enum @ V1n))))) & (( ! [V2n:tp__ty_2Enum_2Enum] : (~~ ((ap@V0s@(inj__ty_2Enum_2Enum @ V2n)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) & (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V4n)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V3m))))))) & ( ! [V5e:tp__ty_2Erealax_2Ereal] : ((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 @ V5e))) => (? [V6n:tp__ty_2Enum_2Enum] : ( ! [V7x:tp__ty_2Erealax_2Ereal] : ( ! [V8y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@(ap@V0s@(inj__ty_2Enum_2Enum @ V6n)))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8y))@(ap@V0s@(inj__ty_2Enum_2Enum @ V6n))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@(inj__ty_2Erealax_2Ereal @ V8y))))@(inj__ty_2Erealax_2Ereal @ V5e)))))))))))) => (? [V9a:tp__ty_2Erealax_2Ereal] : ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(lam@(arr@ty_2Erealax_2Ereal@bool)@(^ [V10t:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@V10t)@(ap@(c_2Ebool_2E_3F@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V11n:$i] : (ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@V10t)@(ap@V0s@V11n)))))))))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__CHAIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ( ! [V1b:$i] : ((mem@V1b@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V0f)) => ((p@(ap@c_2Ereal__topology_2EClosed@V2s)) & (~~ (V2s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) & (( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V3s)@V0f)) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V4t)@V0f))) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3s)@V4t)) | (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4t)@V3s)))))))) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1b)@V0f)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1b))))) => (~~ ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CHAIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V1s)) & (~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) & ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V0f)) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V3t)@V0f))) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2s)@V3t)) | (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t)@V2s))))))))) => (~~ ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__NEST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@c_2Ereal__topology_2Ecompact@(ap@V0s@(inj__ty_2Enum_2Enum @ V1n)))) & (~~ ((ap@V0s@(inj__ty_2Enum_2Enum @ V1n)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) & ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V3n)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V2m)))))))) => (~~ ((ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@V0s@V4n))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V4n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))))))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@(arr@A_27a@ty_2Erealax_2Ereal))) => ((? [V2l:$i] : ((mem@V2l@(arr@A_27a@ty_2Erealax_2Ereal)) & ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) => (? [V4N:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : ( ! [V6x:$i] : ((mem@V6x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V4N))@(inj__ty_2Enum_2Enum @ V5n))) & (p@(ap@V0P@V6x))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V5n))@V6x))@(ap@V2l@V6x))))@(inj__ty_2Erealax_2Ereal @ V3e)))))))))))) <=> ( ! [V7e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V7e))) => (? [V8N:tp__ty_2Enum_2Enum] : ( ! [V9m:tp__ty_2Enum_2Enum] : ( ! [V10n:tp__ty_2Enum_2Enum] : ( ! [V11x:$i] : ((mem@V11x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V8N))@(inj__ty_2Enum_2Enum @ V9m))) & ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V8N))@(inj__ty_2Enum_2Enum @ V10n))) & (p@(ap@V0P@V11x)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V9m))@V11x))@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V10n))@V11x))))@(inj__ty_2Erealax_2Ereal @ V7e)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY__ALT,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@(arr@A_27a@ty_2Erealax_2Ereal))) => ((? [V2l:$i] : ((mem@V2l@(arr@A_27a@ty_2Erealax_2Ereal)) & ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3e))) => (? [V4N:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : ( ! [V6x:$i] : ((mem@V6x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V4N))@(inj__ty_2Enum_2Enum @ V5n))) & (p@(ap@V0P@V6x))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V5n))@V6x))@(ap@V2l@V6x))))@(inj__ty_2Erealax_2Ereal @ V3e)))))))))))) <=> ( ! [V7e:tp__ty_2Erealax_2Ereal] : ((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 @ V7e))) => (? [V8N:tp__ty_2Enum_2Enum] : ( ! [V9m:tp__ty_2Enum_2Enum] : ( ! [V10n:tp__ty_2Enum_2Enum] : ( ! [V11x:$i] : ((mem@V11x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V8N))@(inj__ty_2Enum_2Enum @ V9m))) & ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V8N))@(inj__ty_2Enum_2Enum @ V10n))) & ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V9m))@(inj__ty_2Enum_2Enum @ V10n))) & (p@(ap@V0P@V11x))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V9m))@V11x))@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V10n))@V11x))))@(inj__ty_2Erealax_2Ereal @ V7e)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CAUCHY__IMP__UNIFORMLY__CONVERGENT,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V2l:$i] : ((mem@V2l@(arr@A_27a@ty_2Erealax_2Ereal)) => ((( ! [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) => (? [V4N:tp__ty_2Enum_2Enum] : ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ( ! [V7x:$i] : ((mem@V7x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V4N))@(inj__ty_2Enum_2Enum @ V5m))) & ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V4N))@(inj__ty_2Enum_2Enum @ V6n))) & (p@(ap@V0P@V7x)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V5m))@V7x))@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V6n))@V7x))))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))) & ( ! [V8x:$i] : ((mem@V8x@A_27a) => ((p@(ap@V0P@V8x)) => ( ! [V9e:tp__ty_2Erealax_2Ereal] : ((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 @ V9e))) => (? [V10N:tp__ty_2Enum_2Enum] : ( ! [V11n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V10N))@(inj__ty_2Enum_2Enum @ V11n))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V11n))@V8x))@(ap@V2l@V8x))))@(inj__ty_2Erealax_2Ereal @ V9e)))))))))))) => ( ! [V12e:tp__ty_2Erealax_2Ereal] : ((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 @ V12e))) => (? [V13N:tp__ty_2Enum_2Enum] : ( ! [V14n:tp__ty_2Enum_2Enum] : ( ! [V15x:$i] : ((mem@V15x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V13N))@(inj__ty_2Enum_2Enum @ V14n))) & (p@(ap@V0P@V15x))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@V1s@(inj__ty_2Enum_2Enum @ V14n))@V15x))@(ap@V2l@V15x))))@(inj__ty_2Erealax_2Ereal @ V12e))))))))))))))))))).
thf(ax_thm_2Ereal__topology_2Econtinuous,axiom,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1net:$i] : ((mem@V1net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V1net)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V0f)@(ap@V0f@(ap@(c_2Ereal__topology_2Enetlimit@A_27a)@V1net)))@V1net))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRIVIAL__LIMIT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1net:$i] : ((mem@V1net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V1net)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V1net))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V0s))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V1f)@(ap@V1f@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__CLOSED__NONTRIVIAL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V2s)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V2s)))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4d:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4d))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) & (( ! [V5x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@V2s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@(inj__ty_2Erealax_2Ereal @ V3x))))@(inj__ty_2Erealax_2Ereal @ V4d)))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x_27))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V5x_27)))))) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x)))@V2s)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1g)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x)))@V2s))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__AT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3d))) & (( ! [V4x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x_27))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V3d))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x_27))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V4x_27)))))) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1g)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EOpen@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@V2s)) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V4x)))))) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a))))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1g)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN__IN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@V2s)) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V5x)))))) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V4a)))@V3t)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1g)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V4a)))@V3t)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__SET__IMP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V3t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V2s)))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))@V2s)))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))@V3t)))))))))))).
thf(conj_thm_2Ereal__topology_2Econtinuous__within,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V2s))) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : ((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 @ V4d))) & ( ! [V5x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@V2s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@(inj__ty_2Erealax_2Ereal @ V1x))))@(inj__ty_2Erealax_2Ereal @ V4d)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x_27)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))))@(inj__ty_2Erealax_2Ereal @ V3e)))))))))))))))).
thf(conj_thm_2Ereal__topology_2Econtinuous__at,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (? [V3d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3d))) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x_27))@(inj__ty_2Erealax_2Ereal @ V1x))))@(inj__ty_2Erealax_2Ereal @ V3d))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x_27)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__BALL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V1s))) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4d))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V4d))))@V1s)))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V3e)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__BALL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => (? [V3d:tp__ty_2Erealax_2Ereal] : ((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 @ V3d))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V3d)))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPARISON,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1g)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a)))@V2s))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V2s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3a)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V1g@(inj__ty_2Erealax_2Ereal @ V3a)))@(ap@V1g@(inj__ty_2Erealax_2Ereal @ V4x))))))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3a)))@V2s)))))))))))).
thf(ax_thm_2Ereal__topology_2Econtinuous__on,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4d))) & ( ! [V5x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V4d)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x_27)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))))))))).
thf(ax_thm_2Ereal__topology_2Euniformly__continuous__on,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (? [V3d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3d))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ( ! [V5x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@(inj__ty_2Erealax_2Ereal @ V4x))))@(inj__ty_2Erealax_2Ereal @ V3d))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x_27)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CONTINUOUS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__IMP__CONTINUOUS__ON,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__WITHIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__AT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V1s)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x)))@V1s))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t)@V1s))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3x)))@V2t)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t)@V1s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2t)))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t)@V1s))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V2t)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INTERIOR,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Einterior@V1s)))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V3x)))))) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V3x)))))) & (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V1g)@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SING,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EMPTY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__NO__LIMPT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((~~ (? [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__FINITE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTRACTION__IMP__CONTINUOUS__ON,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V1f@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@V1f@(inj__ty_2Erealax_2Ereal @ V3y)))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y)))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EISOMETRY__ON__IMP__CONTINUOUS__ON,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V1f@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@V1f@(inj__ty_2Erealax_2Ereal @ V3y))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y)))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EFORALL__POS__MONO__1,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V1d:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V2e))) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V1d)))) => (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V2e)))))) & ( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@V0P@(ap@c_2Erealax_2Einv@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V3n)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))) => ( ! [V4e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4e))) => (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V4e))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SEQUENTIALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V1s))) <=> ( ! [V3x:$i] : ((mem@V3x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((( ! [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V3x@(inj__ty_2Enum_2Enum @ V4n)))@V1s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V3x)@(inj__ty_2Erealax_2Ereal @ V2a))@c_2Ereal__topology_2Esequentially))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3x))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2a)))@c_2Ereal__topology_2Esequentially)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__SEQUENTIALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))) <=> ( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V2x)@(inj__ty_2Erealax_2Ereal @ V1a))@c_2Ereal__topology_2Esequentially)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2x))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1a)))@c_2Ereal__topology_2Esequentially)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SEQUENTIALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@V1s)) & (( ! [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V4n)))@V1s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V2x)@(inj__ty_2Erealax_2Ereal @ V3a))@c_2Ereal__topology_2Esequentially)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2x))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3a)))@c_2Ereal__topology_2Esequentially)))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SEQUENTIALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) <=> ( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3y:$i] : ((mem@V3y@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((( ! [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V4n)))@V1s))) & (( ! [V5n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V3y@(inj__ty_2Enum_2Enum @ V5n)))@V1s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V6n:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V2x@V6n))@(ap@V3y@V6n)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V7n:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@(ap@V2x@V7n)))@(ap@V0f@(ap@V3y@V7n))))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__CONTINUOUS__FUNCTION,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1net:$i] : ((mem@V1net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V3l)))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V3l))@V1net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@V0f@(ap@V2g@V4x)))))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3l)))@V1net)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CONST,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V2x:$i] : (inj__ty_2Erealax_2Ereal @ V1c))))@V0net))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CMUL,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2net:$i] : ((mem@V2net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V2net)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@V0f@V3x)))))@V2net)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__NEG,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1net:$i] : ((mem@V1net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V1net)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V0f@V2x)))))@V1net))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ADD,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2net:$i] : ((mem@V2net@(ty_2Ereal__topology_2Enet@A_27a)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V2net)) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1g)@V2net))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2net))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__SUB,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2net:$i] : ((mem@V2net@(ty_2Ereal__topology_2Enet@A_27a)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V2net)) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1g)@V2net))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2net))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ABS,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1net:$i] : ((mem@V1net@(ty_2Ereal__topology_2Enet@A_27a)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V1net)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@c_2Ereal_2Eabs@(ap@V0f@V2x)))))@V1net))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__MAX,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2net:$i] : ((mem@V2net@(ty_2Ereal__topology_2Enet@A_27a)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V2net)) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1g)@V2net))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Emax@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2net))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__MIN,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2net:$i] : ((mem@V2net@(ty_2Ereal__topology_2Enet@A_27a)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V0f)@V2net)) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1g)@V2net))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Emin@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2net))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__SUM,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27b@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2s)) & ( ! [V3a:$i] : ((mem@V3a@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3a)@V2s)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(ap@V1f@V3a))@V0net)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(c_2Eiterate_2ESum@A_27b)@V2s)@(lam@A_27b@(^ [V5a:$i] : (ap@(ap@V1f@V5a)@V4x)))))))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CONST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (inj__ty_2Erealax_2Ereal @ V1c))))@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CMUL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@V0f@V3x)))))@V2s))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__NEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V0f@V2x)))))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ADD,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUB,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ABS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Ereal_2Eabs@(ap@V0f@V2x)))))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MAX,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Emax@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Emin@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUM,lemma,( ! [A_27a:del]: ( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & ( ! [V3a:$i] : ((mem@V3a@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3a)@V2s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@V1f@V3a))@V0t)))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@(lam@A_27a@(^ [V5a:$i] : (ap@(ap@V1f@V5a)@V4x)))))))@V0t))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CONST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (inj__ty_2Erealax_2Ereal @ V1c))))@V0s)))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__UNIFORMLY__CONTINUOUS__ON,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V1g)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s)))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__UNIFORMLY__CONTINUOUS__ON__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V3s)) & ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V1g)@V3s)) & ((p@(ap@c_2Ereal__topology_2Ebilinear@V2h)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3s))) & (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3s))))))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@V2h@(ap@V0f@V4x))@(ap@V1g@V4x)))))@V3s)))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__MUL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V2s)) & ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V1g)@V2s)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))) & (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V2s)))))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CMUL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V2s)) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@(ap@V0f@V3x)))))@V2s))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__VMUL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2v:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V1c)@V0s)) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V1c@V3x))@(inj__ty_2Erealax_2Ereal @ V2v)))))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__NEG,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V0f@V2x)))))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ADD,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V1g)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUB,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V1g)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUM,lemma,( ! [A_27a:del]: ( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) & ( ! [V3a:$i] : ((mem@V3a@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3a)@V2s)) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(ap@V1f@V3a))@V0t)))))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Eiterate_2ESum@A_27a)@V2s)@(lam@A_27a@(^ [V5a:$i] : (ap@(ap@V1f@V5a)@V4x)))))))@V0t))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__ID,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : V2x)))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))@V1s))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__ID,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : V1x)))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0a)))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ID,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : V1x)))@V0s))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ID,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : V1x)))@V0s))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V3s))) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1g)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3s))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V3s)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1g)@(ap@c_2Ereal__topology_2Eat@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s)))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V2u))) <=> ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V3t)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@V3t))) => (? [V4s:$i] : ((mem@V4s@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V4s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V4s)) & ( ! [V5x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@V4s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@V2u))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x_27)))@V3t)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V2t)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))@V2t))) => (? [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V3s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V3s)) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x_27))@V3s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x_27)))@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__GEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@V2t)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__GEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) & (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u)))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__OPEN__IN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@V2t))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED__GEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@V2t)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__GEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) & (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u)))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__CLOSED__IN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@V2t))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@c_2Ereal__topology_2EOpen@V2t))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@c_2Ereal__topology_2EClosed@V2t))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ((p@(ap@c_2Ereal__topology_2EOpen@V1s)) & (p@(ap@c_2Ereal__topology_2EOpen@V2t)))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1s)) & (p@(ap@c_2Ereal__topology_2EClosed@V2t)))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE__UNIV,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x))))) & (p@(ap@c_2Ereal__topology_2EOpen@V1s))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V1s))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__UNIV,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x))))) & (p@(ap@c_2Ereal__topology_2EClosed@V1s))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V1s))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V2t)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V2t)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@V2t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__LIM__0,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2Eat@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__AT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__WITHIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__COMPOSE,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1f)@V0net)) & (p@(ap@c_2Ereal__topology_2Elinear@V2g))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@V2g@(ap@V1f@V3x)))))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (p@(ap@c_2Ereal__topology_2Elinear@V1g))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@V1g@(ap@V0f@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__COMPONENT__COMPOSE,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2i:$i] : ((mem@V2i@A_27b) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1f)@V0net)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@V1f@V3x))))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENT__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@V0f@V2x))))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__COMPOSE,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3h:$i] : ((mem@V3h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1f)@V0net)) & ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V2g)@V0net)) & (p@(ap@c_2Ereal__topology_2Ebilinear@V3h)))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@V3h@(ap@V1f@V4x))@(ap@V2g@V4x)))))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__ON__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V3s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3s)) & (p@(ap@c_2Ereal__topology_2Ebilinear@V2h)))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@V2h@(ap@V0f@V4x))@(ap@V1g@V4x)))))@V3s)))))))))))).
thf(conj_thm_2Ereal__topology_2EBILINEAR__DOT,lemma,(p@(ap@c_2Ereal__topology_2Ebilinear@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V1y:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@V0x)@V1y)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DOT2,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1f)@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V2g)@V0net))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V1f@V3x))@(ap@V2g@V3x)))))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DOT2,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V2s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V0f@V3x))@(ap@V1g@V3x)))))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V1s))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__TRANSLATION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1a))@V2x))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__TRANSLATION__EQ,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@V1s))) <=> (p@(ap@c_2Ereal__topology_2Ecompact@V1s))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V1s)) & (p@(ap@c_2Ereal__topology_2Elinear@V0f))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CONTINUOUS__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@c_2Ereal__topology_2Econnected@V1s))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V1s)) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__TRANSLATION__EQ,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@V1s))) <=> (p@(ap@c_2Ereal__topology_2Econnected@V1s))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & (p@(ap@c_2Ereal__topology_2Elinear@V0f))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EQUASICOMPACT__OPEN__CLOSED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => (( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u)))))) <=> ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5u)))))))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V5u))))))))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) & ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__CLOSED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) & ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__IMP__QUOTIENT__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V2t)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2t))))))) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3t)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@V3t)))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__IMP__QUOTIENT__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V2t)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2t))))))) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3t)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@V3t)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__RIGHT__INVERSE__IMP__QUOTIENT__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V2s)) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V3t)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(ap@V1g@(inj__ty_2Erealax_2Ereal @ V4y)))) = V4y))))))) => ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5u)@V3t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V5u)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LEFT__INVERSE__IMP__QUOTIENT__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))) = V3x))))) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4u)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s)))@V4u)))))))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__CLOSED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => (( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u)))))) <=> ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V5u))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE__QUOTIENT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & (( ! [V5v:$i] : ((mem@V5v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5v)@V3t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V5v)))))) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V2s))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)))))))))))))).
