include('Axioms/measure_2.ax').
tff(tp_ty_2Ereal__topology_2Enet,type,( ty_2Ereal__topology_2Enet: del > del )).
tff(tp_c_2Ereal__topology_2E_2D_2D_3E,type,(c_2Ereal__topology_2E_2D_2D_3E:del > $i)).
tff(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)))))).
tff(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)).
tff(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)).
tff(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 : $i > tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal)).
tff(stp_inj_surj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,( ! [X:tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : (surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X)) = X))).
tff(stp_inj_mem_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,( ! [X:tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : (mem(inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X),ty_2Elist_2Elist(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))).
tff(stp_iso_mem_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,( ! [X:$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)))))).
tff(tp_c_2Ereal__topology_2ECLOSED__interval,type,(c_2Ereal__topology_2ECLOSED__interval:$i)).
tff(mem_c_2Ereal__topology_2ECLOSED__interval,axiom,mem(c_2Ereal__topology_2ECLOSED__interval,arr(ty_2Elist_2Elist(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),arr(ty_2Erealax_2Ereal,bool)))).
tff(stp_fo_c_2Ereal__topology_2ECLOSED__interval,type,(fo__c_2Ereal__topology_2ECLOSED__interval:(tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2ECLOSED__interval,axiom,( ! [X0:tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( ! [X1:tp__ty_2Erealax_2Ereal] : (inj__o(fo__c_2Ereal__topology_2ECLOSED__interval(X0,X1)) = ap(ap(c_2Ereal__topology_2ECLOSED__interval,inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)))))).
tff(tp_c_2Ereal__topology_2EClosed,type,(c_2Ereal__topology_2EClosed:$i)).
tff(mem_c_2Ereal__topology_2EClosed,axiom,mem(c_2Ereal__topology_2EClosed,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2EDist,type,(c_2Ereal__topology_2EDist:$i)).
tff(mem_c_2Ereal__topology_2EDist,axiom,mem(c_2Ereal__topology_2EDist,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal))).
tff(stp_fo_c_2Ereal__topology_2EDist,type,(fo__c_2Ereal__topology_2EDist:(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal) > tp__ty_2Erealax_2Ereal)).
tff(stp_eq_fo_c_2Ereal__topology_2EDist,axiom,( ! [X0:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : (inj__ty_2Erealax_2Ereal(fo__c_2Ereal__topology_2EDist(X0)) = ap(c_2Ereal__topology_2EDist,inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X0))))).
tff(tp_c_2Ereal__topology_2EOPEN__interval,type,(c_2Ereal__topology_2EOPEN__interval:$i)).
tff(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)))).
tff(stp_fo_c_2Ereal__topology_2EOPEN__interval,type,(fo__c_2Ereal__topology_2EOPEN__interval:(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2EOPEN__interval,axiom,( ! [X0:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( ! [X1:tp__ty_2Erealax_2Ereal] : (inj__o(fo__c_2Ereal__topology_2EOPEN__interval(X0,X1)) = ap(ap(c_2Ereal__topology_2EOPEN__interval,inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)))))).
tff(tp_c_2Ereal__topology_2EOpen,type,(c_2Ereal__topology_2EOpen:$i)).
tff(mem_c_2Ereal__topology_2EOpen,axiom,mem(c_2Ereal__topology_2EOpen,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(stp_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,type,(tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal : $tType)).
tff(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)).
tff(stp_surj_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,type,(surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal : $i > tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal)).
tff(stp_inj_surj_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,( ! [X:tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal] : (surj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal(inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal(X)) = X))).
tff(stp_inj_mem_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,( ! [X:tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal] : (mem(inj__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal(X),ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal))))).
tff(stp_iso_mem_c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal,axiom,( ! [X:$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)))))).
tff(tp_c_2Ereal__topology_2Eat,type,(c_2Ereal__topology_2Eat:$i)).
tff(mem_c_2Ereal__topology_2Eat,axiom,mem(c_2Ereal__topology_2Eat,arr(ty_2Erealax_2Ereal,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal)))).
tff(tp_c_2Ereal__topology_2Eat__infinity,type,(c_2Ereal__topology_2Eat__infinity:$i)).
tff(mem_c_2Ereal__topology_2Eat__infinity,axiom,mem(c_2Ereal__topology_2Eat__infinity,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal))).
tff(tp_c_2Ereal__topology_2Eat__neginfinity,type,(c_2Ereal__topology_2Eat__neginfinity:$i)).
tff(mem_c_2Ereal__topology_2Eat__neginfinity,axiom,mem(c_2Ereal__topology_2Eat__neginfinity,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal))).
tff(tp_c_2Ereal__topology_2Eat__posinfinity,type,(c_2Ereal__topology_2Eat__posinfinity:$i)).
tff(mem_c_2Ereal__topology_2Eat__posinfinity,axiom,mem(c_2Ereal__topology_2Eat__posinfinity,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal))).
tff(tp_c_2Ereal__topology_2Eball,type,(c_2Ereal__topology_2Eball:$i)).
tff(mem_c_2Ereal__topology_2Eball,axiom,mem(c_2Ereal__topology_2Eball,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,bool)))).
tff(stp_fo_c_2Ereal__topology_2Eball,type,(fo__c_2Ereal__topology_2Eball:(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2Eball,axiom,( ! [X0:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( ! [X1:tp__ty_2Erealax_2Ereal] : (inj__o(fo__c_2Ereal__topology_2Eball(X0,X1)) = ap(ap(c_2Ereal__topology_2Eball,inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)))))).
tff(tp_c_2Ereal__topology_2Ebetween,type,(c_2Ereal__topology_2Ebetween:$i)).
tff(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)))).
tff(stp_fo_c_2Ereal__topology_2Ebetween,type,(fo__c_2Ereal__topology_2Ebetween:(tp__ty_2Erealax_2Ereal * tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2Ebetween,axiom,( ! [X0:tp__ty_2Erealax_2Ereal] : ( ! [X1:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : (inj__o(fo__c_2Ereal__topology_2Ebetween(X0,X1)) = ap(ap(c_2Ereal__topology_2Ebetween,inj__ty_2Erealax_2Ereal(X0)),inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X1)))))).
tff(tp_c_2Ereal__topology_2Ebilinear,type,(c_2Ereal__topology_2Ebilinear:$i)).
tff(mem_c_2Ereal__topology_2Ebilinear,axiom,mem(c_2Ereal__topology_2Ebilinear,arr(arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),bool))).
tff(tp_c_2Ereal__topology_2Ebounded__def,type,(c_2Ereal__topology_2Ebounded__def:$i)).
tff(mem_c_2Ereal__topology_2Ebounded__def,axiom,mem(c_2Ereal__topology_2Ebounded__def,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Ecauchy,type,(c_2Ereal__topology_2Ecauchy:$i)).
tff(mem_c_2Ereal__topology_2Ecauchy,axiom,mem(c_2Ereal__topology_2Ecauchy,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool))).
tff(tp_c_2Ereal__topology_2Ecball,type,(c_2Ereal__topology_2Ecball:$i)).
tff(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)))).
tff(stp_fo_c_2Ereal__topology_2Ecball,type,(fo__c_2Ereal__topology_2Ecball:(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2Ecball,axiom,( ! [X0:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( ! [X1:tp__ty_2Erealax_2Ereal] : (inj__o(fo__c_2Ereal__topology_2Ecball(X0,X1)) = ap(ap(c_2Ereal__topology_2Ecball,inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)))))).
tff(tp_c_2Ereal__topology_2Eclosed__segment,type,(c_2Ereal__topology_2Eclosed__segment:$i)).
tff(mem_c_2Ereal__topology_2Eclosed__segment,axiom,mem(c_2Ereal__topology_2Eclosed__segment,arr(ty_2Elist_2Elist(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),arr(ty_2Erealax_2Ereal,bool)))).
tff(stp_fo_c_2Ereal__topology_2Eclosed__segment,type,(fo__c_2Ereal__topology_2Eclosed__segment:(tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2Eclosed__segment,axiom,( ! [X0:tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( ! [X1:tp__ty_2Erealax_2Ereal] : (inj__o(fo__c_2Ereal__topology_2Eclosed__segment(X0,X1)) = ap(ap(c_2Ereal__topology_2Eclosed__segment,inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)))))).
tff(tp_c_2Ereal__topology_2Eclosest__point,type,(c_2Ereal__topology_2Eclosest__point:$i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Eclosure,type,(c_2Ereal__topology_2Eclosure:$i)).
tff(mem_c_2Ereal__topology_2Eclosure,axiom,mem(c_2Ereal__topology_2Eclosure,arr(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)))).
tff(tp_c_2Ereal__topology_2Ecollinear,type,(c_2Ereal__topology_2Ecollinear:$i)).
tff(mem_c_2Ereal__topology_2Ecollinear,axiom,mem(c_2Ereal__topology_2Ecollinear,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Ecompact,type,(c_2Ereal__topology_2Ecompact:$i)).