thf(conj_thm_2Ereal__topology_2EFUNCTION__FACTORS__LEFT__GEN,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@A_27b)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@A_27c)) => (( ! [V3x:$i] : ((mem@V3x@A_27a) => ( ! [V4y:$i] : ((mem@V4y@A_27a) => (((p@(ap@V0P@V3x)) & ((p@(ap@V0P@V4y)) & ((ap@V2g@V3x) = (ap@V2g@V4y)))) => ((ap@V1f@V3x) = (ap@V1f@V4y))))))) <=> (? [V5h:$i] : ((mem@V5h@(arr@A_27c@A_27b)) & ( ! [V6x:$i] : ((mem@V6x@A_27a) => ((p@(ap@V0P@V6x)) => ((ap@V1f@V6x) = (ap@V5h@(ap@V2g@V6x))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1h:$i] : ((mem@V1h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s) = V3t) & (( ! [V5v:$i] : ((mem@V5v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5v)@V3t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V5v)))))) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1h)@V2s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1h)@V2s) = V4u) & ( ! [V7x:tp__ty_2Erealax_2Ereal] : ( ! [V8y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8y))@V2s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V7x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V8y)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1h@(inj__ty_2Erealax_2Ereal @ V7x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1h@(inj__ty_2Erealax_2Ereal @ V8y))))))))))) => (? [V9g:$i] : ((mem@V9g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V9g)@V3t)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V9g)@V3t) = V4u) & ( ! [V10x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1h@(inj__ty_2Erealax_2Ereal @ V10x))) = (surj__ty_2Erealax_2Ereal @ (ap@V9g@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V10x))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & (( ! [V5v:$i] : ((mem@V5v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5v)@V3t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V5v)))))) & ( ! [V7v:$i] : ((mem@V7v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V7v)@V4u)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V3t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1g@V8x))@V7v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@V7v)))))))) => ( ! [V9v:$i] : ((mem@V9v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V9v)@V4u)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V10x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V10x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V10x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f)@V10x))@V9v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@V9v)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__COMPOSITION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & ( ! [V5v:$i] : ((mem@V5v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5v)@V4u)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f)@V6x))@V5v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@V5v)))))))))) => ( ! [V7v:$i] : ((mem@V7v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V7v)@V4u)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V3t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1g@V8x))@V7v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@V7v)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2t))@V3u)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1s)@V2t)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V3u) & ( ! [V4v:$i] : ((mem@V4v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4v)@V3u)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3u))@V4v)))))))))) => ( ! [V6v:$i] : ((mem@V6v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V6v)@V3u)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V2t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@V6v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3u))@V6v)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__RESTRICT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3c:$i] : ((mem@V3c@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) & (( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V4u)))))) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3c)) | (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3c))))) => ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V6u)@V3c)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@V3c))))))))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V9x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V9x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V9x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V9x))@V3c))))))))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V8x))@V6u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3c))@V6u)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u)))))) & (( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V2t)) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@(inj__ty_2Erealax_2Ereal @ V5y))))))))))) & (p@(ap@c_2Ereal__topology_2Econnected@V2t)))))) => (p@(ap@c_2Ereal__topology_2Econnected@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE__GEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3c:$i] : ((mem@V3c@(arr@ty_2Erealax_2Ereal@bool)) => ((((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V4u)))))) & (( ! [V6y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6y))@V2t)) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@(inj__ty_2Erealax_2Ereal @ V6y))))))))))) & (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3c)) | (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3c))) & (p@(ap@c_2Ereal__topology_2Econnected@V3c)))))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V8x))@V3c)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__CLOSURES,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V1s)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))))) <=> ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s)))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosure@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__INTERIORS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V1s)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))))) <=> ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Einterior@V2s)))@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__RESTRICT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t_27:$i] : ((mem@V3t_27@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V4u)))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t_27)@V2t))) => ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V3t_27))))))))@V5u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t_27))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5u)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__RESTRICT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t_27:$i] : ((mem@V3t_27@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V4u)))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t_27)@V2t))) => ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V3t_27))))))))@V5u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t_27))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5u)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__MAP__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) & ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u))))))) => (( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V5k)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5k)))))) <=> ( ! [V6k:$i] : ((mem@V6k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V6k)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V6k))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__CLOSED__MAP__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) & ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u))))))) => (( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V5k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5k)))))) <=> ( ! [V6k:$i] : ((mem@V6k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V6k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V6k))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__IMP__OPEN__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3u)))))) & ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V4u)))))))))))))) => ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V6u)))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__IMP__CLOSED__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3u)))))) & ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V4u)))))))))))))) => ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V6u)))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__SURJECTIVE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s) = V3t) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & ( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V5k)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V5k))))))))) => ( ! [V6k:$i] : ((mem@V6k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V6k)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V6k)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__SURJECTIVE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s) = V3t) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & ( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V5k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V5k))))))))) => ( ! [V6k:$i] : ((mem@V6k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V6k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V6k)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__INJECTIVE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V3t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6y))@V3t)) & ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V6y)))))) => (V5x = V6y)))) & ( ! [V7k:$i] : ((mem@V7k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V7k)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V7k)))))))))) => ( ! [V8k:$i] : ((mem@V8k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V8k)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V8k)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__INJECTIVE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V3t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6y))@V3t)) & ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V6y)))))) => (V5x = V6y)))) & ( ! [V7k:$i] : ((mem@V7k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V7k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V7k)))))))))) => ( ! [V8k:$i] : ((mem@V8k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V8k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V8k)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4w:$i] : ((mem@V4w@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V5k)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5k)))))) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3u)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4w)@V2t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V4w)))))))@V3u))))) => (? [V7v:$i] : ((mem@V7v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V7v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4w)@V7v)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V8x))@V7v)))))))@V3u)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => (( ! [V3k:$i] : ((mem@V3k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3k)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3k)))))) <=> ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V5w:$i] : ((mem@V5w@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5w)@V2t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5w)))))))@V4u)))) => (? [V7v:$i] : ((mem@V7v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V7v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5w)@V7v)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V8x))@V7v)))))))@V4u)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4w:$i] : ((mem@V4w@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V5k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5k)))))) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3u)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4w)@V2t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V4w)))))))@V3u))))) => (? [V7v:$i] : ((mem@V7v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V7v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4w)@V7v)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V8x))@V7v)))))))@V3u)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => (( ! [V3k:$i] : ((mem@V3k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3k)))))) <=> ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V5w:$i] : ((mem@V5w@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5w)@V2t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5w)))))))@V4u)))) => (? [V7v:$i] : ((mem@V7v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V7v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5w)@V7v)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V8x))@V7v)))))))@V4u)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EBIGUNION__GSPEC,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [A_27d:del]: ( ! [A_27e:del]: ( ! [A_27f:del]: ( ! [A_27g:del]: ( ! [A_27h:del]: ( ! [A_27i:del]: (( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@A_27b@bool))) => ((ap@(c_2Epred__set_2EBIGUNION@A_27b)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27b@bool)@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27b@bool)@bool)@(ap@V1f@V2x))@(ap@V0P@V2x)))))) = (ap@(c_2Epred__set_2EGSPEC@A_27b@A_27b)@(lam@A_27b@(^ [V3a:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27b@bool)@V3a)@(ap@(c_2Ebool_2E_3F@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@V0P@V4x))@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3a)@(ap@V1f@V4x))))))))))))))) & (( ! [V5P:$i] : ((mem@V5P@(arr@A_27c@(arr@A_27d@bool))) => ( ! [V6f:$i] : ((mem@V6f@(arr@A_27c@(arr@A_27d@(arr@A_27e@bool)))) => ((ap@(c_2Epred__set_2EBIGUNION@A_27e)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27e@bool)@(ty_2Epair_2Eprod@A_27c@A_27d))@(ap@(c_2Epair_2EUNCURRY@A_27c@A_27d@(ty_2Epair_2Eprod@(arr@A_27e@bool)@bool))@(lam@A_27c@(^ [V7x:$i] : (lam@A_27d@(^ [V8y:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27e@bool)@bool)@(ap@(ap@V6f@V7x)@V8y))@(ap@(ap@V5P@V7x)@V8y))))))))) = (ap@(c_2Epred__set_2EGSPEC@A_27e@A_27e)@(lam@A_27e@(^ [V9a:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27e@bool)@V9a)@(ap@(c_2Ebool_2E_3F@A_27c)@(lam@A_27c@(^ [V10x:$i] : (ap@(c_2Ebool_2E_3F@A_27d)@(lam@A_27d@(^ [V11y:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@V5P@V10x)@V11y))@(ap@(ap@(c_2Ebool_2EIN@A_27e)@V9a)@(ap@(ap@V6f@V10x)@V11y)))))))))))))))))) & ( ! [V12P:$i] : ((mem@V12P@(arr@A_27f@(arr@A_27g@(arr@A_27h@bool)))) => ( ! [V13f:$i] : ((mem@V13f@(arr@A_27f@(arr@A_27g@(arr@A_27h@(arr@A_27i@bool))))) => ((ap@(c_2Epred__set_2EBIGUNION@A_27i)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27i@bool)@(ty_2Epair_2Eprod@A_27f@(ty_2Epair_2Eprod@A_27g@A_27h)))@(ap@(c_2Epair_2EUNCURRY@A_27f@(ty_2Epair_2Eprod@A_27g@A_27h)@(ty_2Epair_2Eprod@(arr@A_27i@bool)@bool))@(lam@A_27f@(^ [V14x:$i] : (ap@(c_2Epair_2EUNCURRY@A_27g@A_27h@(ty_2Epair_2Eprod@(arr@A_27i@bool)@bool))@(lam@A_27g@(^ [V15y:$i] : (lam@A_27h@(^ [V16z:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27i@bool)@bool)@(ap@(ap@(ap@V13f@V14x)@V15y)@V16z))@(ap@(ap@(ap@V12P@V14x)@V15y)@V16z)))))))))))) = (ap@(c_2Epred__set_2EGSPEC@A_27i@A_27i)@(lam@A_27i@(^ [V17a:$i] : (ap@(ap@(c_2Epair_2E_2C@A_27i@bool)@V17a)@(ap@(c_2Ebool_2E_3F@A_27f)@(lam@A_27f@(^ [V18x:$i] : (ap@(c_2Ebool_2E_3F@A_27g)@(lam@A_27g@(^ [V19y:$i] : (ap@(c_2Ebool_2E_3F@A_27h)@(lam@A_27h@(^ [V20z:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(ap@V12P@V18x)@V19y)@V20z))@(ap@(ap@(c_2Ebool_2EIN@A_27i)@V17a)@(ap@(ap@(ap@V13f@V18x)@V19y)@V20z))))))))))))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__POINT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => (( ! [V3k:$i] : ((mem@V3k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3k)))))) <=> ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V5y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V2t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@(inj__ty_2Erealax_2Ereal @ V5y))))))))@V4u)))) => (? [V7v:$i] : ((mem@V7v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V7v)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V7v)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V8x))@V7v)))))))@V4u))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__MONOTONE__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (( ! [V3c:$i] : ((mem@V3c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3c)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3c)))))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V2t)) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@(inj__ty_2Erealax_2Ereal @ V4y)))))))))))))) => ( ! [V6c:$i] : ((mem@V6c@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V6c)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V6c)@V2t))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@V6c))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__MONOTONE__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (( ! [V3c:$i] : ((mem@V3c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3c)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3c)))))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V2t)) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@(inj__ty_2Erealax_2Ereal @ V4y)))))))))))))) => ( ! [V6c:$i] : ((mem@V6c@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V6c)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V6c)@V2t))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@V6c))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => (( ! [V3k:$i] : ((mem@V3k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3k)@V2t)) & (p@(ap@c_2Ereal__topology_2Ecompact@V3k))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3k))))))))))) <=> (( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V5k)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5k)))))) & ( ! [V6a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@V2t)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@(inj__ty_2Erealax_2Ereal @ V6a))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))))) => (V2x = V3y)))) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V4t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4t)@V1s))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V4t))))))))))))).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPACT,lemma,( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3k:$i] : ((mem@V3k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V2s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1f)@V2s))@V0t)) & ((p@(ap@c_2Ereal__topology_2Ecompact@V2s)) & (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0t))@V3k))))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1f@V4x))@V3k)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & (( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5k)@V3t)) & (p@(ap@c_2Ereal__topology_2Ecompact@V5k))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5k))))))))))) & ( ! [V7k:$i] : ((mem@V7k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V7k)@V4u)) & (p@(ap@c_2Ereal__topology_2Ecompact@V7k))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V3t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1g@V8x))@V7k))))))))))))) => ( ! [V9k:$i] : ((mem@V9k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V9k)@V4u)) & (p@(ap@c_2Ereal__topology_2Ecompact@V9k))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V10x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V10x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V10x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f)@V10x))@V9k))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__LEFT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s) = V3t) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & ( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5k)@V4u)) & (p@(ap@c_2Ereal__topology_2Ecompact@V5k))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f)@V6x))@V5k))))))))))))))) => ( ! [V7k:$i] : ((mem@V7k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V7k)@V4u)) & (p@(ap@c_2Ereal__topology_2Ecompact@V7k))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V3t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1g@V8x))@V7k))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__RIGHT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & ( ! [V5k:$i] : ((mem@V5k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5k)@V4u)) & (p@(ap@c_2Ereal__topology_2Ecompact@V5k))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f)@V6x))@V5k))))))))))))))) => ( ! [V7k:$i] : ((mem@V7k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V7k)@V3t)) & (p@(ap@c_2Ereal__topology_2Ecompact@V7k))) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V8x))@V7k))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPASTING__LEMMA,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4k:$i] : ((mem@V4k@(arr@A_27a@bool)) => ((( ! [V5i:$i] : ((mem@V5i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5i)@V4k)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3s))@(ap@V2t@V5i))) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@V0f@V5i))@(ap@V2t@V5i))))))) & (( ! [V6i:$i] : ((mem@V6i@A_27a) => ( ! [V7j:$i] : ((mem@V7j@A_27a) => ( ! [V8x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6i)@V4k)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7j)@V4k)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V3s)@(ap@V2t@V6i)))@(ap@V2t@V7j)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V6i)@(inj__ty_2Erealax_2Ereal @ V8x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V7j)@(inj__ty_2Erealax_2Ereal @ V8x)))))))))) & ( ! [V9x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9x))@V3s)) => (? [V10j:$i] : ((mem@V10j@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V10j)@V4k)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9x))@(ap@V2t@V10j))) & ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V9x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V10j)@(inj__ty_2Erealax_2Ereal @ V9x)))))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3s))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3k:$i] : ((mem@V3k@(arr@A_27a@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@A_27a)@(lam@A_27a@(^ [V4i:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@V1t@V4i))@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4i)@V3k)))))))) & (( ! [V5i:$i] : ((mem@V5i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5i)@V3k)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@V1t@V5i))) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@V0f@V5i))@(ap@V1t@V5i))))))) & ( ! [V6i:$i] : ((mem@V6i@A_27a) => ( ! [V7j:$i] : ((mem@V7j@A_27a) => ( ! [V8x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6i)@V3k)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7j)@V3k)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2s)@(ap@V1t@V6i)))@(ap@V1t@V7j)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V6i)@(inj__ty_2Erealax_2Ereal @ V8x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V7j)@(inj__ty_2Erealax_2Ereal @ V8x)))))))))))) => (? [V9g:$i] : ((mem@V9g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V9g)@V2s)) & ( ! [V10x:tp__ty_2Erealax_2Ereal] : ( ! [V11i:$i] : ((mem@V11i@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V11i)@V3k)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10x))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2s)@(ap@V1t@V11i))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V9g@(inj__ty_2Erealax_2Ereal @ V10x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V11i)@(inj__ty_2Erealax_2Ereal @ V10x)))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL__OPEN,lemma,( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V2s)@V0t)))@V2s)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V2s)@V0t)))@V0t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0t))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V2s)@V0t))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V1s)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2t))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1s)@V2t))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL__OPEN,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t)))@V3s)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t)))@V4t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V3s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2g)@V4t)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V3s)) & (~~ (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))))) | ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V4t)) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V5x)))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@V0P@V6x))@(ap@V1f@V6x))@(ap@V2g@V6x)))))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__OPEN,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V3s)) & ((p@(ap@c_2Ereal__topology_2EOpen@V4t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V3s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2g)@V4t)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V3s)) & (~~ (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))))) | ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V4t)) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V5x)))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@V0P@V6x))@(ap@V1f@V6x))@(ap@V2g@V6x)))))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPASTING__LEMMA__CLOSED,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4k:$i] : ((mem@V4k@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V4k)) & (( ! [V5i:$i] : ((mem@V5i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5i)@V4k)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3s))@(ap@V2t@V5i))) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@V0f@V5i))@(ap@V2t@V5i))))))) & (( ! [V6i:$i] : ((mem@V6i@A_27a) => ( ! [V7j:$i] : ((mem@V7j@A_27a) => ( ! [V8x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6i)@V4k)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7j)@V4k)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V3s)@(ap@V2t@V6i)))@(ap@V2t@V7j)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V6i)@(inj__ty_2Erealax_2Ereal @ V8x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V7j)@(inj__ty_2Erealax_2Ereal @ V8x)))))))))) & ( ! [V9x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9x))@V3s)) => (? [V10j:$i] : ((mem@V10j@A_27a) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V10j)@V4k)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9x))@(ap@V2t@V10j))) & ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V9x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V10j)@(inj__ty_2Erealax_2Ereal @ V9x))))))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3s))))))))))))))).
thf(conj_thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS__CLOSED,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3k:$i] : ((mem@V3k@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V3k)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@A_27a)@(lam@A_27a@(^ [V4i:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@V1t@V4i))@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V4i)@V3k)))))))) & (( ! [V5i:$i] : ((mem@V5i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V5i)@V3k)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@V1t@V5i))) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@V0f@V5i))@(ap@V1t@V5i))))))) & ( ! [V6i:$i] : ((mem@V6i@A_27a) => ( ! [V7j:$i] : ((mem@V7j@A_27a) => ( ! [V8x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V6i)@V3k)) & ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V7j)@V3k)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2s)@(ap@V1t@V6i)))@(ap@V1t@V7j)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V6i)@(inj__ty_2Erealax_2Ereal @ V8x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V7j)@(inj__ty_2Erealax_2Ereal @ V8x))))))))))))) => (? [V9g:$i] : ((mem@V9g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V9g)@V2s)) & ( ! [V10x:tp__ty_2Erealax_2Ereal] : ( ! [V11i:$i] : ((mem@V11i@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V11i)@V3k)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10x))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2s)@(ap@V1t@V11i))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V9g@(inj__ty_2Erealax_2Ereal @ V10x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@V0f@V11i)@(inj__ty_2Erealax_2Ereal @ V10x)))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__LIFT__DOT,lemma,( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1net:tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal] : ( ! [V2f:$i] : ((mem@V2f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V2f)@(inj__ty_2Erealax_2Ereal @ V0l))@(inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ V1net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4y:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3a))@(ap@V2f@V4y)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V0l)))@(inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal @ V1net)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__LIFT__DOT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2y))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__LIFT__DOT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2y))))@V1s)))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__LEFT,lemma,( ! [V0b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__lte@V1x)@(inj__ty_2Erealax_2Ereal @ V0b)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__RIGHT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__LE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__GE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__HYPERPLANE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HYPERPLANE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosure@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__STANDARD__HYPERPLANE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V1x)@(inj__ty_2Erealax_2Ereal @ V0a)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__LE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__lte@V1x)@(inj__ty_2Erealax_2Ereal @ V0a)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__GE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__ge@V1x)@(inj__ty_2Erealax_2Ereal @ V0a)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__LT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__LT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Erealax_2Ereal__lt@V1x)@(inj__ty_2Erealax_2Ereal @ V0a)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__GT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__gt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__GT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__gt@V1x)@(inj__ty_2Erealax_2Ereal @ V0a)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__POSITIVE__MULTIPLES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V1c:$i] : (lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__mul@V1c)@V2x))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V1c))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__LEFT,lemma,( ! [V0b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Erealax_2Ereal__lt@V1x)@(inj__ty_2Erealax_2Ereal @ V0b)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__RIGHT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@V1x))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__POSITIVE__ORTHANT,lemma,(p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V0x)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0x)))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__LE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__GE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ereal_2Ereal__gt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__LE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__lte@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Erealax_2Ereal__lt@V2x)@(inj__ty_2Erealax_2Ereal @ V0a))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__GE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__ge@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__gt@V2x)@(inj__ty_2Erealax_2Ereal @ V0a))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__LT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__GT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__gt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__LT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosure@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Erealax_2Ereal__lt@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__lte@V2x)@(inj__ty_2Erealax_2Ereal @ V0a))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__GT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosure@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__gt@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__ge@V2x)@(inj__ty_2Erealax_2Ereal @ V0a))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__HYPERPLANE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (V1b = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => ((ap@c_2Ereal__topology_2Efrontier@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (V1b = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => ((ap@c_2Ereal__topology_2Efrontier@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (V1b = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => ((ap@c_2Ereal__topology_2Efrontier@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (V1b = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => ((ap@c_2Ereal__topology_2Efrontier@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ereal_2Ereal__gt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(inj__ty_2Erealax_2Ereal @ V1b))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__STANDARD__HYPERPLANE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__lte@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))))))).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__ge@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))))))).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Erealax_2Ereal__lt@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))))))).
thf(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ereal_2Ereal__gt@V1x)@(inj__ty_2Erealax_2Ereal @ V0a))))))))))).
thf(conj_thm_2Ereal__topology_2EFORALL__IN__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V2t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@V2t)))))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))@V2t)))))))))))).
thf(conj_thm_2Ereal__topology_2EFORALL__IN__CLOSURE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V2t)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s)))) => (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@V2t)))) <=> ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))@V2t))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LE__ON__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@(inj__ty_2Erealax_2Ereal @ V2a)))))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))@(inj__ty_2Erealax_2Ereal @ V2a)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__GE__ON__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))))))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CONSTANT__ON__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = V2a)))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = V2a)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AGREE__ON__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@(ap@c_2Ereal__topology_2Eclosure@V0s))) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2h)@(ap@c_2Ereal__topology_2Eclosure@V0s))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2h@(inj__ty_2Erealax_2Ereal @ V3x)))))))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(ap@c_2Ereal__topology_2Eclosure@V0s))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2h@(inj__ty_2Erealax_2Ereal @ V4x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__CONSTANT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@(inj__ty_2Erealax_2Ereal @ V2a)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__CONSTANT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V2s)) & (p@(ap@c_2Ereal__topology_2EClosed@V2s))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V2s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V1f@V3x))@(inj__ty_2Erealax_2Ereal @ V0a)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3e)))) => (? [V4d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4d))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V4d)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5y)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))))@(inj__ty_2Erealax_2Ereal @ V3e)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__SEQUENTIALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) <=> ( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@(ap@c_2Ereal__topology_2Eclosure@V1s))) & (( ! [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V4n)))@V1s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V2x)@(inj__ty_2Erealax_2Ereal @ V3a))@c_2Ereal__topology_2Esequentially)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2x))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3a)))@c_2Ereal__topology_2Esequentially)))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s)))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CAUCHY__CONTINUOUS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) => ( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Ecauchy@V2x)) & ( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V3n)))@V1s)))) => (p@(ap@c_2Ereal__topology_2Ecauchy@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IMP__CAUCHY__CONTINUOUS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1s))) => ( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Ecauchy@V2x)) & ( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V3n)))@V1s)))) => (p@(ap@c_2Ereal__topology_2Ecauchy@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__UNIQUENESS__LEMMA,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Ecauchy@V2x)) & ( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V3n)))@V1s)))) => (p@(ap@c_2Ereal__topology_2Ecauchy@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2x)))))) => ( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5x:$i] : ((mem@V5x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((( ! [V6n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V5x@(inj__ty_2Enum_2Enum @ V6n)))@V1s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V5x)@(inj__ty_2Erealax_2Ereal @ V4a))@c_2Ereal__topology_2Esequentially))) => (? [V7l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5x))@(inj__ty_2Erealax_2Ereal @ V7l))@c_2Ereal__topology_2Esequentially)) & ( ! [V8y:$i] : ((mem@V8y@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((( ! [V9n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V8y@(inj__ty_2Enum_2Enum @ V9n)))@V1s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V8y)@(inj__ty_2Erealax_2Ereal @ V4a))@c_2Ereal__topology_2Esequentially))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V8y))@(inj__ty_2Erealax_2Ereal @ V7l))@c_2Ereal__topology_2Esequentially))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__EXTENDS__TO__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Ecauchy@V2x)) & ( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V3n)))@V1s)))) => (p@(ap@c_2Ereal__topology_2Ecauchy@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2x)))))) => (? [V4g:$i] : ((mem@V4g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V4g)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V4g@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))))))))))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__EXTENDS__TO__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) => (? [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V2g)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))))) & ( ! [V4h:$i] : ((mem@V4h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V4h)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V4h@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))))))) => ( ! [V6x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@(ap@c_2Ereal__topology_2Eclosure@V1s))) => ((surj__ty_2Erealax_2Ereal @ (ap@V4h@(inj__ty_2Erealax_2Ereal @ V6x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V6x))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__IMP__CONTINUOUS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2x:$i] : ((mem@V2x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Ecauchy@V2x)) & ( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2x@(inj__ty_2Enum_2Enum @ V3n)))@V1s)))) => (p@(ap@c_2Ereal__topology_2Ecauchy@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2x)))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__UNIFORMLY__CONTINUOUS__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1s))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3h:$i] : ((mem@V3h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V2g)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0x)))) & ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V3h)@(ap@c_2Ereal__topology_2Eat@(ap@V2g@(inj__ty_2Erealax_2Ereal @ V0x))))) & (( ! [V4y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V2g@(ap@V3h@(inj__ty_2Erealax_2Ereal @ V4y)))) = V4y)) & ((surj__ty_2Erealax_2Ereal @ (ap@V3h@(ap@V2g@(inj__ty_2Erealax_2Ereal @ V0x)))) = V0x)))) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V1f)@(ap@c_2Ereal__topology_2Eat@(ap@V2g@(inj__ty_2Erealax_2Ereal @ V0x))))) <=> (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@V1f@(ap@V2g@V5x)))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1z:tp__ty_2Erealax_2Ereal] : ( ! [V2f:$i] : ((mem@V2f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V2f)@(ap@c_2Ereal__topology_2Eat@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1z))))) <=> (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@V2f@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3x)))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1z)))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__IMAGE__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x))))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))) => (V3x = V4y))))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Einterior@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__AVOID,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))@V2s))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V2s)) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = V3a)))) => (? [V4e:tp__ty_2Erealax_2Ereal] : ((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 @ V4e))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V2s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V5y))))@(inj__ty_2Erealax_2Ereal @ V4e)))) => (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5y))) = V3a)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__AVOID,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = V2a))) => (? [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3e))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V4y))))@(inj__ty_2Erealax_2Ereal @ V3e))) => (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y))) = V2a)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__AVOID,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V2s)) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = V3a)))) => (? [V4e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4e))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V2s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V5y))))@(inj__ty_2Erealax_2Ereal @ V4e)))) => (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5y))) = V3a)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__AVOID,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V2s)) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = V3a))))) => (? [V4e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4e))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V5y))))@(inj__ty_2Erealax_2Ereal @ V4e))) => (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5y))) = V3a)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN__CASES,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@(inj__ty_2Erealax_2Ereal @ V2a))))))))))) => (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s)) => (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = V2a)))) | ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))) = V2a))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@(inj__ty_2Erealax_2Ereal @ V2a))))))))) & (? [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = V2a)))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))) = V2a)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ((p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V3x))@(inj__ty_2Erealax_2Ereal @ V2a))))))))) & (? [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))) = V2a)))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))) = V2a)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SCALING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : (((~~ (V1c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) & (p@(ap@c_2Ereal__topology_2EOpen@V0s))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@V2x))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__NEGATIONS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__TRANSLATION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1a))@V2x))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__TRANSLATION__EQ,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@V1s))) <=> (p@(ap@c_2Ereal__topology_2EOpen@V1s))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__AFFINITY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (~~ (V2c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@V3x)))))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@V1s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))))@(ap@c_2Ereal__topology_2Einterior@V1s))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SUMS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) | (p@(ap@c_2Ereal__topology_2EOpen@V1t))) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EUPPER__HEMICONTINUOUS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@V1t)))) => (( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V2s))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4u))))))))))) <=> ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V2s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@V6u))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELOWER__HEMICONTINUOUS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@V1t)))) => (( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V2s))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4u))))))))))) <=> ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V2s))@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@V6u))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__MAP__IFF__LOWER__HEMICONTINUOUS__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => (( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3u)))))) <=> ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5y)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5y)@V2t))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5y)))))))@V4u)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__MAP__IFF__UPPER__HEMICONTINUOUS__PREIMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) => (( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3u)))))) <=> ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5y)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5y)@V2t))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5y)))))))@V4u)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS__EXPLICIT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@V1t)))) & (( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V2s))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4u))))))))))) & ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V2s))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@V6u))))))))))))) => ( ! [V8x:tp__ty_2Erealax_2Ereal] : ( ! [V9e:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@V2s)) & ((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 @ V9e))) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V8x)))) & (~~ ((ap@V0f@(inj__ty_2Erealax_2Ereal @ V8x)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) => (? [V10d:tp__ty_2Erealax_2Ereal] : ((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 @ V10d))) & ( ! [V11x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11x_27))@V2s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@(inj__ty_2Erealax_2Ereal @ V11x_27))))@(inj__ty_2Erealax_2Ereal @ V10d)))) => (( ! [V12y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12y))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V8x)))) => (? [V13y_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V13y_27))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V11x_27)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12y))@(inj__ty_2Erealax_2Ereal @ V13y_27))))@(inj__ty_2Erealax_2Ereal @ V9e))))))) & ( ! [V14y_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V14y_27))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V11x_27)))) => (? [V15y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V15y))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V8x)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V14y_27))@(inj__ty_2Erealax_2Ereal @ V15y))))@(inj__ty_2Erealax_2Ereal @ V9e))))))))))))))))))))))).
thf(ax_thm_2Ereal__topology_2Econnected__component,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))) <=> (? [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2Econnected@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t)@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V3t)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V3t))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__IN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V1x)))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V1x))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__TRANS,lemma,( ! [V0z:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V1s)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V1s)@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V0z)))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V1s)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V0z)))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__OF__SUBSET,lemma,( ! [V0y:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1s)@V2t)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V1s)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V0y)))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V2t)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V0y))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2y)@(ap@(c_2Ebool_2E_3F@(arr@ty_2Erealax_2Ereal@bool))@(lam@(arr@ty_2Erealax_2Ereal@bool)@(^ [V3t:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@c_2Ereal__topology_2Econnected@V3t))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t)@V0s))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V3t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2y)@V3t)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__BIGUNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)) = (ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(lam@(arr@ty_2Erealax_2Ereal@bool)@(^ [V2t:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@V2t)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@c_2Ereal__topology_2Econnected@V2t))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V2t))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2t)@V0s))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)))@V0s)))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT__SET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)) = V0s))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIV,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((ap@(ap@c_2Ereal__topology_2Econnected__component@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(inj__ty_2Erealax_2Ereal @ V0x)) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__UNIV,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) <=> (V0s = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__SELF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)) = V0s)))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTED__COMPONENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__MAXIMAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1t)) & ((p@(ap@c_2Ereal__topology_2Econnected@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__MONO,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V1t)@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EMPTY,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((ap@(ap@c_2Ereal__topology_2Econnected__component@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@(inj__ty_2Erealax_2Ereal @ V0x)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)))) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2y)) = (ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__CONNECTED__COMPONENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__DISJOINT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2b)))) <=> (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2b))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__NONOVERLAP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s))) | ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V0s))) | (~~ ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)) = (ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2b)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__OVERLAP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) <=> ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V0s)) & ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)) = (ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2b))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))) <=> (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)) = (ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2y))) <=> (((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s)))) | ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENT__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s))) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)) = (ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2y)))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__IDEMP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((ap@(ap@c_2Ereal__topology_2Econnected__component@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V1x)) = (ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIQUE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1c)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1c)@V0s)) & ((p@(ap@c_2Ereal__topology_2Econnected@V1c)) & ( ! [V3c_27:$i] : ((mem@V3c_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V3c_27)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3c_27)@V0s)) & (p@(ap@c_2Ereal__topology_2Econnected@V3c_27)))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3c_27)@V1c)))))))) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2x)) = V1c)))))))).
thf(conj_thm_2Ereal__topology_2EJOINABLE__CONNECTED__COMPONENT__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V1t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & ((~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2x)))@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V3y)))@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V2x)) = (ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V3y))))))))))).
thf(conj_thm_2Ereal__topology_2EBIGUNION__CONNECTED__COMPONENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@V1x))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V1x)@V0s)))))) = V0s)))).
thf(conj_thm_2Ereal__topology_2ECOMPLEMENT__CONNECTED__COMPONENT__BIGUNION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))) = (ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EDELETE@(arr@ty_2Erealax_2Ereal@bool))@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@V2y))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2y)@V0s))))))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__CONNECTED__COMPONENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)))))))).
thf(conj_thm_2Ereal__topology_2EBIGUNION__DIFF,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@(arr@A_27a@bool)@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@A_27a@bool)) => ((ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@(ap@(c_2Epred__set_2EBIGUNION@A_27a)@V0s))@V1t) = (ap@(c_2Epred__set_2EBIGUNION@A_27a)@(ap@(c_2Epred__set_2EGSPEC@(arr@A_27a@bool)@(arr@A_27a@bool))@(lam@(arr@A_27a@bool)@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@A_27a@bool)@bool)@(ap@(ap@(c_2Epred__set_2EDIFF@A_27a)@V2x)@V1t))@(ap@(ap@(c_2Ebool_2EIN@(arr@A_27a@bool))@V2x)@V0s))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IN__CONNECTED__COMPONENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@V2x))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))))))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQUIVALENCE__RELATION,lemma,( ! [V0R:$i] : ((mem@V0R@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))) => (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V3y))@(inj__ty_2Erealax_2Ereal @ V2x)))))) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ( ! [V5y:tp__ty_2Erealax_2Ereal] : ( ! [V6z:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5y))) & (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V5y))@(inj__ty_2Erealax_2Ereal @ V6z)))) => (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V6z))))))) & ( ! [V7a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7a))@V1s)) => (? [V8t:$i] : ((mem@V8t@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V8t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7a))@V8t)) & ( ! [V9x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9x))@V8t)) => (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V7a))@(inj__ty_2Erealax_2Ereal @ V9x))))))))))))) => ( ! [V10a:tp__ty_2Erealax_2Ereal] : ( ! [V11b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Econnected__component@V1s)@(inj__ty_2Erealax_2Ereal @ V10a))@(inj__ty_2Erealax_2Ereal @ V11b))) => (p@(ap@(ap@V0R@(inj__ty_2Erealax_2Ereal @ V10a))@(inj__ty_2Erealax_2Ereal @ V11b)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__INTERMEDIATE__SUBSET,lemma,( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1u:$i] : ((mem@V1u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V1u)@(inj__ty_2Erealax_2Ereal @ V2a)))@V0t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0t)@V1u))) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0t)@(inj__ty_2Erealax_2Ereal @ V2a)) = (ap@(ap@c_2Ereal__topology_2Econnected__component@V1u)@(inj__ty_2Erealax_2Ereal @ V2a)))))))))).
thf(ax_thm_2Ereal__topology_2Ecomponents,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Ecomponents@V0s) = (ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@(ap@c_2Ereal__topology_2Econnected__component@V0s)@V1x))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V1x)@V0s))))))))).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@(ap@c_2Ereal__topology_2Ecomponents@V0u))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0u)) & (V1s = (ap@(ap@c_2Ereal__topology_2Econnected__component@V0u)@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2EBIGUNION__COMPONENTS,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => (V0u = (ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecomponents@V0u)))))).
thf(conj_thm_2Ereal__topology_2EPAIRWISE__DISJOINT__COMPONENTS,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Ereal__topology_2Epairwise@(arr@ty_2Erealax_2Ereal@bool))@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal))@(ap@c_2Ereal__topology_2Ecomponents@V0u)))))).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__NONEMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) => (~~ (V1c = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1c)@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__CONNECTED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) => (p@(ap@c_2Ereal__topology_2Econnected@V1c)))))))).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__MAXIMAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) <=> ((~~ (V1c = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1c)@V0s)) & ((p@(ap@c_2Ereal__topology_2Econnected@V1c)) & ( ! [V2c_27:$i] : ((mem@V2c_27@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V2c_27 = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1c)@V2c_27)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2c_27)@V0s)) & (p@(ap@c_2Ereal__topology_2Econnected@V2c_27))))) => (V2c_27 = V1c))))))))))))).
thf(conj_thm_2Ereal__topology_2EJOINABLE__COMPONENTS__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2c1:$i] : ((mem@V2c1@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3c2:$i] : ((mem@V3c2@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V1t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c1)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V3c2)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) & ((~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2c1)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V3c2)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) => (V2c1 = V3c2))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPONENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0s))@V1c)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__COMPONENTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s)))) => (p@(ap@c_2Ereal__topology_2EClosed@V1c)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__COMPONENTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s)))) => (p@(ap@c_2Ereal__topology_2Ecompact@V1c)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__GEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V2c:$i] : ((mem@V2c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c)@(ap@c_2Ereal__topology_2Ecomponents@V1s))) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V2c)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2c)))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__FINITE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Ecomponents@V1s))) & ( ! [V2c:$i] : ((mem@V2c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c)@(ap@c_2Ereal__topology_2Ecomponents@V1s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2c)))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__NONOVERLAP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2c_27:$i] : ((mem@V2c_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c_27)@(ap@c_2Ereal__topology_2Ecomponents@V0s)))) => (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1c)@V2c_27) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> (~~ (V1c = V2c_27))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2c_27:$i] : ((mem@V2c_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c_27)@(ap@c_2Ereal__topology_2Ecomponents@V0s)))) => ((V1c = V2c_27) <=> (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1c)@V2c_27) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__EMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Ecomponents@V0s) = (c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool))) <=> (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EMPTY,lemma,((ap@c_2Ereal__topology_2Ecomponents@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) = (c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool)))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENTS__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2c_27:$i] : ((mem@V2c_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c_27)@(ap@c_2Ereal__topology_2Ecomponents@V0s)))) => (V1c = V2c_27)))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__N__EXISTS,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Ecomponents@V0s) = (ap@(ap@(c_2Epred__set_2EINSERT@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool)))) <=> ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) & ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((? [V2a:$i] : ((mem@V2a@(arr@ty_2Erealax_2Ereal@bool)) & ((ap@c_2Ereal__topology_2Ecomponents@V1s) = (ap@(ap@(c_2Epred__set_2EINSERT@(arr@ty_2Erealax_2Ereal@bool))@V2a)@(c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool)))))) <=> ((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & (~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((ap@c_2Ereal__topology_2Ecomponents@V0s) = (ap@(ap@(c_2Epred__set_2EINSERT@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool)))) <=> ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__EXISTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((? [V1a:$i] : ((mem@V1a@(arr@ty_2Erealax_2Ereal@bool)) & ((ap@c_2Ereal__topology_2Ecomponents@V0s) = (ap@(ap@(c_2Epred__set_2EINSERT@(arr@ty_2Erealax_2Ereal@bool))@V1a)@(c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool)))))) <=> ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIV,lemma,((ap@c_2Ereal__topology_2Ecomponents@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) = (ap@(ap@(c_2Epred__set_2EINSERT@(arr@ty_2Erealax_2Ereal@bool))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Ecomponents@V0s))@(ap@(ap@(c_2Epred__set_2EINSERT@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING__EXISTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (? [V1a:$i] : ((mem@V1a@(arr@ty_2Erealax_2Ereal@bool)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Ecomponents@V0s))@(ap@(ap@(c_2Epred__set_2EINSERT@(arr@ty_2Erealax_2Ereal@bool))@V1a)@(c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool))))))))))).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__SELF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) <=> ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__MAXIMAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2c:$i] : ((mem@V2c@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) & ((p@(ap@c_2Ereal__topology_2Econnected@V1t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2c)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2c)))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIQUE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1k:$i] : ((mem@V1k@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((((ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1k) = V0s) & ( ! [V2c:$i] : ((mem@V2c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c)@V1k)) => ((p@(ap@c_2Ereal__topology_2Econnected@V2c)) & ((~~ (V2c = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V3c_27:$i] : ((mem@V3c_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V3c_27)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2c)@V3c_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3c_27)@V0s)))) => (V3c_27 = V2c)))))))))) => ((ap@c_2Ereal__topology_2Ecomponents@V0s) = V1k))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIQUE__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1k:$i] : ((mem@V1k@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((ap@c_2Ereal__topology_2Ecomponents@V0s) = V1k) <=> (((ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1k) = V0s) & ( ! [V2c:$i] : ((mem@V2c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c)@V1k)) => ((p@(ap@c_2Ereal__topology_2Econnected@V2c)) & ((~~ (V2c = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V3c_27:$i] : ((mem@V3c_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V3c_27)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2c)@V3c_27)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3c_27)@V0s)))) => (V3c_27 = V2c)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EEXISTS__COMPONENT__SUPERSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V0s)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (p@(ap@c_2Ereal__topology_2Econnected@V1t)))) => (? [V2c:$i] : ((mem@V2c@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2c))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPONENTS__INTERMEDIATE__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(ap@c_2Ereal__topology_2Ecomponents@V2u))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2u)))) => (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(ap@c_2Ereal__topology_2Ecomponents@V1t))))))))))).
thf(conj_thm_2Ereal__topology_2EIN__COMPONENTS__BIGUNION__COMPLEMENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1c)@(ap@c_2Ereal__topology_2Ecomponents@V0s))) => ((ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V0s)@V1c) = (ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EDELETE@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Ecomponents@V0s))@V1c))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__SUBSET__CLOPEN,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2c:$i] : ((mem@V2c@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) & ((p@(ap@c_2Ereal__topology_2Econnected@V2c)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2c)@V0u)) & (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2c)@V1s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2c)@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOPEN__BIGUNION__COMPONENTS,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) & (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s))) => (? [V2k:$i] : ((mem@V2k@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@(arr@ty_2Erealax_2Ereal@bool))@V2k)@(ap@c_2Ereal__topology_2Ecomponents@V0u))) & (V1s = (ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V2k))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOPEN__IN__COMPONENTS,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) & ((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & (~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) => (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@(ap@c_2Ereal__topology_2Ecomponents@V0u))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__EQUICONTINUOUS,lemma,( ! [V0fs:$i] : ((mem@V0fs@(arr@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) & (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 @ V3e)))) => (? [V4d:tp__ty_2Erealax_2Ereal] : ((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 @ V4d))) & ( ! [V5f:$i] : ((mem@V5f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V6x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V5f)@V0fs)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x_27))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x_27))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V4d))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V5f@(inj__ty_2Erealax_2Ereal @ V6x_27)))@(ap@V5f@(inj__ty_2Erealax_2Ereal @ V2x)))))@(inj__ty_2Erealax_2Ereal @ V3e)))))))))))) & (p@(ap@c_2Ereal__topology_2Ecompact@V1s))) => ( ! [V7e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V7e))) => (? [V8d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V8d))) & ( ! [V9f:$i] : ((mem@V9f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V10x:tp__ty_2Erealax_2Ereal] : ( ! [V11x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V9f)@V0fs)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11x_27))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11x_27))@(inj__ty_2Erealax_2Ereal @ V10x))))@(inj__ty_2Erealax_2Ereal @ V8d)))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V9f@(inj__ty_2Erealax_2Ereal @ V11x_27)))@(ap@V9f@(inj__ty_2Erealax_2Ereal @ V10x)))))@(inj__ty_2Erealax_2Ereal @ V7e)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__CONTINUOUS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V1s))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@V0f)@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EABS__TRIANGLE__LE,lemma,( ! [V0e:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2y))))@(inj__ty_2Erealax_2Ereal @ V0e))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))))@(inj__ty_2Erealax_2Ereal @ V0e)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__UNIFORM__LIMIT,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & ((p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4n:$i] : (ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@V1f@V4n))@V3s))))@V0net)) & ( ! [V5e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V5e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V6n:$i] : (ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V3s))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V1f@V6n)@V7x))@(ap@V2g@V7x))))@(inj__ty_2Erealax_2Ereal @ V5e)))))))))@V0net)))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2g)@V3s))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (? [V2e:tp__ty_2Erealax_2Ereal] : ((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))) & ( ! [V3x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3x_27))@(inj__ty_2Erealax_2Ereal @ V1x))))@(inj__ty_2Erealax_2Ereal @ V2e))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x_27))@V0s)))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : (( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (? [V3x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x_27))@V0s)) & ((~~ (V3x_27 = V1x)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3x_27))@(inj__ty_2Erealax_2Ereal @ V1x))))@(inj__ty_2Erealax_2Ereal @ V2e)))))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__RANGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => (? [V3d:tp__ty_2Erealax_2Ereal] : ((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 @ V3d))) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V4x_27))@(inj__ty_2Erealax_2Ereal @ V1x))))@(inj__ty_2Erealax_2Ereal @ V3d))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x_27)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x)))))@(inj__ty_2Erealax_2Ereal @ V2e)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__RANGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : ((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 @ V4d))) & ( ! [V5x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x_27))@V1s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V5x_27))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V4d)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x_27)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ABS__COMPOSE,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1f)@V0net)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V2x:$i] : (ap@c_2Ereal_2Eabs@(ap@V1f@V2x)))))@V0net))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ABS__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Ereal_2Eabs@(ap@V0f@V2x)))))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__ABS,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@c_2Ereal_2Eabs)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V0x)))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__DIST,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V2x)))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DIST,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@V2x)))))@V1s)))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__ATTAINS__SUP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2y))@(inj__ty_2Erealax_2Ereal @ V1x))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__ATTAINS__INF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__SUP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V1s)) & ((~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))) => (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__INF,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V1s)) & ((~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)))) => (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))))))))))))).