tff(mem_c_2Ereal__topology_2Ecompact,axiom,mem(c_2Ereal__topology_2Ecompact,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Ecomplete,type,(c_2Ereal__topology_2Ecomplete:$i)).
tff(mem_c_2Ereal__topology_2Ecomplete,axiom,mem(c_2Ereal__topology_2Ecomplete,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Ecomponents,type,(c_2Ereal__topology_2Ecomponents:$i)).
tff(mem_c_2Ereal__topology_2Ecomponents,axiom,mem(c_2Ereal__topology_2Ecomponents,arr(arr(ty_2Erealax_2Ereal,bool),arr(arr(ty_2Erealax_2Ereal,bool),bool)))).
tff(tp_c_2Ereal__topology_2Econdensation__point__of,type,(c_2Ereal__topology_2Econdensation__point__of:$i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Econnected,type,(c_2Ereal__topology_2Econnected:$i)).
tff(mem_c_2Ereal__topology_2Econnected,axiom,mem(c_2Ereal__topology_2Econnected,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Econnected__component,type,(c_2Ereal__topology_2Econnected__component:$i)).
tff(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))))).
tff(tp_c_2Ereal__topology_2Econtinuous,type,(c_2Ereal__topology_2Econtinuous:del > $i)).
tff(mem_c_2Ereal__topology_2Econtinuous,axiom,( ! [A_27a:del] : mem(c_2Ereal__topology_2Econtinuous(A_27a),arr(arr(A_27a,ty_2Erealax_2Ereal),arr(ty_2Ereal__topology_2Enet(A_27a),bool))))).
tff(tp_c_2Ereal__topology_2Econtinuous__on,type,(c_2Ereal__topology_2Econtinuous__on:$i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Edependent,type,(c_2Ereal__topology_2Edependent:$i)).
tff(mem_c_2Ereal__topology_2Edependent,axiom,mem(c_2Ereal__topology_2Edependent,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Ediameter,type,(c_2Ereal__topology_2Ediameter:$i)).
tff(mem_c_2Ereal__topology_2Ediameter,axiom,mem(c_2Ereal__topology_2Ediameter,arr(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal))).
tff(tp_c_2Ereal__topology_2Edim,type,(c_2Ereal__topology_2Edim:$i)).
tff(mem_c_2Ereal__topology_2Edim,axiom,mem(c_2Ereal__topology_2Edim,arr(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum))).
tff(tp_c_2Ereal__topology_2Eeuclidean,type,(c_2Ereal__topology_2Eeuclidean:$i)).
tff(mem_c_2Ereal__topology_2Eeuclidean,axiom,mem(c_2Ereal__topology_2Eeuclidean,ty_2Etopology_2Etopology(ty_2Erealax_2Ereal))).
tff(tp_c_2Ereal__topology_2Eeventually,type,(c_2Ereal__topology_2Eeventually:del > $i)).
tff(mem_c_2Ereal__topology_2Eeventually,axiom,( ! [A_27a:del] : mem(c_2Ereal__topology_2Eeventually(A_27a),arr(arr(A_27a,bool),arr(ty_2Ereal__topology_2Enet(A_27a),bool))))).
tff(tp_c_2Ereal__topology_2Efrom,type,(c_2Ereal__topology_2Efrom:$i)).
tff(mem_c_2Ereal__topology_2Efrom,axiom,mem(c_2Ereal__topology_2Efrom,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool)))).
tff(stp_fo_c_2Ereal__topology_2Efrom,type,(fo__c_2Ereal__topology_2Efrom:(tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2Efrom,axiom,( ! [X0:tp__ty_2Enum_2Enum] : ( ! [X1:tp__ty_2Enum_2Enum] : (inj__o(fo__c_2Ereal__topology_2Efrom(X0,X1)) = ap(ap(c_2Ereal__topology_2Efrom,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)))))).
tff(tp_c_2Ereal__topology_2Efrontier,type,(c_2Ereal__topology_2Efrontier:$i)).
tff(mem_c_2Ereal__topology_2Efrontier,axiom,mem(c_2Ereal__topology_2Efrontier,arr(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)))).
tff(tp_c_2Ereal__topology_2Efsigma,type,(c_2Ereal__topology_2Efsigma:$i)).
tff(mem_c_2Ereal__topology_2Efsigma,axiom,mem(c_2Ereal__topology_2Efsigma,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Egdelta,type,(c_2Ereal__topology_2Egdelta:$i)).
tff(mem_c_2Ereal__topology_2Egdelta,axiom,mem(c_2Ereal__topology_2Egdelta,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Ehausdist,type,(c_2Ereal__topology_2Ehausdist:$i)).