thf(conj_thm_2Ereal__topology_2EDISTANCE__ATTAINS__SUP,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V3y))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x)))))))))))))).
thf(conj_thm_2Ereal__topology_2EDISTANCE__ATTAINS__INF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V3y)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__MUL,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3c:$i] : ((mem@V3c@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V4d:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V3c)@(inj__ty_2Erealax_2Ereal @ V4d))@V0net)) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V3c@V5x))@(ap@V1f@V5x)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V4d))@(inj__ty_2Erealax_2Ereal @ V2l)))@V0net))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__VMUL,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1c:$i] : ((mem@V1c@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2d:tp__ty_2Erealax_2Ereal] : ( ! [V3v:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1c)@(inj__ty_2Erealax_2Ereal @ V2d))@V0net)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V1c@V4x))@(inj__ty_2Erealax_2Ereal @ V3v)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2d))@(inj__ty_2Erealax_2Ereal @ V3v)))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__VMUL,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1c:$i] : ((mem@V1c@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2v:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1c)@V0net)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V1c@V3x))@(inj__ty_2Erealax_2Ereal @ V2v)))))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__MUL,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2c:$i] : ((mem@V2c@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V2c)@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1f)@V0net))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2c@V3x))@(ap@V1f@V3x)))))@V0net))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__VMUL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2v:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1c)@V0s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V1c@V3x))@(inj__ty_2Erealax_2Ereal @ V2v)))))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MUL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2f:$i] : ((mem@V2f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1c)@V0s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2f)@V0s))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V1c@V3x))@(ap@V2f@V3x)))))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__POW,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V3x:$i] : (ap@V1f@V3x))))@V0net)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Epow@(ap@V1f@V4x))@(inj__ty_2Enum_2Enum @ V2n)))))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__POW,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@V0f@V3x))))@V1s)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Epow@(ap@V0f@V4x))@(inj__ty_2Enum_2Enum @ V2n)))))@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__PRODUCT,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@A_27b@ty_2Erealax_2Ereal))) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27b@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27b)@V2t)) & ( ! [V3i:$i] : ((mem@V3i@A_27b) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27b)@V3i)@V2t)) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@V1f@V4x)@V3i))))@V0net)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@(c_2Eproduct_2Eproduct@A_27b)@V2t)@(ap@V1f@V5x)))))@V0net)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__PRODUCT,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@(arr@A_27a@ty_2Erealax_2Ereal))) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2t)) & ( ! [V3i:$i] : ((mem@V3i@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3i)@V2t)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@V0f@V4x)@V3i))))@V1s)))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Eproduct_2Eproduct@A_27a)@V2t)@(ap@V0f@V5x)))))@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__INV,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) & (~~ (V2l = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@c_2Erealax_2Einv)@V1f))@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V2l)))@V0net)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__INV,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@V1f)@V0net)) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(ap@(c_2Ereal__topology_2Enetlimit@A_27a)@V0net))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@A_27a)@(ap@(ap@(c_2Ecombin_2Eo@A_27a@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@c_2Erealax_2Einv)@V1f))@V0net))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN__INV,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V1s))) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2a))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@c_2Erealax_2Einv)@V0f))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2a)))@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__INV,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a)))) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1a))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@c_2Erealax_2Einv)@V0f))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1a))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INV,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@c_2Erealax_2Einv)@V0f))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__SCALING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@V2x))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__NEGATIONS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__SCALING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@V2x))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__NEGATIONS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__AFFINITY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) => (p@(ap@c_2Ereal__topology_2Ecompact@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@V3x)))))@V0s))))))))).
thf(ax_thm_2Ereal__topology_2Ediameter,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@V0s)) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Esup@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@V1x)@V2y)))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V1x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2y)@V0s)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))))@(ap@c_2Ereal__topology_2Ediameter@V0s)))))) & ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3d))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V3d))@(ap@c_2Ereal__topology_2Ediameter@V0s)))) => (? [V4x:tp__ty_2Erealax_2Ereal] : (? [V5y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V0s)) & (p@(ap@(ap@c_2Ereal_2Ereal__gt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5y))))@(inj__ty_2Erealax_2Ereal @ V3d)))))))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__BOUNDED__BOUND,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))))@(ap@c_2Ereal__topology_2Ediameter@V0s))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V2x)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__EMPTY,lemma,((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__POS__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2Ediameter@V0s))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Ediameter@V0s))@(ap@c_2Ereal__topology_2Ediameter@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@(ap@c_2Ereal__topology_2Eclosure@V0s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@V0s))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL__NONEMPTY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V1z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1z))@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1z))@(ap@c_2Ereal__topology_2Ediameter@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (? [V1z:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1z))@(ap@c_2Ereal__topology_2Ediameter@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__EQ__0,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@V0s)) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (? [V1a:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1d:tp__ty_2Erealax_2Ereal] : ((((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1d)))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))))@(inj__ty_2Erealax_2Ereal @ V1d))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Ediameter@V0s))@(inj__ty_2Erealax_2Ereal @ V1d)))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__CBALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V1r)))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__BALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V1r)))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__SUMS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Ediameter@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t)))))))))))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2Ediameter@V0s))@(ap@c_2Ereal__topology_2Ediameter@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2ELEBESGUE__COVERING__LEMMA,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:$i] : ((mem@V1c@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & ((~~ (V1c = (c_2Epred__set_2EEMPTY@(arr@ty_2Erealax_2Ereal@bool)))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@V1c))) & ( ! [V2b:$i] : ((mem@V2b@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2b)@V1c)) => (p@(ap@c_2Ereal__topology_2EOpen@V2b)))))))) => (? [V3d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3d))) & ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4t)@V0s)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Ediameter@V4t))@(inj__ty_2Erealax_2Ereal @ V3d)))) => (? [V5b:$i] : ((mem@V5b@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V5b)@V1c)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4t)@V5b)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__SCALING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@V2x))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__NEGATIONS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CLOSED__SUMS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__COMPACT__SUMS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V1t))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__SUMS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) | (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t)))))))))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V5y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V4x)@V5y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@(ap@c_2Ereal__topology_2Eclosure@V0s)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5y)@(ap@c_2Ereal__topology_2Eclosure@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__CLOSED__DIFFERENCES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Ereal_2Ereal__sub@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__COMPACT__DIFFERENCES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V1t))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Ereal_2Ereal__sub@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ETRANSLATION__DIFF,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))))@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@V1s)@V2t)) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V4x))))@V1s))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V5x))))@V2t))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATE__POINT__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)))) => (? [V2d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2d))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V3x)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATE__COMPACT__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V2d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2d))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATE__CLOSED__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & ((p@(ap@c_2Ereal__topology_2Ecompact@V1t)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V2d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2d))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => (? [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@bool))) & (( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@V1f@(inj__ty_2Enum_2Enum @ V2n))))) & (( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V1f@(inj__ty_2Enum_2Enum @ V3n)))@V0s))) & (( ! [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V1f@(inj__ty_2Enum_2Enum @ V4n)))@(ap@V1f@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V4n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) & (((ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@V1f@V5n))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V5n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))))))) = V0s) & ( ! [V6k:$i] : ((mem@V6k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V6k)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V6k)@V0s))) => (? [V7N:tp__ty_2Enum_2Enum] : ( ! [V8n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V8n))@(inj__ty_2Enum_2Enum @ V7N))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V6k)@(ap@V1f@(inj__ty_2Enum_2Enum @ V8n)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__UNION__COMPACT__SUBSETS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => (? [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@bool))) & (( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@V1f@(inj__ty_2Enum_2Enum @ V2n))))) & (( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V1f@(inj__ty_2Enum_2Enum @ V3n)))@V0s))) & (( ! [V4n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V1f@(inj__ty_2Enum_2Enum @ V4n)))@(ap@c_2Ereal__topology_2Einterior@(ap@V1f@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V4n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))) & (((ap@(c_2Epred__set_2EBIGUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@V1f@V5n))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V5n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))))))) = V0s) & ( ! [V6k:$i] : ((mem@V6k@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V6k)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V6k)@V0s))) => (? [V7N:tp__ty_2Enum_2Enum] : ( ! [V8n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V8n))@(inj__ty_2Enum_2Enum @ V7N))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V6k)@(ap@V1f@(inj__ty_2Enum_2Enum @ V8n)))))))))))))))))))).
thf(ax_thm_2Ereal__topology_2EOPEN__interval,axiom,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(ap@(ap@c_2Erealax_2Ereal__lt@V2x)@(inj__ty_2Erealax_2Ereal @ V1b))))))))))).
thf(ax_thm_2Ereal__topology_2ECLOSED__interval,axiom,( ! [V0l:tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2ECLOSED__interval@(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l)) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V1x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(c_2Epair_2EFST@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(c_2Elist_2EHD@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l))))@V1x))@(ap@(ap@c_2Ereal_2Ereal__lte@V1x)@(ap@(c_2Epair_2ESND@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(c_2Elist_2EHD@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ V0l)))))))))))).
thf(conj_thm_2Ereal__topology_2Einterval,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))@(ap@(ap@c_2Erealax_2Ereal__lt@V2x)@(inj__ty_2Erealax_2Ereal @ V1b)))))))) & ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(ap@(ap@c_2Ereal_2Ereal__lte@V3x)@(inj__ty_2Erealax_2Ereal @ V1b)))))))))))).
thf(conj_thm_2Ereal__topology_2EIN__INTERVAL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b))))) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V0x))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2b))))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V0x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))).
thf(conj_thm_2Ereal__topology_2EIN__INTERVAL__REFLECT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1b)))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0a))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))) & ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V5x)))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V4b)))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V3a)))))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V4b))))))))))).
thf(conj_thm_2Ereal__topology_2EREFLECT__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Erealax_2Ereal__neg@V2x))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1b)))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0a))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))) & ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@c_2Erealax_2Ereal__neg@V5x))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V4b)))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V4b)))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V3a))))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__EQ__EMPTY,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a))) <=> ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a))) <=> ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__NE__EMPTY,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (((~~ ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) & ((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__INTERVAL__IMP,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2d:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1c))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(inj__ty_2Erealax_2Ereal @ V3b)))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V2d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))) & ((((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1c))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2d))@(inj__ty_2Erealax_2Ereal @ V3b)))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V2d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V3b)))))) & ((((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1c))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(inj__ty_2Erealax_2Ereal @ V3b)))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V2d))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))) & (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1c))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(inj__ty_2Erealax_2Ereal @ V3b)))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V2d))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V3b)))))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V0a)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V0a))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ESUBSET__INTERVAL,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : ( ! [V1d:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1d))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V0c))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V3b)))))) & (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1d))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V0c))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V3b)))))) & (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1d))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1d))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V0c))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V3b)))))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1d))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1d))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V0c))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1d))@(inj__ty_2Erealax_2Ereal @ V3b)))))))))))))).
thf(conj_thm_2Ereal__topology_2EDISJOINT__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : ((((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a))) | ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V3d))@(inj__ty_2Erealax_2Ereal @ V2c))) | ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c))) | (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V3d))@(inj__ty_2Erealax_2Ereal @ V0a))))))) & ((((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a))) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3d))@(inj__ty_2Erealax_2Ereal @ V2c))) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3d))@(inj__ty_2Erealax_2Ereal @ V0a))))))) & ((((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a))) | ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V3d))@(inj__ty_2Erealax_2Ereal @ V2c))) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3d))@(inj__ty_2Erealax_2Ereal @ V0a))))))) & (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a))) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3d))@(inj__ty_2Erealax_2Ereal @ V2c))) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2c))) | (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3d))@(inj__ty_2Erealax_2Ereal @ V0a))))))))))))))).
thf(conj_thm_2Ereal__topology_2EENDS__IN__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> (~~ ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & (( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3b))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> (~~ ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) & (( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5b:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V5b)))))))) & ( ! [V6a:tp__ty_2Erealax_2Ereal] : ( ! [V7b:tp__ty_2Erealax_2Ereal] : (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7b))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@(inj__ty_2Erealax_2Ereal @ V7b)))))))))))).
thf(conj_thm_2Ereal__topology_2EENDS__IN__UNIT__INTERVAL,lemma,((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) & ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))))).
thf(conj_thm_2Ereal__topology_2EINTER__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Emax@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2c)))@(ap@(ap@c_2Ereal_2Emin@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3d))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__OPEN__SUBSET__CLOSED,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LT__MIN,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2z))@(ap@(ap@c_2Ereal_2Emin@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2z))@(inj__ty_2Erealax_2Ereal @ V0x))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2z))@(inj__ty_2Erealax_2Ereal @ V1y))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__LEMMA,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2x))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V1b)))) => (? [V3d:tp__ty_2Erealax_2Ereal] : ((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 @ V3d))) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V4x_27))@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V3d))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V4x_27))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V4x_27))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Einterior@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))))))).
thf(conj_thm_2Ereal__topology_2ENOT__INTERVAL__UNIV,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (~~ ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))) = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__MIDPOINT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL__CONVEX,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ( ! [V4e:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) & ((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 @ V4e))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4e))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V4e))@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V4e)))@(inj__ty_2Erealax_2Ereal @ V3y))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__INV2,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((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 @ V0x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0x)))))))).
thf(conj_thm_2Ereal__topology_2EREAL__INV__LE__1,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V0x))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosure@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL__SYMMETRIC,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (? [V1a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1a)))@(inj__ty_2Erealax_2Ereal @ V1a)))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (? [V1a:tp__ty_2Erealax_2Ereal] : (? [V2b:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL__SYMMETRIC,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (? [V1a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1a)))@(inj__ty_2Erealax_2Ereal @ V1a)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) => (? [V1a:tp__ty_2Erealax_2Ereal] : (? [V2b:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__CLOSED__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))).
thf(conj_thm_2Ereal__topology_2EFRONTIER__OPEN__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Efrontier@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))))))).
thf(conj_thm_2Ereal__topology_2EINTER__INTERVAL__MIXED__EQ__EMPTY,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => (((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) <=> ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__TRANSLATION,lemma,(( ! [V0c:tp__ty_2Erealax_2Ereal] : ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V1a)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0c))@(inj__ty_2Erealax_2Ereal @ V2b))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0c))@V3x))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))) & ( ! [V4c:tp__ty_2Erealax_2Ereal] : ( ! [V5a:tp__ty_2Erealax_2Ereal] : ( ! [V6b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V4c))@(inj__ty_2Erealax_2Ereal @ V5a)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V4c))@(inj__ty_2Erealax_2Ereal @ V6b)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V4c))@V7x))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5a))@(inj__ty_2Erealax_2Ereal @ V6b)))))))))).
thf(conj_thm_2Ereal__topology_2EEMPTY__AS__INTERVAL,lemma,((c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2EUNIT__INTERVAL__NONEMPTY,lemma,((~~ ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2EIMAGE__STRETCH__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2m:$i] : ((mem@V2m@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4f:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V4f)@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2m@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@V3x))))))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5f:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V5f)@(ap@(ap@c_2Ereal_2Emin@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2m@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2m@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V1b))))))))@(ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6f:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V6f)@(ap@(ap@c_2Ereal_2Emax@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2m@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V0a)))@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2m@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V1b)))))))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__IMAGE__STRETCH__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2m:$i] : ((mem@V2m@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (? [V3u:tp__ty_2Erealax_2Ereal] : (? [V4v:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6f:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V6f)@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@V2m@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@V5x))))))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3u))@(inj__ty_2Erealax_2Ereal @ V4v)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__IMAGE__UNIT__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4f:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V4f)@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a)))@V3x))))))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__INTERVALS,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((~~ ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V5y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V4x)@V5y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5y)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2c)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3d))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))) & ( ! [V6a:tp__ty_2Erealax_2Ereal] : ( ! [V7b:tp__ty_2Erealax_2Ereal] : ( ! [V8c:tp__ty_2Erealax_2Ereal] : ( ! [V9d:tp__ty_2Erealax_2Ereal] : (((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@(inj__ty_2Erealax_2Ereal @ V7b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8c))@(inj__ty_2Erealax_2Ereal @ V9d))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V10x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V11y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V10x)@V11y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V10x)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@(inj__ty_2Erealax_2Ereal @ V7b)))))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V11y)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8c))@(inj__ty_2Erealax_2Ereal @ V9d)))))))))))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V6a))@(inj__ty_2Erealax_2Ereal @ V8c)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V7b))@(inj__ty_2Erealax_2Ereal @ V9d)))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL__OPEN__INTERVAL,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (? [V2a:tp__ty_2Erealax_2Ereal] : (? [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@V0s)))))))))) & ( ! [V4s:$i] : ((mem@V4s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V4s)) <=> ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V4s)) => (? [V6a:tp__ty_2Erealax_2Ereal] : (? [V7b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@(inj__ty_2Erealax_2Ereal @ V7b))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6a))@(inj__ty_2Erealax_2Ereal @ V7b))))@V4s)))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (? [V2a:tp__ty_2Erealax_2Ereal] : (? [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__OPEN__INTERVAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (? [V2a:tp__ty_2Erealax_2Ereal] : (? [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2EDIAMETER__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a)))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ediameter@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3b))@(inj__ty_2Erealax_2Ereal @ V2a)))))))))).
thf(conj_thm_2Ereal__topology_2EIMAGE__TWIZZLE__INTERVAL,lemma,( ! [A_27a:del]: ( ! [V0p:$i] : ((mem@V0p@A_27a) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : V3x)))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2EEQ__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) <=> ((((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) | ((V0a = V2c) & (V1b = V3d)))))))) & (( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5b:tp__ty_2Erealax_2Ereal] : ( ! [V6c:tp__ty_2Erealax_2Ereal] : ( ! [V7d:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V5b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6c))@(inj__ty_2Erealax_2Ereal @ V7d)))) <=> (((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V5b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6c))@(inj__ty_2Erealax_2Ereal @ V7d))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) & (( ! [V8a:tp__ty_2Erealax_2Ereal] : ( ! [V9b:tp__ty_2Erealax_2Ereal] : ( ! [V10c:tp__ty_2Erealax_2Ereal] : ( ! [V11d:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V9b))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10c))@(inj__ty_2Erealax_2Ereal @ V11d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) <=> (((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V9b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10c))@(inj__ty_2Erealax_2Ereal @ V11d)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) & ( ! [V12a:tp__ty_2Erealax_2Ereal] : ( ! [V13b:tp__ty_2Erealax_2Ereal] : ( ! [V14c:tp__ty_2Erealax_2Ereal] : ( ! [V15d:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V13b))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V14c))@(inj__ty_2Erealax_2Ereal @ V15d)))) <=> ((((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V13b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V14c))@(inj__ty_2Erealax_2Ereal @ V15d))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) | ((V12a = V14c) & (V13b = V15d)))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__EQ,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EClosed@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2EClosed@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) <=> ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__EQ,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2EOpen@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPACT__INTERVAL__EQ,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) <=> ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2EEQ__BALLS,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1a_27:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ( ! [V3r_27:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2r))) = (ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a_27))@(inj__ty_2Erealax_2Ereal @ V3r_27)))) <=> (((V0a = V1a_27) & (V2r = V3r_27)) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3r_27))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))) & (( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5a_27:tp__ty_2Erealax_2Ereal] : ( ! [V6r:tp__ty_2Erealax_2Ereal] : ( ! [V7r_27:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V6r))) = (ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5a_27))@(inj__ty_2Erealax_2Ereal @ V7r_27)))) <=> ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V6r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V7r_27))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))) & (( ! [V8a:tp__ty_2Erealax_2Ereal] : ( ! [V9a_27:tp__ty_2Erealax_2Ereal] : ( ! [V10r:tp__ty_2Erealax_2Ereal] : ( ! [V11r_27:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V10r))) = (ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9a_27))@(inj__ty_2Erealax_2Ereal @ V11r_27)))) <=> ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V10r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V11r_27))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))) & ( ! [V12a:tp__ty_2Erealax_2Ereal] : ( ! [V13a_27:tp__ty_2Erealax_2Ereal] : ( ! [V14r:tp__ty_2Erealax_2Ereal] : ( ! [V15r_27:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V14r))) = (ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V13a_27))@(inj__ty_2Erealax_2Ereal @ V15r_27)))) <=> (((V12a = V13a_27) & (V14r = V15r_27)) | ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V14r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V15r_27))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__CASES,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) | ((V2x = V0a) | (V2x = V1b)))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__OPEN__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) => ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2EBALL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : (((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1r))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1r)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1r))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) & ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1r))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1r)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1r))))))))).
thf(conj_thm_2Ereal__topology_2ESPHERE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__SPHERE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r)))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a)))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(inj__ty_2Erealax_2Ereal @ V2a)))))))).
thf(conj_thm_2Ereal__topology_2EBALL__INTERVAL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))))))).
thf(conj_thm_2Ereal__topology_2ECBALL__INTERVAL,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e)))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1e))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2EBALL__INTERVAL__0,lemma,( ! [V0e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0e))) = (ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0e)))@(inj__ty_2Erealax_2Ereal @ V0e)))))).
thf(conj_thm_2Ereal__topology_2ECBALL__INTERVAL__0,lemma,( ! [V0e:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0e))) = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0e)))@(inj__ty_2Erealax_2Ereal @ V0e)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__DIFF__OPEN__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EDIFF@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@c_2Ereal_2E_2F@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Emidpoint@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@(ap@(ap@c_2Ereal_2E_2F@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V0a)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3b))@(inj__ty_2Erealax_2Ereal @ V2a))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SEGMENT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2EOpen@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT__SCALAR__MULTIPLE,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2v:tp__ty_2Erealax_2Ereal] : ((ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2v)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V2v))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__mul@V3x)@(inj__ty_2Erealax_2Ereal @ V2v)))@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(ap@(ap@c_2Ereal_2Ereal__lte@V3x)@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1b))@V3x))@(ap@(ap@c_2Ereal_2Ereal__lte@V3x)@(inj__ty_2Erealax_2Ereal @ V0a)))))))))))) & ( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5b:tp__ty_2Erealax_2Ereal] : ( ! [V6v:tp__ty_2Erealax_2Ereal] : ((~~ (V6v = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V6v)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V5b))@(inj__ty_2Erealax_2Ereal @ V6v)))) = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__mul@V7x)@(inj__ty_2Erealax_2Ereal @ V6v)))@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V4a))@V7x))@(ap@(ap@c_2Erealax_2Ereal__lt@V7x)@(inj__ty_2Erealax_2Ereal @ V5b))))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V5b))@V7x))@(ap@(ap@c_2Erealax_2Ereal__lt@V7x)@(inj__ty_2Erealax_2Ereal @ V4a))))))))))))))).
thf(ax_thm_2Ereal__topology_2Eis__interval,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) <=> ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V0s))) => ((((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V3x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b)))) | ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2b))@(inj__ty_2Erealax_2Ereal @ V3x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V1a))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Eis__interval@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) & (p@(ap@c_2Ereal__topology_2Eis__interval@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__EMPTY,lemma,(p@(ap@c_2Ereal__topology_2Eis__interval@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__UNIV,lemma,(p@(ap@c_2Ereal__topology_2Eis__interval@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__POINTWISE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) => ((? [V2a:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@V0s)) & (V2a = V1x))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V0s))) <=> (? [V1a:tp__ty_2Erealax_2Ereal] : (? [V2b:tp__ty_2Erealax_2Ereal] : (V0s = (ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) <=> ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V0s)) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V3x))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V2b)))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__CASES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) <=> ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | ((V0s = (c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)) | ((? [V1a:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V2x)@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1a))@V2x))))))) | ((? [V3a:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3a))@V4x))))))) | ((? [V5b:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ereal_2Ereal__lte@V6x)@(inj__ty_2Erealax_2Ereal @ V5b)))))))) | ((? [V7b:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Erealax_2Ereal__lt@V8x)@(inj__ty_2Erealax_2Ereal @ V7b)))))))) | ((? [V9a:tp__ty_2Erealax_2Ereal] : (? [V10b:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V11x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V11x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V9a))@V11x))@(ap@(ap@c_2Erealax_2Ereal__lt@V11x)@(inj__ty_2Erealax_2Ereal @ V10b)))))))))) | ((? [V12a:tp__ty_2Erealax_2Ereal] : (? [V13b:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V14x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V14x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V12a))@V14x))@(ap@(ap@c_2Ereal_2Ereal__lte@V14x)@(inj__ty_2Erealax_2Ereal @ V13b)))))))))) | ((? [V15a:tp__ty_2Erealax_2Ereal] : (? [V16b:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V17x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V17x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V15a))@V17x))@(ap@(ap@c_2Erealax_2Ereal__lt@V17x)@(inj__ty_2Erealax_2Ereal @ V16b)))))))))) | (? [V18a:tp__ty_2Erealax_2Ereal] : (? [V19b:tp__ty_2Erealax_2Ereal] : (V0s = (ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V20x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V20x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V18a))@V20x))@(ap@(ap@c_2Ereal_2Ereal__lte@V20x)@(inj__ty_2Erealax_2Ereal @ V19b))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__INTER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) & (p@(ap@c_2Ereal__topology_2Eis__interval@V1t))) => (p@(ap@c_2Ereal__topology_2Eis__interval@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__SUBSET__IS__INTERVAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@V0s)) <=> (((ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V0s))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERVAL__CONTAINS__COMPACT__NEIGHBOURHOOD,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))) => (? [V2a:tp__ty_2Erealax_2Ereal] : (? [V3b:tp__ty_2Erealax_2Ereal] : (? [V4d:tp__ty_2Erealax_2Ereal] : ((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 @ V4d))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@V0s)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V4d))))@V0s))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : (p@(ap@c_2Ereal__topology_2Eis__interval@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__SCALING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) => (p@(ap@c_2Ereal__topology_2Eis__interval@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@V2x))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EIS__INTERVAL__SCALING__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2Eis__interval@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@V2x))))@V0s))) <=> ((V1c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | (p@(ap@c_2Ereal__topology_2Eis__interval@V0s)))))))).
thf(conj_thm_2Ereal__topology_2ECARD__FRONTIER__INTERVAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Eis__interval@V0s)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Efrontier@V0s)))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT__UBOUND,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ( ! [V4k:$i] : ((mem@V4k@A_27b) => (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(ap@V1f@V5x))@(inj__ty_2Erealax_2Ereal @ V3b)))))@V0net)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2l))@(inj__ty_2Erealax_2Ereal @ V3b))))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT__LBOUND,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(ap@V1f@V4x)))))@V0net)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(inj__ty_2Erealax_2Ereal @ V2l)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT__EQ,lemma,( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2i:$i] : ((mem@V2i@A_27b) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) & ((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V1f@V5x))@(inj__ty_2Erealax_2Ereal @ V4b)))))@V0net)))) => (V3l = V4b))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__COMPONENT__LE,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V4m))@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(ap@V1f@V5x))@(ap@V2g@V5x)))))@V0net))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3l))@(inj__ty_2Erealax_2Ereal @ V4m)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__DROP__LE,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : (((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V3l))@V0net)) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V2g)@(inj__ty_2Erealax_2Ereal @ V4m))@V0net)) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V5x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(ap@V1f@V5x))@(ap@V2g@V5x)))))@V0net))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3l))@(inj__ty_2Erealax_2Ereal @ V4m)))))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__DROP__UBOUND,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) & ((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(ap@V1f@V4x))@(inj__ty_2Erealax_2Ereal @ V3b)))))@V0net)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2l))@(inj__ty_2Erealax_2Ereal @ V3b)))))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__DROP__LBOUND,lemma,( ! [A_27a:del]: ( ! [V0net:$i] : ((mem@V0net@(ty_2Ereal__topology_2Enet@A_27a)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@A_27a)@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0net)) & ((~~ (p@(ap@(c_2Ereal__topology_2Etrivial__limit@A_27a)@V0net))) & (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(ap@V1f@V4x)))))@V0net)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(inj__ty_2Erealax_2Ereal @ V2l)))))))))))).
thf(conj_thm_2Ereal__topology_2EIMAGE__CLOSURE__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & ((p@(ap@c_2Ereal__topology_2EClosed@V2t)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s)))@V2t)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__IMAGE__CLOSURE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) = (ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__IMAGE__BOUNDED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1s))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__ABS__LE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & (( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y))))@(inj__ty_2Erealax_2Ereal @ V3b))))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosure@V1s))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))))@(inj__ty_2Erealax_2Ereal @ V3b))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__LE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & (( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))@(inj__ty_2Erealax_2Ereal @ V3b))))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosure@V1s))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V3b))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__GE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))) & (( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosure@V1s))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3b))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__MAP__CLOSURES,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))) <=> ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s)))@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ELIM__WITHIN__UNION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t)))) <=> ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V3s))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Erealax_2Ereal)@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Ereal__topology_2Ewithin@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V2x)))@V4t)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V1s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V2t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2t))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V1s)@V2t))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V3s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V4t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V3s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2g)@V4t)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V3s)) & (~~ (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))))) | ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V4t)) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V5x)))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@V0P@V6x))@(ap@V1f@V6x))@(ap@V2g@V6x)))))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL,lemma,( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V2s)@V0t)))@V2s)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V2s)@V0t)))@V0t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V2s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0t))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V2s)@V0t))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t)))@V3s)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t)))@V4t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V3s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2g)@V4t)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V3s)) & (~~ (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))))) | ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V4t)) & (p@(ap@V0P@(inj__ty_2Erealax_2Ereal @ V5x))))) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Erealax_2Ereal @ V5x)))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@V0P@V6x))@(ap@V1f@V6x))@(ap@V2g@V6x)))))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V3s)@V4t))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5t:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5t)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5t)@V3s))@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V2h@V5t))@(inj__ty_2Erealax_2Ereal @ V4a))))))))) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6t:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6t)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6t)@V3s))@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4a))@(ap@V2h@V6t))))))))) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2h)@V3s)) & ( ! [V7t:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7t))@V3s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V2h@(inj__ty_2Erealax_2Ereal @ V7t))) = V4a)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V7t))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V7t))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V8t:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V2h@V8t))@(inj__ty_2Erealax_2Ereal @ V4a)))@(ap@V0f@V8t))@(ap@V1g@V8t)))))@V3s))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__1,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4t:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4t)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4t)@V2s))@(ap@(ap@c_2Ereal_2Ereal__lte@V4t)@(inj__ty_2Erealax_2Ereal @ V3a))))))))) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5t:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5t)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5t)@V2s))@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3a))@V5t)))))))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3a))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V3a))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V6t:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal_2Ereal__lte@V6t)@(inj__ty_2Erealax_2Ereal @ V3a)))@(ap@V0f@V6t))@(ap@V1g@V6t)))))@V2s))))))))))).
thf(conj_thm_2Ereal__topology_2EEXTENSION__FROM__CLOPEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V2t)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V2t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2t))@V3u)) & ((V3u = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) => (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) => (? [V4g:$i] : ((mem@V4g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V4g)@V1s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V4g)@V1s))@V3u)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V2t)) => ((surj__ty_2Erealax_2Ereal @ (ap@V4g@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IVT__HYPERPLANE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s)) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Erealax_2Ereal @ V4b))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4b))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V2y)))))))) => (? [V5z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5z))@V0s)) & ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V5z))) = V4b))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__IVT__COMPONENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s)) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V3a))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V2y))))))) => (? [V4z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4z))@V0s)) & (V4z = V3a)))))))))).
thf(conj_thm_2Ereal__topology_2EMAPPING__CONNECTED__ONTO__SEGMENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Econnected@V0s)) & (~~ (? [V3a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) => (? [V4f:$i] : ((mem@V4f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V4f)@V0s)) & ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V4f)@V0s) = (ap@c_2Ereal__topology_2Eclosed__segment@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__INCREASING__CONVERGENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V1n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@V0s@V1n))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V1n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))))))) & ( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0s@(inj__ty_2Enum_2Enum @ V2n)))@(ap@V0s@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2n))))))) => (? [V3l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0s)@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__DECREASING__CONVERGENT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V1n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@V0s@V1n))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V1n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))))))) & ( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0s@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V2n))))@(ap@V0s@(inj__ty_2Enum_2Enum @ V2n)))))) => (? [V3l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0s)@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially))))))).
thf(ax_thm_2Ereal__topology_2Ehomeomorphism,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3g:$i] : ((mem@V3g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V2f)@V3g))) <=> (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V3g@(ap@V2f@(inj__ty_2Erealax_2Ereal @ V4x)))) = V4x))) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2f)@V0s) = V1t) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2f)@V0s)) & (( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V1t)) => ((surj__ty_2Erealax_2Ereal @ (ap@V2f@(ap@V3g@(inj__ty_2Erealax_2Ereal @ V5y)))) = V5y))) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V3g)@V1t) = V0s) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V3g)@V1t))))))))))))))))).
thf(ax_thm_2Ereal__topology_2Ehomeomorphic,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) <=> (? [V2f:$i] : ((mem@V2f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (? [V3g:$i] : ((mem@V3g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V2f)@V3g))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3g:$i] : ((mem@V3g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V2f)@V3g))) <=> ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2f)@V0s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2f)@V0s))@V1t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V3g)@V1t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V3g)@V1t))@V0s)) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V3g@(ap@V2f@(inj__ty_2Erealax_2Ereal @ V4x)))) = V4x))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V1t)) => ((surj__ty_2Erealax_2Ereal @ (ap@V2f@(ap@V3g@(inj__ty_2Erealax_2Ereal @ V5y)))) = V5y)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__OF__SUBSETS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4s_27:$i] : ((mem@V4s_27@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V5t_27:$i] : ((mem@V5t_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V1g))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4s_27)@V2s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5t_27)@V3t)) & ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V4s_27) = V5t_27)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V4s_27)@V5t_27))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V1g))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__ID,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V0s))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : V1x)))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : V2x)))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__REFL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V0s))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__SYM,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V1g))) <=> (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V3t)@V2s))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V1g)@V0f))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SYM,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) <=> (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1t)@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__COMPOSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V3k:$i] : ((mem@V3k@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V4s:$i] : ((mem@V4s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V5t:$i] : ((mem@V5t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V4s)@V5t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V1g))) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V5t)@V6u))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V2h)@V3k)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V4s)@V6u))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2h)@V0f))@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3k)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1t)@V2u))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V2u)))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__IMP__CARD__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) => (p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0s)@V1t)))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITENESS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) <=> (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__EMPTY,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) <=> (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) & ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@V1s)) <=> (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__MINIMAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) <=> (? [V2f:$i] : ((mem@V2f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (? [V3g:$i] : ((mem@V3g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V2f@(inj__ty_2Erealax_2Ereal @ V4x)))@V1t)) & ((surj__ty_2Erealax_2Ereal @ (ap@V3g@(ap@V2f@(inj__ty_2Erealax_2Ereal @ V4x)))) = V4x)))) & (( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V1t)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V3g@(inj__ty_2Erealax_2Ereal @ V5y)))@V0s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V2f@(ap@V3g@(inj__ty_2Erealax_2Ereal @ V5y)))) = V5y)))) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V2f)@V0s)) & (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V3g)@V1t))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__SELF,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__LEFT__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))) => (V3x = V4y))))) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V2t)) <=> (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@V2t))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__RIGHT__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))) => (V3x = V4y))))) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2t))) <=> (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@V2t))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__SELF,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V2x))))@V1s))@V1s)))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__LEFT__EQ,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))))@V1s))@V2t)) <=> (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@V2t))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__RIGHT__EQ,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))))@V2t))) <=> (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@V2t))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__QUOTIENT__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V1g))) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4u)@V3t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V4u)))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__LEFT,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0c))) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@V3x))))@V1s))@V2t)) <=> (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@V2t)))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__RIGHT,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : ((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 @ V0c))) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@V3x))))@V2t))) <=> (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V1s)@V2t)))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t))) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) <=> ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V0s)) = (surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE__STRONG,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) | (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t))) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) <=> ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V1t)) & ((surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V0s)) = (surj__ty_2Enum_2Enum @ (ap@(c_2Epred__set_2ECARD@ty_2Erealax_2Ereal)@V1t)))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SING,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE__UNIQUE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s) = V3t) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V2s) = V4u) & (( ! [V5v:$i] : ((mem@V5v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V5v)@V3t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V6x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V6x))@V5v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@V5v)))))) & (( ! [V7v:$i] : ((mem@V7v@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V7v)@V4u)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V8x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V2s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V1g@V8x))@V7v)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V4u))@V7v)))))) & ( ! [V9x:tp__ty_2Erealax_2Ereal] : ( ! [V10y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9x))@V2s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10y))@V2s))) => (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V9x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V10y)))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V9x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V10y)))))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V3t)@V4u)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__OPEN__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s) = V3t) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))) = V4x))) & ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V5u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5u))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__CLOSED__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s) = V3t) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x)))) = V4x))) & ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V5u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5u))))))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))))) => (V3x = V4y)))) & ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V5u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5u))))))))) => (? [V6g:$i] : ((mem@V6g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V6g))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))))) => (V3x = V4y)))) & ( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V5u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5u))))))))) => (? [V6g:$i] : ((mem@V6g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V6g))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__OPEN__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V1g))) & (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V4u))) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V4u))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__CLOSED__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V1g))) & (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@V4u))) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V3t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V4u))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))))) => (V3x = V4y)))))) => ((? [V5g:$i] : ((mem@V5g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V5g))))) <=> ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V6u))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))))) => (V3x = V4y)))))) => ((? [V5g:$i] : ((mem@V5g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V5g))))) <=> ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V6u))))))))))))))).
thf(conj_thm_2Ereal__topology_2EINJECTIVE__MAP__OPEN__IFF__CLOSED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))))) => (V3x = V4y)))))) => (( ! [V5u:$i] : ((mem@V5u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V5u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V5u)))))) <=> ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V6u))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__IMP__CLOSED__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (p@(ap@c_2Ereal__topology_2Ecompact@V1s)))) => ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@V3u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3u)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__IMP__QUOTIENT__MAP,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s) = V2t) & (p@(ap@c_2Ereal__topology_2Ecompact@V1s)))) => ( ! [V3u:$i] : ((mem@V3u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3u)@V2t)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4x)@V1s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@V4x))@V3u)))))))) <=> (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2t))@V3u)))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & ((p@(ap@c_2Ereal__topology_2Ecompact@V2s)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))) = V3x))))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1f)@V0s) = V2t) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V0s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V4y)))))) => (V3x = V4y))))))) => (? [V5g:$i] : ((mem@V5g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V2t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V1f)@V5g))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1f)@V0s) = V2t) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V0s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V4y)))))) => (V3x = V4y))))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V2t)))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__SURJECTIVE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & (((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s) = V3t) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & (? [V5h:$i] : ((mem@V5h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V4u))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V5h))))))))) => ((? [V6f_27:$i] : ((mem@V6f_27@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V6f_27))))) & (? [V7g_27:$i] : ((mem@V7g_27@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V3t)@V4u))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V1g)@V7g_27)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__INJECTIVE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V2s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))@V3t)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V3t)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V3t))@V4u)) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V3t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6y))@V3t)) & ((surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V5x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Erealax_2Ereal @ V6y)))))) => (V5x = V6y)))) & (? [V7h:$i] : ((mem@V7h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V4u))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1g)@V0f))@V7h)))))))))) => ((? [V8f_27:$i] : ((mem@V8f_27@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V0f)@V8f_27))))) & (? [V9g_27:$i] : ((mem@V9g_27@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphism@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V3t)@V4u))@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@V1g)@V9g_27)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACTNESS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) <=> (p@(ap@c_2Ereal__topology_2Ecompact@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__CONNECTEDNESS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> (p@(ap@c_2Ereal__topology_2Econnected@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((~~ (V1c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V1c))@V2x))))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1a))@V2x))))@V0s))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__AFFINITY,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((~~ (V2c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@V3x)))))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS__CBALL__SPHERE,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2d:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : (((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 @ V2d))) & (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 @ V3e)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2d))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3e)))))))))) & (( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5b:tp__ty_2Erealax_2Ereal] : ( ! [V6d:tp__ty_2Erealax_2Ereal] : ( ! [V7e:tp__ty_2Erealax_2Ereal] : (((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 @ V6d))) & (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 @ V7e)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V6d))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5b))@(inj__ty_2Erealax_2Ereal @ V7e)))))))))) & ( ! [V8a:tp__ty_2Erealax_2Ereal] : ( ! [V9b:tp__ty_2Erealax_2Ereal] : ( ! [V10d:tp__ty_2Erealax_2Ereal] : ( ! [V11e:tp__ty_2Erealax_2Ereal] : (((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 @ V10d))) & (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 @ V11e)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V10d))))@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9b))@(inj__ty_2Erealax_2Ereal @ V11e))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2d:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : (((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 @ V2d))) & (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 @ V3e)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2d))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__CBALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2d:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : (((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 @ V2d))) & (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 @ V3e)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2d))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SPHERE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2d:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : (((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 @ V2d))) & (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 @ V3e)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2d))))@(ap@c_2Ereal__topology_2Esphere@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__ONE__POINT__COMPACTIFICATIONS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & ((p@(ap@c_2Ereal__topology_2Ecompact@V1t)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3b))@V1t)) & (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V0s)@(inj__ty_2Erealax_2Ereal @ V2a)))@(ap@(ap@(c_2Epred__set_2EDELETE@ty_2Erealax_2Ereal)@V1t)@(inj__ty_2Erealax_2Ereal @ V3b)))))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@V0s)@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVALS,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d)))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V3d))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVAL__UNIV,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__REAL,lemma,(p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@(arr@ty_2Enum_2Enum@bool))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(c_2Epred__set_2EUNIV@(arr@ty_2Enum_2Enum@bool))))).
thf(conj_thm_2Ereal__topology_2EUNCOUNTABLE__REAL,lemma,(~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__REAL__IMP__UNCOUNTABLE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0s)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))) => (~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__EUCLIDEAN,lemma,(p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))).
thf(conj_thm_2Ereal__topology_2EUNCOUNTABLE__EUCLIDEAN,lemma,(~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => (p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => (p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))))))).
thf(conj_thm_2Ereal__topology_2EUNCOUNTABLE__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => (~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))) & ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((~~ ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => (~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))))))))).
thf(conj_thm_2Ereal__topology_2ECOUNTABLE__OPEN__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))) <=> ((ap@c_2Ereal__topology_2EOPEN__interval@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0s)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2EUNCOUNTABLE__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EOpen@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (~~ (p@(ap@(c_2Epred__set_2Ecountable@ty_2Erealax_2Ereal)@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__BALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1r))) => (p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ECARD__EQ__CBALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1r))) => (p@(ap@(ap@(c_2Ecardinal_2Ecardeq@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__IMP__NOT__OPEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (~~ (p@(ap@c_2Ereal__topology_2EOpen@V0s))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__IMP__INFINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V0s)) => ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (~~ (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EEMPTY__INTERIOR__FINITE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@V0s)) => ((ap@c_2Ereal__topology_2Einterior@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))).