tff(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))).
tff(tp_c_2Ereal__topology_2Ehomeomorphic,type,(c_2Ereal__topology_2Ehomeomorphic:$i)).
tff(mem_c_2Ereal__topology_2Ehomeomorphic,axiom,mem(c_2Ereal__topology_2Ehomeomorphic,arr(arr(ty_2Erealax_2Ereal,bool),arr(arr(ty_2Erealax_2Ereal,bool),bool)))).
tff(tp_c_2Ereal__topology_2Ehomeomorphism,type,(c_2Ereal__topology_2Ehomeomorphism:$i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Ein__direction,type,(c_2Ereal__topology_2Ein__direction:$i)).
tff(mem_c_2Ereal__topology_2Ein__direction,axiom,mem(c_2Ereal__topology_2Ein__direction,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal))))).
tff(tp_c_2Ereal__topology_2Eindependent,type,(c_2Ereal__topology_2Eindependent:$i)).
tff(mem_c_2Ereal__topology_2Eindependent,axiom,mem(c_2Ereal__topology_2Eindependent,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Einfsum,type,(c_2Ereal__topology_2Einfsum:$i)).
tff(mem_c_2Ereal__topology_2Einfsum,axiom,mem(c_2Ereal__topology_2Einfsum,arr(arr(ty_2Enum_2Enum,bool),arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)))).
tff(tp_c_2Ereal__topology_2Einterior,type,(c_2Ereal__topology_2Einterior:$i)).
tff(mem_c_2Ereal__topology_2Einterior,axiom,mem(c_2Ereal__topology_2Einterior,arr(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)))).
tff(tp_c_2Ereal__topology_2Eis__interval,type,(c_2Ereal__topology_2Eis__interval:$i)).
tff(mem_c_2Ereal__topology_2Eis__interval,axiom,mem(c_2Ereal__topology_2Eis__interval,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Eisnet,type,(c_2Ereal__topology_2Eisnet:(del * del) > $i)).
tff(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))))).
tff(tp_c_2Ereal__topology_2Elim__def,type,(c_2Ereal__topology_2Elim__def:del > $i)).
tff(mem_c_2Ereal__topology_2Elim__def,axiom,( ! [A_27a:del] : mem(c_2Ereal__topology_2Elim__def(A_27a),arr(ty_2Ereal__topology_2Enet(A_27a),arr(arr(A_27a,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal))))).
tff(tp_c_2Ereal__topology_2Elimit__point__of,type,(c_2Ereal__topology_2Elimit__point__of:$i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Elinear,type,(c_2Ereal__topology_2Elinear:$i)).
tff(mem_c_2Ereal__topology_2Elinear,axiom,mem(c_2Ereal__topology_2Elinear,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool))).
tff(tp_c_2Ereal__topology_2Elocally,type,(c_2Ereal__topology_2Elocally:$i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Emidpoint,type,(c_2Ereal__topology_2Emidpoint:$i)).
tff(mem_c_2Ereal__topology_2Emidpoint,axiom,mem(c_2Ereal__topology_2Emidpoint,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal))).
tff(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)).
tff(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))))).
tff(tp_c_2Ereal__topology_2Emk__net,type,(c_2Ereal__topology_2Emk__net:del > $i)).
tff(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))))).
tff(tp_c_2Ereal__topology_2Enetlimit,type,(c_2Ereal__topology_2Enetlimit:del > $i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Enetord,type,(c_2Ereal__topology_2Enetord:del > $i)).
tff(mem_c_2Ereal__topology_2Enetord,axiom,( ! [A_27a:del] : mem(c_2Ereal__topology_2Enetord(A_27a),arr(ty_2Ereal__topology_2Enet(A_27a),arr(A_27a,arr(A_27a,bool)))))).
tff(tp_c_2Ereal__topology_2Eopen__segment,type,(c_2Ereal__topology_2Eopen__segment:$i)).
tff(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)))).
tff(stp_fo_c_2Ereal__topology_2Eopen__segment,type,(fo__c_2Ereal__topology_2Eopen__segment:(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2Eopen__segment,axiom,( ! [X0:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( ! [X1:tp__ty_2Erealax_2Ereal] : (inj__o(fo__c_2Ereal__topology_2Eopen__segment(X0,X1)) = ap(ap(c_2Ereal__topology_2Eopen__segment,inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)))))).
tff(tp_c_2Ereal__topology_2Epairwise,type,(c_2Ereal__topology_2Epairwise:del > $i)).
tff(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))))).