thf(conj_thm_2Ereal__topology_2EFINITE__CBALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__BALL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1r))))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V1r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__DISCRETE__FINITE__RANGE__CONSTANT__EQ,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1f)@V0s))@V2t)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V2t)) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V2t)@(inj__ty_2Erealax_2Ereal @ V3y)) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) => (? [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = V4a)))))))))))) & (( ! [V6s:$i] : ((mem@V6s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V6s)) <=> ( ! [V7f:$i] : ((mem@V7f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V7f)@V6s)) & ( ! [V8x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@V6s)) => (? [V9e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V9e))) & ( ! [V10y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V10y))@V6s)) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V7f@(inj__ty_2Erealax_2Ereal @ V10y))) = (surj__ty_2Erealax_2Ereal @ (ap@V7f@(inj__ty_2Erealax_2Ereal @ V8x)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V9e))@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V7f@(inj__ty_2Erealax_2Ereal @ V10y)))@(ap@V7f@(inj__ty_2Erealax_2Ereal @ V8x))))))))))))) => (? [V11a:tp__ty_2Erealax_2Ereal] : ( ! [V12x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12x))@V6s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V7f@(inj__ty_2Erealax_2Ereal @ V12x))) = V11a)))))))))) & ( ! [V13s:$i] : ((mem@V13s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V13s)) <=> ( ! [V14f:$i] : ((mem@V14f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V14f)@V13s)) & (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V14f)@V13s)))) => (? [V15a:tp__ty_2Erealax_2Ereal] : ( ! [V16x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V16x))@V13s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V14f@(inj__ty_2Erealax_2Ereal @ V16x))) = V15a))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1f)@V0s))@V2t)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V2t)) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V2t)@(inj__ty_2Erealax_2Ereal @ V3y)) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) => (? [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = V4a))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0s)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V0s)) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V4y))) = (surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V2x)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3e))@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1f@(inj__ty_2Erealax_2Ereal @ V4y)))@(ap@V1f@(inj__ty_2Erealax_2Ereal @ V2x))))))))))))) => (? [V5a:tp__ty_2Erealax_2Ereal] : ( ! [V6x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V6x))) = V5a))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Econnected@V0s)) <=> ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V0s)) & (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1f)@V0s)))) => (? [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V3x))) = V2a))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT,lemma,( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V2s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1f)@V2s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1f)@V2s))@V0t)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V0t)) => ((ap@(ap@c_2Ereal__topology_2Econnected__component@V0t)@(inj__ty_2Erealax_2Ereal @ V3y)) = (ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))) => (? [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V5x))) = V4a)))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V3e))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s)) & (~~ ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3e))@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x)))))))))))))) => (? [V5a:tp__ty_2Erealax_2Ereal] : ( ! [V6x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V6x))) = V5a)))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Econnected@V1s)) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V0f)@V1s)) & (p@(ap@(c_2Epred__set_2EFINITE@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))) => (? [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = V2a)))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANES,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((~~ (V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) & (~~ (V2c = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V4x))@(inj__ty_2Erealax_2Ereal @ V1b)))))))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@V5x))@(inj__ty_2Erealax_2Ereal @ V3d)))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANE__STANDARD__HYPERPLANE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))@(inj__ty_2Erealax_2Ereal @ V1b)))))))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V4x)@(inj__ty_2Erealax_2Ereal @ V2c))))))))))))).
thf(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__STANDARD__HYPERPLANE__HYPERPLANE,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((~~ (V0a = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Ereal__topology_2Ehomeomorphic@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V3x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V3x)@(inj__ty_2Erealax_2Ereal @ V2c)))))))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V4x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0a))@V4x))@(inj__ty_2Erealax_2Ereal @ V1b))))))))))))).
thf(conj_thm_2Ereal__topology_2ECAUCHY__ISOMETRIC,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2e:tp__ty_2Erealax_2Ereal] : ( ! [V3x:$i] : ((mem@V3x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (((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@c_2Ereal__topology_2Esubspace@V1s)) & ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2e))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V4x))))))) & (( ! [V5n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V3x@(inj__ty_2Enum_2Enum @ V5n)))@V1s))) & (p@(ap@c_2Ereal__topology_2Ecauchy@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3x)))))))) => (p@(ap@c_2Ereal__topology_2Ecauchy@V3x))))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__ISOMETRIC__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2e:tp__ty_2Erealax_2Ereal] : (((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@c_2Ereal__topology_2Esubspace@V1s)) & ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2e))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V3x))))))) & (p@(ap@c_2Ereal__topology_2Ecomplete@V1s)))))) => (p@(ap@c_2Ereal__topology_2Ecomplete@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2EINJECTIVE__IMP__ISOMETRIC,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V1s)) & ((p@(ap@c_2Ereal__topology_2Esubspace@V1s)) & ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (V2x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))) => (? [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__ge@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4x))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V3e))@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V4x))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__IMAGE__SUBSPACE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Esubspace@V1s)) & ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s)) & ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (V2x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) & (p@(ap@c_2Ereal__topology_2EClosed@V1s))))) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__SURJECTIVE__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V1y:tp__ty_2Erealax_2Ereal] : (? [V2x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = V1y)))) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V3s)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3s)))))))))).
thf(conj_thm_2Ereal__topology_2EOPEN__BIJECTIVE__LINEAR__IMAGE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y)))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : (? [V5x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))) = V4y))))) => ((p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))) <=> (p@(ap@c_2Ereal__topology_2EOpen@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2y)))) => (V1x = V2y))))) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EClosed@V3s)) => (p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3s)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y))))) => ((p@(ap@c_2Ereal__topology_2EClosed@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))) <=> (p@(ap@c_2Ereal__topology_2EClosed@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__LINEAR__IMAGE__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Elinear@V0f)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s)))@(ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__INJECTIVE__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y))))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__BOUNDED__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1s))) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Eclosure@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__INTERIOR__IMAGE__SUBSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y))))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Einterior@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__IMAGE__SUBSET__INTERIOR,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2y:tp__ty_2Erealax_2Ereal] : (? [V3x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x))) = V2y)))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Einterior@V1s)))@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__BIJECTIVE__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y)))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : (? [V5x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V5x))) = V4y))))) => ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Einterior@V1s))))))))).
thf(conj_thm_2Ereal__topology_2EIN__INTERIOR__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ((p@(ap@c_2Ereal__topology_2Elinear@V1g)) & (((ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1g) = (c_2Ecombin_2EI@ty_2Erealax_2Ereal)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal__topology_2Einterior@V2s)))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@(ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s))))))))))))).
thf(conj_thm_2Ereal__topology_2ELINEAR__OPEN__MAPPING,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ((p@(ap@c_2Ereal__topology_2Elinear@V1g)) & ((ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1g) = (c_2Ecombin_2EI@ty_2Erealax_2Ereal)))) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2EOpen@V2s)) => (p@(ap@c_2Ereal__topology_2EOpen@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2s)))))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__INJECTIVE__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y))))) => ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@(ap@c_2Ereal__topology_2Einterior@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V1x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2y)))) => (V1x = V2y))))) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@c_2Ereal__topology_2Ecomplete@V3s)) => (p@(ap@c_2Ereal__topology_2Ecomplete@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V3s)))))))))).
thf(conj_thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V2x))) = (surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Erealax_2Ereal @ V3y)))) => (V2x = V3y))))) => ((p@(ap@c_2Ereal__topology_2Ecomplete@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))) <=> (p@(ap@c_2Ereal__topology_2Ecomplete@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ELIMPT__INJECTIVE__LINEAR__IMAGE__EQ,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V1f)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Erealax_2Ereal @ V4y)))) => (V3x = V4y))))) => ((p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(ap@V1f@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V1f)@V2s))) <=> (p@(ap@(ap@c_2Ereal__topology_2Elimit__point__of@(inj__ty_2Erealax_2Ereal @ V0x))@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2EINTERIOR__NEGATIONS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Einterior@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@V0s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Erealax_2Ereal__neg@V2x))))@(ap@c_2Ereal__topology_2Einterior@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESYMMETRIC__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1x)))@V0s)))) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Einterior@V0s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@c_2Ereal__topology_2Einterior@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSURE__NEGATIONS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((ap@c_2Ereal__topology_2Eclosure@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1x:$i] : (ap@c_2Erealax_2Ereal__neg@V1x))))@V0s)) = (ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Erealax_2Ereal__neg@V2x))))@(ap@c_2Ereal__topology_2Eclosure@V0s)))))).
thf(conj_thm_2Ereal__topology_2ESYMMETRIC__CLOSURE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1x)))@V0s)))) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@c_2Ereal__topology_2Eclosure@V0s))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V2x)))@(ap@c_2Ereal__topology_2Eclosure@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ESUBSPACE__SUBSTANDARD,lemma,(p@(ap@c_2Ereal__topology_2Esubspace@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V0x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V0x)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSED__SUBSTANDARD,lemma,(p@(ap@c_2Ereal__topology_2EClosed@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V0x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V0x)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))).
thf(conj_thm_2Ereal__topology_2EDIM__SUBSTANDARD,lemma,((surj__ty_2Enum_2Enum @ (ap@c_2Ereal__topology_2Edim@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V0x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V0x)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@V0x)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))) = fo__c_2Enum_2E0)).
thf(conj_thm_2Ereal__topology_2EAFFINITY__INVERSES,lemma,( ! [V0m:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ((~~ (V0m = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (((ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0m))@V2x))@(inj__ty_2Erealax_2Ereal @ V1c)))))@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@V3x))@(ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@(inj__ty_2Erealax_2Ereal @ V1c))))))) = (lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : V4x))) & ((ap@(ap@(c_2Ecombin_2Eo@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@V5x))@(ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@(inj__ty_2Erealax_2Ereal @ V1c)))))))@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0m))@V6x))@(inj__ty_2Erealax_2Ereal @ V1c))))) = (lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : V7x)))))))).
thf(conj_thm_2Ereal__topology_2EREAL__AFFINITY__LE,lemma,( ! [V0m:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((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 @ V0m))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0m))@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1c)))@(inj__ty_2Erealax_2Ereal @ V3y))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0m))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__AFFINITY,lemma,( ! [V0m:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((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 @ V0m))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3y))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0m))@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1c)))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0m)))))@(inj__ty_2Erealax_2Ereal @ V2x)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__AFFINITY__LT,lemma,( ! [V0m:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((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 @ V0m))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0m))@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1c)))@(inj__ty_2Erealax_2Ereal @ V3y))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2x))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0m))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LT__AFFINITY,lemma,( ! [V0m:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((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 @ V0m))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V3y))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0m))@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1c)))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0m)))))@(inj__ty_2Erealax_2Ereal @ V2x)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__AFFINITY__EQ,lemma,( ! [V0m:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((~~ (V0m = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0m))@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1c))) = V3y) <=> (V2x = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0m))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__EQ__AFFINITY,lemma,( ! [V0m:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((~~ (V0m = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((V3y = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0m))@(inj__ty_2Erealax_2Ereal @ V2x)))@(inj__ty_2Erealax_2Ereal @ V1c)))) <=> ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0m)))@(inj__ty_2Erealax_2Ereal @ V3y)))@(ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1c))@(inj__ty_2Erealax_2Ereal @ V0m))))) = V2x)))))))).
thf(conj_thm_2Ereal__topology_2EIMAGE__AFFINITY__INTERVAL,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2m:tp__ty_2Erealax_2Ereal] : ( ! [V3c:tp__ty_2Erealax_2Ereal] : ((ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2m))@V4x))@(inj__ty_2Erealax_2Ereal @ V3c)))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))))) = (ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b)))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@(ap@(ap@(ap@(c_2Ebool_2ECOND@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2m)))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2m))@(inj__ty_2Erealax_2Ereal @ V0a)))@(inj__ty_2Erealax_2Ereal @ V3c)))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2m))@(inj__ty_2Erealax_2Ereal @ V1b)))@(inj__ty_2Erealax_2Ereal @ V3c))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2ECLOSED__interval@(ap@(ap@(c_2Elist_2ECONS@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2m))@(inj__ty_2Erealax_2Ereal @ V1b)))@(inj__ty_2Erealax_2Ereal @ V3c)))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2m))@(inj__ty_2Erealax_2Ereal @ V0a)))@(inj__ty_2Erealax_2Ereal @ V3c))))@(c_2Elist_2ENIL@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)))))))))))).
thf(ax_thm_2Ereal__topology_2Esums,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@V2s)) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V2s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@V3n)))@V0f))))@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially))))))))).
thf(ax_thm_2Ereal__topology_2Einfsum,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V0s)@V1f)) = (surj__ty_2Erealax_2Ereal @ (ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2l:$i] : (ap@(ap@(ap@c_2Ereal__topology_2Esums@V1f)@V2l)@V0s))))))))))).
thf(ax_thm_2Ereal__topology_2Esummable,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V0s)@V1f)) <=> (? [V2l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1f)@(inj__ty_2Erealax_2Ereal @ V2l))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__SUMMABLE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@V2s)) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V0f))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__INFSUM,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@(ap@c_2Ereal__topology_2Einfsum@V1s)@V0f))@V1s)) <=> (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0f)))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__LIM,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@(ap@(c_2Ereal__topology_2Elim__def@ty_2Enum_2Enum)@c_2Ereal__topology_2Esequentially)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@V2n)))@V0f)))))@V1s)) <=> (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0f)))))))).
thf(conj_thm_2Ereal__topology_2EFINITE__INTER__NUMSEG,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V0s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n)))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__FROM,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2k:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V2k)))) <=> (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2k))@V3n))@V0f))))@(inj__ty_2Erealax_2Ereal @ V1l))@c_2Ereal__topology_2Esequentially)))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__UNIQUE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2l_27:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@V3s)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V2l_27))@V3s))) => (V1l = V2l_27))))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__UNIQUE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@V2s)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@V0f)) = V1l)))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__TERMS__TOZERO,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V2n)))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially)))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__FINITE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@V1s)) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@V1s)@V0f))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__LINEAR,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1h:$i] : ((mem@V1h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@V3s)) & (p@(ap@c_2Ereal__topology_2Elinear@V1h))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@V1h@(ap@V0f@V4n)))))@(ap@V1h@(inj__ty_2Erealax_2Ereal @ V2l)))@V3s))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__0,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V1n:$i] : (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V0s))))).
thf(conj_thm_2Ereal__topology_2ESERIES__ADD,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1x0:tp__ty_2Erealax_2Ereal] : ( ! [V2y:$i] : ((mem@V2y@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3y0:tp__ty_2Erealax_2Ereal] : ( ! [V4s:$i] : ((mem@V4s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0x)@(inj__ty_2Erealax_2Ereal @ V1x0))@V4s)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V2y)@(inj__ty_2Erealax_2Ereal @ V3y0))@V4s))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0x@V5n))@(ap@V2y@V5n)))))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1x0))@(inj__ty_2Erealax_2Ereal @ V3y0)))@V4s)))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__SUB,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1x0:tp__ty_2Erealax_2Ereal] : ( ! [V2y:$i] : ((mem@V2y@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3y0:tp__ty_2Erealax_2Ereal] : ( ! [V4s:$i] : ((mem@V4s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0x)@(inj__ty_2Erealax_2Ereal @ V1x0))@V4s)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V2y)@(inj__ty_2Erealax_2Ereal @ V3y0))@V4s))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0x@V5n))@(ap@V2y@V5n)))))@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1x0))@(inj__ty_2Erealax_2Ereal @ V3y0)))@V4s)))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__CMUL,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1x0:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0x)@(inj__ty_2Erealax_2Ereal @ V1x0))@V3s)) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@V0x@V4n)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(inj__ty_2Erealax_2Ereal @ V1x0)))@V3s)))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__NEG,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1x0:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0x)@(inj__ty_2Erealax_2Ereal @ V1x0))@V2s)) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V0x@V3n)))))@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V1x0)))@V2s))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__IFF,lemma,( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3k:$i] : ((mem@V3k@(arr@ty_2Enum_2Enum@bool)) => (( ! [V4x:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4x))@V3k)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Enum_2Enum @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Enum_2Enum @ V4x)))))) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1f)@(inj__ty_2Erealax_2Ereal @ V0l))@V3k)) <=> (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V2g)@(inj__ty_2Erealax_2Ereal @ V0l))@V3k)))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__EQ,lemma,( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : ((mem@V1f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2g:$i] : ((mem@V2g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V3k:$i] : ((mem@V3k@(arr@ty_2Enum_2Enum@bool)) => ((( ! [V4x:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4x))@V3k)) => ((surj__ty_2Erealax_2Ereal @ (ap@V1f@(inj__ty_2Enum_2Enum @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@V2g@(inj__ty_2Enum_2Enum @ V4x)))))) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1f)@(inj__ty_2Erealax_2Ereal @ V0l))@V3k))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V2g)@(inj__ty_2Erealax_2Ereal @ V0l))@V3k))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__0,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => (( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V2n))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V2n))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@V1s)))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__FINITE__SUPPORT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2k:$i] : ((mem@V2k@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@V2k))) & ( ! [V3x:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3x))@V2k)) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3x))@V1s)))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@V2k))@V0f))@V2k)))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__COMPONENT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@V1s)) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@V0f@V3i))))@(inj__ty_2Erealax_2Ereal @ V2l))@V1s))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__DIFFS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V0f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially)) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@V2n))@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@V2n)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))@(ap@V0f@(inj__ty_2Enum_2Enum @ V1k)))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V1k))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__TRIVIAL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(c_2Epred__set_2EEMPTY@ty_2Enum_2Enum)))))).
thf(conj_thm_2Ereal__topology_2ESERIES__RESTRICT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : ((mem@V1k@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V3n)@V1k))@(ap@V0f@V3n))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))@(inj__ty_2Erealax_2Ereal @ V2l))@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))) <=> (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@V1k))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__SUM,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2k:$i] : ((mem@V2k@(arr@ty_2Enum_2Enum@bool)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@V3s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Enum_2Enum)@V3s)@V2k)) & (( ! [V4x:tp__ty_2Enum_2Enum] : ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4x))@V3s))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V4x))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) & ((surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@V3s)@V0f)) = V1l)))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@V2k))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__REINDEX,lemma,( ! [V0k:tp__ty_2Enum_2Enum] : ( ! [V1a:$i] : ((mem@V1a@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V4x:$i] : (ap@V1a@(ap@(ap@c_2Earithmetic_2E_2B@V4x)@(inj__ty_2Enum_2Enum @ V0k))))))@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V3n)))) <=> (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1a)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@c_2Ereal__topology_2Efrom@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V3n))@(inj__ty_2Enum_2Enum @ V0k)))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__REINDEX__GEN,lemma,( ! [V0k:tp__ty_2Enum_2Enum] : ( ! [V1a:$i] : ((mem@V1a@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V4x:$i] : (ap@V1a@(ap@(ap@c_2Earithmetic_2E_2B@V4x)@(inj__ty_2Enum_2Enum @ V0k))))))@(inj__ty_2Erealax_2Ereal @ V2l))@V3s)) <=> (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1a)@(inj__ty_2Erealax_2Ereal @ V2l))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5i:$i] : (ap@(ap@c_2Earithmetic_2E_2B@V5i)@(inj__ty_2Enum_2Enum @ V0k)))))@V3s))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__LINEAR,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1h:$i] : ((mem@V1h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V0f)) & (p@(ap@c_2Ereal__topology_2Elinear@V1h))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@V1h@(ap@V0f@V3n)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__0,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V0s)@(lam@ty_2Enum_2Enum@(^ [V1n:$i] : (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__ADD,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1y:$i] : ((mem@V1y@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V0x)) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V1y))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0x@V3n))@(ap@V1y@V3n)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__SUB,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1y:$i] : ((mem@V1y@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V0x)) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V1y))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0x@V3n))@(ap@V1y@V3n)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__CMUL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1x:$i] : ((mem@V1x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V0s)@V1x)) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V0s)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@V1x@V3n))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__NEG,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0x)) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V0x@V2n)))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IFF,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2k:$i] : ((mem@V2k@(arr@ty_2Enum_2Enum@bool)) => (( ! [V3x:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3x))@V2k)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Enum_2Enum @ V3x)))))) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V0f)) <=> (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V1g))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2k:$i] : ((mem@V2k@(arr@ty_2Enum_2Enum@bool)) => ((( ! [V3x:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3x))@V2k)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Enum_2Enum @ V3x)))))) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V0f))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V1g)))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__COMPONENT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0f)) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@(lam@ty_2Enum_2Enum@(^ [V2i:$i] : (ap@V0f@V2i))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__SUBSET,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Enum_2Enum@bool)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Enum_2Enum)@V1s)@V2t)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(lam@ty_2Enum_2Enum@(^ [V4i:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V4i)@V1s))@(ap@V0x@V4i))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))@(inj__ty_2Erealax_2Ereal @ V3l))@V2t))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0x)@(inj__ty_2Erealax_2Ereal @ V3l))@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__SUBSET,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Enum_2Enum)@V1s)@V2t)) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2t)@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V3i)@V1s))@(ap@V0x@V3i))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0x)))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__TRIVIAL,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@(c_2Epred__set_2EEMPTY@ty_2Enum_2Enum))@V0f))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__RESTRICT,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : ((mem@V1k@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V2n)@V1k))@(ap@V0f@V2n))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))) <=> (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1k)@V0f)))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__FINITE__DIFF,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Enum_2Enum)@V1t)@V2s)) & ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@V1t)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V3l))@V2s)))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V3l))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@V1t)@V0f)))@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Enum_2Enum)@V2s)@V1t)))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__FINITE__UNION,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Enum_2Enum@bool)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@V2t)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V3l))@V1s))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V3l))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Enum_2Enum)@V2t)@V1s))@V0f)))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Enum_2Enum)@V1s)@V2t)))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__OFFSET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V2m)))) & ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V3n))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V3n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@V0f)))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V3n))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMS__OFFSET__REV,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V2m)))) & ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V2m))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3n))@(inj__ty_2Enum_2Enum @ V2m))))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V1l))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V3n))@(ap@(ap@c_2Earithmetic_2E_2D@(inj__ty_2Enum_2Enum @ V2m))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@V0f)))@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V3n))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__REINDEX,lemma,( ! [V0k:tp__ty_2Enum_2Enum] : ( ! [V1a:$i] : ((mem@V1a@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V2n)))@(lam@ty_2Enum_2Enum@(^ [V3x:$i] : (ap@V1a@(ap@(ap@c_2Earithmetic_2E_2B@V3x)@(inj__ty_2Enum_2Enum @ V0k))))))) <=> (p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2n))@(inj__ty_2Enum_2Enum @ V0k))))@V1a)))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__DROP__LE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V3a))@V2s)) & ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1g)@(inj__ty_2Erealax_2Ereal @ V4b))@V2s)) & ( ! [V5x:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V5x))@V2s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Enum_2Enum @ V5x)))@(ap@V1g@(inj__ty_2Enum_2Enum @ V5x)))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3a))@(inj__ty_2Erealax_2Ereal @ V4b))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__DROP__POS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V2a))@V1s)) & ( ! [V3x:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3x))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@(inj__ty_2Enum_2Enum @ V3x))))))) => (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 @ V2a)))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__BOUND,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V3a))@V2s)) & ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1g)@(inj__ty_2Erealax_2Ereal @ V4b))@V2s)) & ( ! [V5i:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V5i))@V2s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Enum_2Enum @ V5i))))@(ap@V1g@(inj__ty_2Enum_2Enum @ V5i)))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V3a)))@(inj__ty_2Erealax_2Ereal @ V4b))))))))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__LINEAR,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1h:$i] : ((mem@V1h@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V0f)) & (p@(ap@c_2Ereal__topology_2Elinear@V1h))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@V1h@(ap@V0f@V3n)))))) = (surj__ty_2Erealax_2Ereal @ (ap@V1h@(ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@V0f)))))))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__0,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V0s)@(lam@ty_2Enum_2Enum@(^ [V1i:$i] : (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__ADD,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1y:$i] : ((mem@V1y@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V0x)) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V1y))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(ap@V0x@V3i))@(ap@V1y@V3i)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__add@(ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@V0x))@(ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@V1y)))))))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__SUB,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1y:$i] : ((mem@V1y@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V0x)) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V1y))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@(lam@ty_2Enum_2Enum@(^ [V3i:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0x@V3i))@(ap@V1y@V3i)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@V0x))@(ap@(ap@c_2Ereal__topology_2Einfsum@V2s)@V1y)))))))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__CMUL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1x:$i] : ((mem@V1x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V0s)@V1x)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V0s)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@V1x@V3n)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@(ap@c_2Ereal__topology_2Einfsum@V0s)@V1x))))))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__NEG,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1x:$i] : ((mem@V1x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V0s)@V1x)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V0s)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@c_2Erealax_2Ereal__neg@(ap@V1x@V2n)))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Erealax_2Ereal__neg@(ap@(ap@c_2Ereal__topology_2Einfsum@V0s)@V1x)))))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__EQ,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2k:$i] : ((mem@V2k@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V0f)) & ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V1g)) & ( ! [V3x:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3x))@V2k)) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V3x))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Enum_2Enum @ V3x)))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V2k)@V0f)) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V2k)@V1g))))))))))).
thf(conj_thm_2Ereal__topology_2EINFSUM__RESTRICT,lemma,( ! [V0k:$i] : ((mem@V0k@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1a:$i] : ((mem@V1a@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V2n)@V0k))@(ap@V1a@V2n))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Einfsum@V0k)@V1a)))))))).
thf(conj_thm_2Ereal__topology_2EPARTIAL__SUMS__COMPONENT__LE__INFSUM,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2n:tp__ty_2Enum_2Enum] : ((( ! [V3i:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3i))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@(inj__ty_2Enum_2Enum @ V3i)))))) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0f))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V2n))))@V0f))@(ap@(ap@c_2Ereal__topology_2Einfsum@V1s)@V0f)))))))))).
thf(conj_thm_2Ereal__topology_2EPARTIAL__SUMS__DROP__LE__INFSUM,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2n:tp__ty_2Enum_2Enum] : ((( ! [V3i:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3i))@V1s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@(inj__ty_2Enum_2Enum @ V3i)))))) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0f))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V2n))))@V0f))@(ap@(ap@c_2Ereal__topology_2Einfsum@V1s)@V0f)))))))))).
thf(conj_thm_2Ereal__topology_2ESEQUENCE__CAUCHY__WLOG,lemma,( ! [V0e:tp__ty_2Erealax_2Ereal] : ( ! [V1P:$i] : ((mem@V1P@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : (((p@(ap@V1P@(inj__ty_2Enum_2Enum @ V3m))) & (p@(ap@V1P@(inj__ty_2Enum_2Enum @ V4n)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V2s@(inj__ty_2Enum_2Enum @ V3m)))@(ap@V2s@(inj__ty_2Enum_2Enum @ V4n)))))@(inj__ty_2Erealax_2Ereal @ V0e)))))) <=> ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : (((p@(ap@V1P@(inj__ty_2Enum_2Enum @ V5m))) & ((p@(ap@V1P@(inj__ty_2Enum_2Enum @ V6n))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V2s@(inj__ty_2Enum_2Enum @ V5m)))@(ap@V2s@(inj__ty_2Enum_2Enum @ V6n)))))@(inj__ty_2Erealax_2Ereal @ V0e))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUM__DIFF__LEMMA,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : ((mem@V1k@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1k)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V3n))))@V0f))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1k)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V2m))))@V0f))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1k)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V2m))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V3n))))@V0f))))))))))).
thf(conj_thm_2Ereal__topology_2EABS__SUM__TRIVIAL__LEMMA,lemma,( ! [V0P:$i] : ((mem@V0P@bool) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ( ! [V4f:$i] : ((mem@V4f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V5e:tp__ty_2Erealax_2Ereal] : ((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 @ V5e))) => (((p@V0P) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))))@V4f)))@(inj__ty_2Erealax_2Ereal @ V5e)))) <=> ((p@V0P) => ((p@(ap@(ap@c_2Eprim__rec_2E_3C@(inj__ty_2Enum_2Enum @ V3n))@(inj__ty_2Enum_2Enum @ V2m))) | (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))))@V4f)))@(inj__ty_2Erealax_2Ereal @ V5e))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__CAUCHY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ((? [V2l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V2l))@V1s))) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((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 @ V3e))) => (? [V4N:tp__ty_2Enum_2Enum] : ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V4N))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V5m))@(inj__ty_2Enum_2Enum @ V6n))))@V0f)))@(inj__ty_2Erealax_2Ereal @ V3e))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__CAUCHY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0f)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((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))) => (? [V3N:tp__ty_2Enum_2Enum] : ( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V4m))@(inj__ty_2Enum_2Enum @ V3N))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V4m))@(inj__ty_2Enum_2Enum @ V5n))))@V0f)))@(inj__ty_2Erealax_2Ereal @ V2e))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IFF__EVENTUALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2k:$i] : ((mem@V2k@(arr@ty_2Enum_2Enum@bool)) => ((? [V3N:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3N))@(inj__ty_2Enum_2Enum @ V4n))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4n))@V2k))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V4n))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Enum_2Enum @ V4n))))))) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V0f)) <=> (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V1g))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__EQ__EVENTUALLY,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2k:$i] : ((mem@V2k@(arr@ty_2Enum_2Enum@bool)) => (((? [V3N:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3N))@(inj__ty_2Enum_2Enum @ V4n))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4n))@V2k))) => ((surj__ty_2Erealax_2Ereal @ (ap@V0f@(inj__ty_2Enum_2Enum @ V4n))) = (surj__ty_2Erealax_2Ereal @ (ap@V1g@(inj__ty_2Enum_2Enum @ V4n))))))) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V0f))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2k)@V1g)))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IFF__COFINITE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Enum_2Enum)@V1s)@V2t))@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Enum_2Enum)@V2t)@V1s)))) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0f)) <=> (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2t)@V0f))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__EQ__COFINITE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Enum_2Enum)@V1s)@V2t))@(ap@(ap@(c_2Epred__set_2EDIFF@ty_2Enum_2Enum)@V2t)@V1s)))) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@V0f))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2t)@V0f)))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__FROM__ELSEWHERE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V1m)))@V0f)) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V2n)))@V0f)))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__CAUCHY__UNIFORM,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal))) => ( ! [V2k:$i] : ((mem@V2k@(arr@ty_2Enum_2Enum@bool)) => ((? [V3l:$i] : ((mem@V3l@(arr@A_27a@ty_2Erealax_2Ereal)) & ( ! [V4e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V4e))) => (? [V5N:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ( ! [V7x:$i] : ((mem@V7x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V5N))@(inj__ty_2Enum_2Enum @ V6n))) & (p@(ap@V0P@V7x))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V2k)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V6n))))@(ap@V1f@V7x)))@(ap@V3l@V7x))))@(inj__ty_2Erealax_2Ereal @ V4e)))))))))))) <=> ( ! [V8e:tp__ty_2Erealax_2Ereal] : ((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 @ V8e))) => (? [V9N:tp__ty_2Enum_2Enum] : ( ! [V10m:tp__ty_2Enum_2Enum] : ( ! [V11n:tp__ty_2Enum_2Enum] : ( ! [V12x:$i] : ((mem@V12x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V9N))@(inj__ty_2Enum_2Enum @ V10m))) & (p@(ap@V0P@V12x))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V2k)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V10m))@(inj__ty_2Enum_2Enum @ V11n))))@(ap@V1f@V12x))))@(inj__ty_2Erealax_2Ereal @ V8e)))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__GOESTOZERO,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V1x:$i] : ((mem@V1x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V0s)@V1x)) => ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V3n)@V0s))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@V1x@V3n)))@(inj__ty_2Erealax_2Ereal @ V2e))))))@c_2Ereal__topology_2Esequentially)))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__TOZERO,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : ((mem@V1k@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1k)@V0f)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V2n)@V1k))@(ap@V0f@V2n))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially)))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__BOUNDED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : ((mem@V1k@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1k)@V0f)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@V0f)@V1k))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__SUMS__BOUNDED,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V1k)))@V0f)) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1k))@V2n))@V0f))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V2n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__COMPARISON,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => (((? [V3l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1g)@(inj__ty_2Erealax_2Ereal @ V3l))@V2s))) & (? [V4N:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V5n))@(inj__ty_2Enum_2Enum @ V4N))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V5n))@V2s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Enum_2Enum @ V5n))))@(ap@V1g@(inj__ty_2Enum_2Enum @ V5n)))))))) => (? [V6l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V6l))@V2s))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__COMPARISON,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V1g)) & (? [V3N:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V4n))@(inj__ty_2Enum_2Enum @ V3N))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4n))@V2s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Enum_2Enum @ V4n))))@(ap@V1g@(inj__ty_2Enum_2Enum @ V4n)))))))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2s)@V0f)))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__ABSCONV__IMP__CONV,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : ((mem@V1k@(arr@ty_2Enum_2Enum@bool)) => ((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1k)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@c_2Ereal_2Eabs@(ap@V0x@V2n)))))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1k)@V0x)))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__SUBSET__ABSCONV,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Enum_2Enum@bool)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@c_2Ereal_2Eabs@(ap@V0x@V3n)))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Enum_2Enum)@V2t)@V1s))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V2t)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@c_2Ereal_2Eabs@(ap@V0x@V4n)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__COMPARISON__BOUND,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1g)@(inj__ty_2Erealax_2Ereal @ V3a))@V2s)) & ( ! [V4i:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4i))@V2s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Enum_2Enum @ V4i))))@(ap@V1g@(inj__ty_2Enum_2Enum @ V4i))))))) => (? [V5l:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0f)@(inj__ty_2Erealax_2Ereal @ V5l))@V2s)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V5l)))@(inj__ty_2Erealax_2Ereal @ V3a)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__COMPARISON__UNIFORM,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal))) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2P:$i] : ((mem@V2P@(arr@A_27a@bool)) => ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Enum_2Enum@bool)) => (((? [V4l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1g)@(inj__ty_2Erealax_2Ereal @ V4l))@V3s))) & (? [V5N:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ( ! [V7x:$i] : ((mem@V7x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V5N))@(inj__ty_2Enum_2Enum @ V6n))) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V6n))@V3s)) & (p@(ap@V2P@V7x)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@V0f@V7x)@(inj__ty_2Enum_2Enum @ V6n))))@(ap@V1g@(inj__ty_2Enum_2Enum @ V6n)))))))))) => (? [V8l:$i] : ((mem@V8l@(arr@A_27a@ty_2Erealax_2Ereal)) & ( ! [V9e:tp__ty_2Erealax_2Ereal] : ((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 @ V9e))) => (? [V10N:tp__ty_2Enum_2Enum] : ( ! [V11n:tp__ty_2Enum_2Enum] : ( ! [V12x:$i] : ((mem@V12x@A_27a) => (((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V10N))@(inj__ty_2Enum_2Enum @ V11n))) & (p@(ap@V2P@V12x))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V3s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(inj__ty_2Enum_2Enum @ V11n))))@(ap@V0f@V12x)))@(ap@V8l@V12x))))@(inj__ty_2Erealax_2Ereal @ V9e))))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__RATIO,lemma,( ! [V0c:tp__ty_2Erealax_2Ereal] : ( ! [V1a:$i] : ((mem@V1a@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V3N:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0c))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) & ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3E_3D@(inj__ty_2Enum_2Enum @ V4n))@(inj__ty_2Enum_2Enum @ V3N))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@V1a@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V4n)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0c))@(ap@c_2Ereal_2Eabs@(ap@V1a@(inj__ty_2Enum_2Enum @ V4n))))))))) => (? [V5l:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V1a)@(inj__ty_2Erealax_2Ereal @ V5l))@V2s))))))))))).
thf(conj_thm_2Ereal__topology_2EBOUNDED__PARTIAL__SUMS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1k:tp__ty_2Enum_2Enum] : ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V2n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V1k))@V2n))@V0f))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V2n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))))))) => (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Enum_2Enum@ty_2Enum_2Enum))@(ap@(c_2Epair_2EUNCURRY@ty_2Enum_2Enum@ty_2Enum_2Enum@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Enum_2Enum@(^ [V3m:$i] : (lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@V3m)@V4n))@V0f))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V3m)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V4n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__BILINEAR__PARTIAL__PRE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4k:tp__ty_2Enum_2Enum] : (((p@(ap@c_2Ereal__topology_2Ebilinear@V2h)) & ((p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@V2h@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@V5n)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V1g@V5n)))))@(inj__ty_2Erealax_2Ereal @ V3l))@c_2Ereal__topology_2Esequentially)) & (p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V4k)))@(lam@ty_2Enum_2Enum@(^ [V6n:$i] : (ap@(ap@V2h@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@V6n)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V0f@V6n)))@(ap@V1g@V6n)))))))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V4k)))@(lam@ty_2Enum_2Enum@(^ [V7n:$i] : (ap@(ap@V2h@(ap@V0f@V7n))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1g@V7n))@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2D@V7n)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__DIRICHLET__BILINEAR,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2h:$i] : ((mem@V2h@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V3k:tp__ty_2Enum_2Enum] : ( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5p:tp__ty_2Enum_2Enum] : ( ! [V6l:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ebilinear@V2h)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V7n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V4m))@V7n))@V0f))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V7n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))))))) & ((p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V5p)))@(lam@ty_2Enum_2Enum@(^ [V8n:$i] : (ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2B@V8n)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V1g@V8n))))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V9n:$i] : (ap@(ap@V2h@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2B@V9n)@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@V9n))@V0f)))))@(inj__ty_2Erealax_2Ereal @ V6l))@c_2Ereal__topology_2Esequentially))))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V3k)))@(lam@ty_2Enum_2Enum@(^ [V10n:$i] : (ap@(ap@V2h@(ap@V1g@V10n))@(ap@V0f@V10n)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__DIRICHLET,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V2N:tp__ty_2Enum_2Enum] : ( ! [V3k:tp__ty_2Enum_2Enum] : ( ! [V4m:tp__ty_2Enum_2Enum] : (((p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ V4m))@V5n))@V0f))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V5n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))))))) & (( ! [V6n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2N))@(inj__ty_2Enum_2Enum @ V6n))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V1g@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V6n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@V1g@(inj__ty_2Enum_2Enum @ V6n)))))) & (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@V1g)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially)))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@c_2Ereal__topology_2Efrom@(inj__ty_2Enum_2Enum @ V3k)))@(lam@ty_2Enum_2Enum@(^ [V7n:$i] : (ap@(ap@c_2Erealax_2Ereal__mul@(ap@V1g@V7n))@(ap@V0f@V7n))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE__STRONG,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Enum_2Enum)@V2f)@V1s))@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@c_2Ereal_2Eabs@(ap@V0x@V3n)))))) & ( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V4m))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V5n))@V1s)) & ((surj__ty_2Enum_2Enum @ (ap@V2f@(inj__ty_2Enum_2Enum @ V4m))) = (surj__ty_2Enum_2Enum @ (ap@V2f@(inj__ty_2Enum_2Enum @ V5n)))))) => (V4m = V5n))))) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V6n:$i] : (ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Enum_2Enum)@V2f)@V1s))@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@V6n)))@V0x))@(ap@(ap@(c_2Eiterate_2ESum@ty_2Enum_2Enum)@(ap@(ap@(c_2Epred__set_2EINTER@ty_2Enum_2Enum)@V1s)@(ap@(ap@c_2Eiterate_2E_2E_2E@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@V6n)))@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@V0x)@V2f))))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@c_2Ereal__topology_2Esequentially)))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Esummable@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Enum_2Enum)@V2f)@V1s))@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@c_2Ereal_2Eabs@(ap@V0x@V4n)))))) & ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V5m))@V1s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V6n))@V1s)) & ((surj__ty_2Enum_2Enum @ (ap@V2f@(inj__ty_2Enum_2Enum @ V5m))) = (surj__ty_2Enum_2Enum @ (ap@V2f@(inj__ty_2Enum_2Enum @ V6n)))))) => (V5m = V6n))))) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@V0x)@V2f))@(inj__ty_2Erealax_2Ereal @ V3l))@V1s)) <=> (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0x)@(inj__ty_2Erealax_2Ereal @ V3l))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Enum_2Enum)@V2f)@V1s))))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__REARRANGE__EQ,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2p:$i] : ((mem@V2p@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@c_2Ereal_2Eabs@(ap@V0x@V4n)))))) & (p@(ap@(ap@(c_2Ereal__topology_2Epermutes@ty_2Enum_2Enum)@V2p)@V1s))) => ((p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@V0x)@V2p))@(inj__ty_2Erealax_2Ereal @ V3l))@V1s)) <=> (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0x)@(inj__ty_2Erealax_2Ereal @ V3l))@V1s)))))))))))).
thf(conj_thm_2Ereal__topology_2ESERIES__REARRANGE,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2p:$i] : ((mem@V2p@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@c_2Ereal_2Eabs@(ap@V0x@V4n)))))) & ((p@(ap@(ap@(c_2Ereal__topology_2Epermutes@ty_2Enum_2Enum)@V2p)@V1s)) & (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@V0x)@(inj__ty_2Erealax_2Ereal @ V3l))@V1s)))) => (p@(ap@(ap@(ap@c_2Ereal__topology_2Esums@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@V0x)@V2p))@(inj__ty_2Erealax_2Ereal @ V3l))@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2ESUMMABLE__REARRANGE,lemma,( ! [V0x:$i] : ((mem@V0x@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Enum_2Enum@bool)) => ( ! [V2p:$i] : ((mem@V2p@(arr@ty_2Enum_2Enum@ty_2Enum_2Enum)) => (((p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@(lam@ty_2Enum_2Enum@(^ [V3n:$i] : (ap@c_2Ereal_2Eabs@(ap@V0x@V3n)))))) & (p@(ap@(ap@(c_2Ereal__topology_2Epermutes@ty_2Enum_2Enum)@V2p)@V1s))) => (p@(ap@(ap@c_2Ereal__topology_2Esummable@V1s)@(ap@(ap@(c_2Ecombin_2Eo@ty_2Enum_2Enum@ty_2Erealax_2Ereal@ty_2Enum_2Enum)@V0x)@V2p))))))))))).
thf(conj_thm_2Ereal__topology_2EBANACH__FIX,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2c:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ecomplete@V1s)) & ((~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2c))) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@V1s)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V4y)))))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2c))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y)))))))))))))) => (p@(ap@(c_2Ebool_2E_3F_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V1s))@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V5x))))))))))))).
thf(conj_thm_2Ereal__topology_2EDINI,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))) => ( ! [V1g:$i] : ((mem@V1g@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V2s)) & (( ! [V3n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(ap@V0f@(inj__ty_2Enum_2Enum @ V3n)))@V2s))) & ((p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@V1g)@V2s)) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V2s)) => (p@(ap@(ap@(ap@(c_2Ereal__topology_2E_2D_2D_3E@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@V0f@V5n)@(inj__ty_2Erealax_2Ereal @ V4x)))))@(ap@V1g@(inj__ty_2Erealax_2Ereal @ V4x)))@c_2Ereal__topology_2Esequentially)))) & ( ! [V6n:tp__ty_2Enum_2Enum] : ( ! [V7x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@V2s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@V0f@(inj__ty_2Enum_2Enum @ V6n))@(inj__ty_2Erealax_2Ereal @ V7x)))@(ap@(ap@V0f@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V6n))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V7x))))))))))) => ( ! [V8e:tp__ty_2Erealax_2Ereal] : ((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 @ V8e))) => (p@(ap@(ap@(c_2Ereal__topology_2Eeventually@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V9n:$i] : (ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V10x:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V10x)@V2s))@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@(ap@V0f@V9n)@V10x))@(ap@V1g@V10x))))@(inj__ty_2Erealax_2Ereal @ V8e)))))))))@c_2Ereal__topology_2Esequentially)))))))))))).
thf(ax_thm_2Ereal__topology_2Eclosest__point,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1a))) = (surj__ty_2Erealax_2Ereal @ (ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V0s))@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V2x)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@V3y))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__EXISTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)))@V0s)) & ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2y))))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__SET,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)))@V0s))))))).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2x))))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__SELF,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)) => ((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))) = V1x)))))).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__REFL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (((surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1x))) = V1x) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s)))))))).
thf(conj_thm_2Ereal__topology_2EDIST__CLOSEST__POINT__LIPSCHITZ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V2y)))))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__DIST__CLOSEST__POINT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2x)@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@V2x))))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DIST__CLOSEST__POINT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2x)@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@V2x))))))@V1t)))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__DIST__CLOSEST__POINT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2x)@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@V2x))))))@V1t)))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT__TO__CLOSEST__POINT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1a)))))@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))).
thf(conj_thm_2Ereal__topology_2ESEGMENT__TO__POINT__EXISTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V2b:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2b))@V0s)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eopen__segment@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(inj__ty_2Erealax_2Ereal @ V2b))))@V0s) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__INTERIOR,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@c_2Ereal__topology_2Einterior@V0s))) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Einterior@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Einterior@V0s)))))) => (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V0s)@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@c_2Ereal__topology_2Efrontier@V0s)))))))).
thf(ax_thm_2Ereal__topology_2Esetdist,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@V1t)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eiterate_2Einf@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V2x)@V3y)))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__EMPTY,lemma,(( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@V0t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) & ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__POS__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SUBSETS__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2s_27:$i] : ((mem@V2s_27@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t_27:$i] : ((mem@V3t_27@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2s_27)@V0s)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t_27)@V1t)) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ( ! [V5y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5y))@V1t))) => (? [V6x_27:tp__ty_2Erealax_2Ereal] : (? [V7y_27:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x_27))@V2s_27)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7y_27))@V3t_27)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x_27))@(inj__ty_2Erealax_2Ereal @ V7y_27))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4x))@(inj__ty_2Erealax_2Ereal @ V5y)))))))))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s_27)@V3t_27))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__SETDIST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2d:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y)))))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2d))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__LE__DIST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__SETDIST__EQ,lemma,( ! [V0d:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0d))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t)))) <=> (( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V2t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0d))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y)))))))) & (((V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (V2t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0d))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__SETDIST__LT__EXISTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(inj__ty_2Erealax_2Ereal @ V2b))))) => (? [V3x:tp__ty_2Erealax_2Ereal] : (? [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1t)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(inj__ty_2Erealax_2Ereal @ V4y))))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__REFL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V0s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SYM,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__TRIANGLE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V2t)))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V2t))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SINGS,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__LIPSCHITZ,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@A_27a) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__SETDIST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@(c_2Ereal__topology_2Econtinuous@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V2y)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))))@(ap@c_2Ereal__topology_2Eat@(inj__ty_2Erealax_2Ereal @ V1x)))))))).
thf(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SETDIST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Econtinuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V2y)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))))@V1t))))))).
thf(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SETDIST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal__topology_2Euniformly__continuous__on@(lam@ty_2Erealax_2Ereal@(^ [V2y:$i] : (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V2y)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))))@V1t))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__DIFFERENCES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Ereal_2Ereal__sub@V2x)@V3y))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SUBSET__RIGHT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V2u))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V2u)))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SUBSET__LEFT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V2u)))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V2u)))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__CLOSURE,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Eclosure@V0s))@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))))))) & ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@(ap@c_2Ereal__topology_2Eclosure@V3t)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__FRONTIER,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal)@V0s)@V1t)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Efrontier@V0s))@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))))))))) & ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal)@V2s)@V3t)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@(ap@c_2Ereal__topology_2Efrontier@V3t)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__COMPACT__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) => (? [V2x:tp__ty_2Erealax_2Ereal] : (? [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1t)) & ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__CLOSED__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & ((p@(ap@c_2Ereal__topology_2Ecompact@V1t)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) => (? [V2x:tp__ty_2Erealax_2Ereal] : (? [V3y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3y))@V1t)) & ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V3y)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__COMPACT__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & (p@(ap@c_2Ereal__topology_2EClosed@V1t))) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | ((V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & (p@(ap@c_2Ereal__topology_2Ecompact@V1t))) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | ((V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__BOUNDED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) | (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | ((V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (~~ ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))))@V1s))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V4x))))@V2t)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V3x)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2t)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t)))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__UNIQUE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ( ! [V4d:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@V0s)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3b))@V1t)) & (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b)))) = V4d) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V0s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6y))@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2a))@(inj__ty_2Erealax_2Ereal @ V3b))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(inj__ty_2Erealax_2Ereal @ V6y))))))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = V4d)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__UNIV,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) & ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(c_2Epred__set_2EUNIV@ty_2Erealax_2Ereal))@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__ZERO,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((~~ (p@(ap@(ap@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal)@V0s)@V1t))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__ZERO__STRONG,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((~~ (p@(ap@(ap@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal)@(ap@c_2Ereal__topology_2Eclosure@V0s))@(ap@c_2Ereal__topology_2Eclosure@V1t)))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__FRONTIERS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal)@V0s)@V1t))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Efrontier@V0s))@(ap@c_2Ereal__topology_2Efrontier@V1t))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__FRONTIER,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@c_2Ereal__topology_2Efrontier@V0s)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__CLOSEST__POINT,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V1s)) & (~~ (V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(ap@(ap@c_2Ereal__topology_2Eclosest__point@V1s)@(inj__ty_2Erealax_2Ereal @ V0a))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__SING,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal__topology_2Eclosure@V0s)))))))) & ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V2s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(ap@c_2Ereal__topology_2Eclosure@V2s)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@c_2Ereal__topology_2EClosed@V0s)) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__IN,lemma,( ! [V0u:$i] : ((mem@V0u@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V0u))@V1s)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0u))) => (((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V1s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) | (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V1s))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__IN__SET,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V1s)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__FRONTIER__CASES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@V0s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@c_2Ereal__topology_2Efrontier@V0s)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__TRIANGLE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Eabs@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))))@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Erealax_2Ereal @ V2y)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__LE__SING,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__BALLS,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ( ! [V3s:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V2r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Erealax_2Ereal @ V3s)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V3s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Emax@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Erealax_2Ereal @ V1b))))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V2r))@(inj__ty_2Erealax_2Ereal @ V3s))))))))))) & (( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5b:tp__ty_2Erealax_2Ereal] : ( ! [V6r:tp__ty_2Erealax_2Ereal] : ( ! [V7s:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V6r))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5b))@(inj__ty_2Erealax_2Ereal @ V7s)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V6r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V7s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Emax@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4a))@(inj__ty_2Erealax_2Ereal @ V5b))))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V6r))@(inj__ty_2Erealax_2Ereal @ V7s))))))))))) & (( ! [V8a:tp__ty_2Erealax_2Ereal] : ( ! [V9b:tp__ty_2Erealax_2Ereal] : ( ! [V10r:tp__ty_2Erealax_2Ereal] : ( ! [V11s:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V10r))))@(ap@c_2Ereal__topology_2Eball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V9b))@(inj__ty_2Erealax_2Ereal @ V11s)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V10r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V11s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Emax@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8a))@(inj__ty_2Erealax_2Ereal @ V9b))))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V10r))@(inj__ty_2Erealax_2Ereal @ V11s))))))))))) & ( ! [V12a:tp__ty_2Erealax_2Ereal] : ( ! [V13b:tp__ty_2Erealax_2Ereal] : ( ! [V14r:tp__ty_2Erealax_2Ereal] : ( ! [V15s:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V14r))))@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V13b))@(inj__ty_2Erealax_2Ereal @ V15s)))))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V14r))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V15s))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Emax@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Ereal__sub@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V12a))@(inj__ty_2Erealax_2Ereal @ V13b))))@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V14r))@(inj__ty_2Erealax_2Ereal @ V15s))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATION__CLOSURES,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@(ap@c_2Ereal__topology_2Eclosure@V1t)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V1t)@(ap@c_2Ereal__topology_2Eclosure@V0s)) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) => (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) & (? [V3v:$i] : ((mem@V3v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Epred__set_2EDISJOINT@ty_2Erealax_2Ereal)@V2u)@V3v)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2u)) & ((p@(ap@c_2Ereal__topology_2EOpen@V3v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V2u)) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V3v)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2EClosed@V0s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) & (? [V3v:$i] : ((mem@V3v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2u)) & ((p@(ap@c_2Ereal__topology_2EOpen@V3v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V2u)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V3v)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2u)@V3v) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL__LOCAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V0s)) & ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V1t)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V3s_27:$i] : ((mem@V3s_27@(arr@ty_2Erealax_2Ereal@bool)) & (? [V4t_27:$i] : ((mem@V4t_27@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V3s_27)) & ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2u))@V4t_27)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V3s_27)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V4t_27)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V3s_27)@V4t_27) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL__COMPACT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ecompact@V0s)) & ((p@(ap@c_2Ereal__topology_2EClosed@V1t)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V0s)@V1t) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) => (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) & (? [V3v:$i] : ((mem@V3v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2u)) & ((p@(ap@c_2Ereal__topology_2Ecompact@(ap@c_2Ereal__topology_2Eclosure@V2u))) & ((p@(ap@c_2Ereal__topology_2EOpen@V3v)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@V2u)) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@V3v)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2u)@V3v) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATION__HAUSDORFF,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((~~ (V0x = V1y)) => (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) & (? [V3v:$i] : ((mem@V3v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2u)) & ((p@(ap@c_2Ereal__topology_2EOpen@V3v)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2u)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@V3v)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2u)@V3v) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATION__T2,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((~~ (V0x = V1y)) <=> (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) & (? [V3v:$i] : ((mem@V3v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2u)) & ((p@(ap@c_2Ereal__topology_2EOpen@V3v)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2u)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@V3v)) & ((ap@(ap@(c_2Epred__set_2EINTER@ty_2Erealax_2Ereal)@V2u)@V3v) = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATION__T1,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((~~ (V0x = V1y)) <=> (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) & (? [V3v:$i] : ((mem@V3v@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2u)) & ((p@(ap@c_2Ereal__topology_2EOpen@V3v)) & ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2u)) & ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@V2u))) & ((~~ (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V3v))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@V3v))))))))))))))).
thf(conj_thm_2Ereal__topology_2ESEPARATION__T0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((~~ (V0x = V1y)) <=> (? [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@c_2Ereal__topology_2EOpen@V2u)) & (~~ ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@V2u)) <=> (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@V2u))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CHAIN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) => ((p@(ap@c_2Ereal__topology_2Ecompact@V1s)) & (p@(ap@c_2Ereal__topology_2Econnected@V1s)))))) & ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V0f)) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V3t)@V0f))) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V2s)@V3t)) | (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3t)@V2s))))))))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__CHAIN__GEN,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@(arr@ty_2Erealax_2Ereal@bool)@bool)) => ((( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V0f)) => ((p@(ap@c_2Ereal__topology_2EClosed@V1s)) & (p@(ap@c_2Ereal__topology_2Econnected@V1s)))))) & ((? [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) & ((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V0f)) & (p@(ap@c_2Ereal__topology_2Ecompact@V2s))))) & ( ! [V3s:$i] : ((mem@V3s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V4t:$i] : ((mem@V4t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V3s)@V0f)) & (p@(ap@(ap@(c_2Ebool_2EIN@(arr@ty_2Erealax_2Ereal@bool))@V4t)@V0f))) => ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V3s)@V4t)) | (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V4t)@V3s)))))))))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@V0f))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__NEST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@c_2Ereal__topology_2Ecompact@(ap@V0s@(inj__ty_2Enum_2Enum @ V1n)))) & (p@(ap@c_2Ereal__topology_2Econnected@(ap@V0s@(inj__ty_2Enum_2Enum @ V1n)))))) & ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V2m))@(inj__ty_2Enum_2Enum @ V3n))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V3n)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V2m)))))))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@V0s@V4n))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V4n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))))))))))))).
thf(conj_thm_2Ereal__topology_2ECONNECTED__NEST__GEN,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Enum_2Enum@(arr@ty_2Erealax_2Ereal@bool))) => ((( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@c_2Ereal__topology_2EClosed@(ap@V0s@(inj__ty_2Enum_2Enum @ V1n)))) & (p@(ap@c_2Ereal__topology_2Econnected@(ap@V0s@(inj__ty_2Enum_2Enum @ V1n)))))) & ((? [V2n:tp__ty_2Enum_2Enum] : (p@(ap@c_2Ereal__topology_2Ecompact@(ap@V0s@(inj__ty_2Enum_2Enum @ V2n))))) & ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V3m))@(inj__ty_2Enum_2Enum @ V4n))) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0s@(inj__ty_2Enum_2Enum @ V4n)))@(ap@V0s@(inj__ty_2Enum_2Enum @ V3m))))))))) => (p@(ap@c_2Ereal__topology_2Econnected@(ap@(c_2Epred__set_2EBIGINTER@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@(arr@ty_2Erealax_2Ereal@bool)@ty_2Enum_2Enum)@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@bool)@(ap@V0s@V5n))@(ap@(ap@(c_2Ebool_2EIN@ty_2Enum_2Enum)@V5n)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum))))))))))))).
thf(ax_thm_2Ereal__topology_2Ehausdist,axiom,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@c_2Ebool_2E_7E@(ap@(ap@(c_2Emin_2E_3D@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V2x)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))))))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V3y)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t)))))))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))@(ap@(c_2Ebool_2E_3F@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4b:$i] : (ap@(c_2Ebool_2E_21@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5d:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5d)@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V6x)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6x)@V0s))))))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V7y)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7y)@V1t))))))))@(ap@(ap@c_2Ereal_2Ereal__lte@V5d)@V4b))))))))))@(ap@c_2Ereal_2Esup@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V8x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V8x)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V8x)@V0s))))))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V9y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V9y)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V9y)@V1t))))))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__POS__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__REFL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V0s))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__SYM,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V0s))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__EMPTY,lemma,(( ! [V0t:$i] : ((mem@V0t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))@V0t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) & ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__SINGS,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2s_27:$i] : ((mem@V2s_27@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t_27:$i] : ((mem@V3t_27@(arr@ty_2Erealax_2Ereal@bool)) => (( ! [V4b:tp__ty_2Erealax_2Ereal] : ((( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(inj__ty_2Erealax_2Ereal @ V4b))))) & ( ! [V6y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6y))@V1t)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(inj__ty_2Erealax_2Ereal @ V4b)))))) <=> (( ! [V7x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@V2s_27)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V3t_27)))@(inj__ty_2Erealax_2Ereal @ V4b))))) & ( ! [V8y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8y))@V3t_27)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V2s_27)))@(inj__ty_2Erealax_2Ereal @ V4b)))))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s_27)@V3t_27)))))))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__TRANSLATION,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V3x))))@V1s))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V4x:$i] : (ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0a))@V4x))))@V2t)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t)))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__LINEAR__IMAGE,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2t:$i] : ((mem@V2t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Elinear@V0f)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V3x)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V1s))@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@V0f)@V2t)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1s)@V2t)))))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__CLOSURE,lemma,(( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@c_2Ereal__topology_2Eclosure@V0s))@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))))))) & ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V3t:$i] : ((mem@V3t@(arr@ty_2Erealax_2Ereal@bool)) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@(ap@c_2Ereal__topology_2Eclosure@V3t)))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V2s)@V3t)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(inj__ty_2Erealax_2Ereal @ V2b))))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1t)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(inj__ty_2Erealax_2Ereal @ V2b)))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE__SUMS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V0s)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (lam@ty_2Erealax_2Ereal@(^ [V4z:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V3y)@V4z))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V4z)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))))))) & (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@V1t)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal))@(ap@(c_2Epair_2EUNCURRY@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal@(ty_2Epair_2Eprod@ty_2Erealax_2Ereal@bool))@(lam@ty_2Erealax_2Ereal@(^ [V5y:$i] : (lam@ty_2Erealax_2Ereal@(^ [V6z:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@(ap@c_2Erealax_2Ereal__add@V5y)@V6z))@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5y)@V0s))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V6z)@(ap@c_2Ereal__topology_2Ecball@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2b))))))))))))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LE__HAUSDIST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ( ! [V4c:tp__ty_2Erealax_2Ereal] : ( ! [V5z:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (( ! [V6x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V6x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(inj__ty_2Erealax_2Ereal @ V3b))))) & (( ! [V7y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7y))@V1t)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V7y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(inj__ty_2Erealax_2Ereal @ V4c))))) & (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5z))@V0s)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5z))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t))))) | ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5z))@V1t)) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V5z))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V2a))@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__LE__HAUSDIST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__SING__LE__HAUSDIST,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))))))))))).
thf(conj_thm_2Ereal__topology_2ESETDIST__HAUSDIST__TRIANGLE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V2u)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V2u)))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V2u))))))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__SETDIST__TRIANGLE,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V2u)))@(ap@(ap@c_2Erealax_2Ereal__add@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V2u))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__LT__HAUSDIST__POINT__EXISTS,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3d:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(inj__ty_2Erealax_2Ereal @ V3d))) & (p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@V0s)))))) => (? [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1t)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V2x))@(inj__ty_2Erealax_2Ereal @ V4y))))@(inj__ty_2Erealax_2Ereal @ V3d))))))))))))).
thf(conj_thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Erealax_2Ereal@(arr@ty_2Erealax_2Ereal@bool))) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2s:$i] : ((mem@V2s@(arr@ty_2Erealax_2Ereal@bool)) => ((( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V2s)) => (p@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V3x)))@V1t)))) & (( ! [V4u:$i] : ((mem@V4u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V4u)) => (p@(ap@(ap@(c_2Etopology_2Eopen__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V5x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V5x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V5x)@V2s))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@V5x))@V4u))))))))))) & ( ! [V6u:$i] : ((mem@V6u@(arr@ty_2Erealax_2Ereal@bool)) => ((p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V1t))@V6u)) => (p@(ap@(ap@(c_2Etopology_2Eclosed__in@ty_2Erealax_2Ereal)@(ap@(ap@(c_2Ereal__topology_2Esubtopology@ty_2Erealax_2Ereal)@c_2Ereal__topology_2Eeuclidean)@V2s))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V7x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@V7x)@(ap@(ap@c_2Ebool_2E_2F_5C@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V7x)@V2s))@(ap@(ap@(c_2Epred__set_2ESUBSET@ty_2Erealax_2Ereal)@(ap@V0f@V7x))@V6u))))))))))))) => ( ! [V8x:tp__ty_2Erealax_2Ereal] : ( ! [V9e:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@V2s)) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V9e))) & (p@(ap@c_2Ereal__topology_2Ebounded__def@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V8x)))))) => (? [V10d:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V10d))) & ( ! [V11x_27:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V11x_27))@V2s)) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2EDist@(ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V8x))@(inj__ty_2Erealax_2Ereal @ V11x_27))))@(inj__ty_2Erealax_2Ereal @ V10d)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V8x)))@(ap@V0f@(inj__ty_2Erealax_2Ereal @ V11x_27)))))@(inj__ty_2Erealax_2Ereal @ V9e)))))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V2x)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s))))))@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V3y)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL__ALT,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)) & ((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))))) => ((surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@c_2Ereal_2Emax@(ap@c_2Ereal_2Esup@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V2x:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V2x)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V2x)@V0s)))))))@(ap@c_2Ereal_2Esup@(ap@(c_2Epred__set_2EGSPEC@ty_2Erealax_2Ereal@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V3y:$i] : (ap@(ap@(c_2Epair_2E_2C@ty_2Erealax_2Ereal@bool)@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@V3y)@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@V3y)@V1t))))))))))))))).
thf(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE__EQ,lemma,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((~~ (V0s = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & (p@(ap@c_2Ereal__topology_2Ebounded__def@V1t))))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V0s)@V1t)))@(inj__ty_2Erealax_2Ereal @ V2b))) <=> (( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@V0s)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V3x))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V1t)))@(inj__ty_2Erealax_2Ereal @ V2b))))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@(c_2Ebool_2EIN@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@V1t)) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Esetdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EINSERT@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V4y))@(c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal)))@V0s)))@(inj__ty_2Erealax_2Ereal @ V2b)))))))))))))).
thf(conj_thm_2Ereal__topology_2EHAUSDIST__UNION__LE,conjecture,( ! [V0s:$i] : ((mem@V0s@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V1t:$i] : ((mem@V1t@(arr@ty_2Erealax_2Ereal@bool)) => ( ! [V2u:$i] : ((mem@V2u@(arr@ty_2Erealax_2Ereal@bool)) => (((p@(ap@c_2Ereal__topology_2Ebounded__def@V0s)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V1t)) & ((p@(ap@c_2Ereal__topology_2Ebounded__def@V2u)) & ((~~ (V1t = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))) & (~~ (V2u = (c_2Epred__set_2EEMPTY@ty_2Erealax_2Ereal))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V1t))@(ap@(ap@(c_2Epred__set_2EUNION@ty_2Erealax_2Ereal)@V0s)@V2u))))@(ap@c_2Ereal__topology_2Ehausdist@(ap@(ap@(c_2Epair_2E_2C@(arr@ty_2Erealax_2Ereal@bool)@(arr@ty_2Erealax_2Ereal@bool))@V1t)@V2u)))))))))))).