tff(tp_c_2Ereal__topology_2Epermutes,type,(c_2Ereal__topology_2Epermutes:del > $i)).
tff(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))))).
tff(stp_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,type,(tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum : $tType)).
tff(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)).
tff(stp_surj_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,type,(surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum : $i > tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum)).
tff(stp_inj_surj_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,( ! [X:tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum] : (surj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum(inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum(X)) = X))).
tff(stp_inj_mem_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,( ! [X:tp__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum] : (mem(inj__c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum(X),ty_2Ereal__topology_2Enet(ty_2Enum_2Enum))))).
tff(stp_iso_mem_c_ty_2Ereal__topology_2Enet_ty_2Enum_2Enum,axiom,( ! [X:$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)))))).
tff(tp_c_2Ereal__topology_2Esequentially,type,(c_2Ereal__topology_2Esequentially:$i)).
tff(mem_c_2Ereal__topology_2Esequentially,axiom,mem(c_2Ereal__topology_2Esequentially,ty_2Ereal__topology_2Enet(ty_2Enum_2Enum))).
tff(tp_c_2Ereal__topology_2Esetdist,type,(c_2Ereal__topology_2Esetdist:$i)).
tff(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))).
tff(tp_c_2Ereal__topology_2Espan,type,(c_2Ereal__topology_2Espan:$i)).
tff(mem_c_2Ereal__topology_2Espan,axiom,mem(c_2Ereal__topology_2Espan,arr(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)))).
tff(tp_c_2Ereal__topology_2Esphere,type,(c_2Ereal__topology_2Esphere:$i)).
tff(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)))).
tff(stp_fo_c_2Ereal__topology_2Esphere,type,(fo__c_2Ereal__topology_2Esphere:(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal) > tp__o)).
tff(stp_eq_fo_c_2Ereal__topology_2Esphere,axiom,( ! [X0:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( ! [X1:tp__ty_2Erealax_2Ereal] : (inj__o(fo__c_2Ereal__topology_2Esphere(X0,X1)) = ap(ap(c_2Ereal__topology_2Esphere,inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)))))).
tff(tp_c_2Ereal__topology_2Esubspace,type,(c_2Ereal__topology_2Esubspace:$i)).
tff(mem_c_2Ereal__topology_2Esubspace,axiom,mem(c_2Ereal__topology_2Esubspace,arr(arr(ty_2Erealax_2Ereal,bool),bool))).
tff(tp_c_2Ereal__topology_2Esubtopology,type,(c_2Ereal__topology_2Esubtopology:del > $i)).
tff(mem_c_2Ereal__topology_2Esubtopology,axiom,( ! [A_27a:del] : mem(c_2Ereal__topology_2Esubtopology(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(arr(A_27a,bool),ty_2Etopology_2Etopology(A_27a)))))).
tff(tp_c_2Ereal__topology_2Esummable,type,(c_2Ereal__topology_2Esummable:$i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Esums,type,(c_2Ereal__topology_2Esums:$i)).
tff(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))))).
tff(tp_c_2Ereal__topology_2Etrivial__limit,type,(c_2Ereal__topology_2Etrivial__limit:del > $i)).
tff(mem_c_2Ereal__topology_2Etrivial__limit,axiom,( ! [A_27a:del] : mem(c_2Ereal__topology_2Etrivial__limit(A_27a),arr(ty_2Ereal__topology_2Enet(A_27a),bool)))).
tff(tp_c_2Ereal__topology_2Euniformly__continuous__on,type,(c_2Ereal__topology_2Euniformly__continuous__on:$i)).
tff(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)))).
tff(tp_c_2Ereal__topology_2Ewithin,type,(c_2Ereal__topology_2Ewithin:del > $i)).
tff(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)))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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)))))).
tff(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:$i] : (mem(V3s_27,arr(A_27a,bool)) & (p(ap(c_2Epred__set_2EFINITE(A_27a),V3s_27)) & (p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3s_27),V1s)) & (V2t = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V3s_27))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EEXISTS__FINITE__SUBSET__IMAGE,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0P:$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)))))))))))))))))).
tff(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:$i] : (mem(V4t,arr(A_27b,bool)) => ((p(ap(c_2Epred__set_2EFINITE(A_27b),V4t)) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V4t),V2s))) => p(ap(V0P,ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27a),V1f),V4t))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EFORALL__IN__GSPEC,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [A_27d:del]: ( ! [A_27e:del]: ( ! [A_27f:del]: ( ! [A_27g:del]: ( ! [V0Q:$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:$i] : (mem(V3z,A_27b) => (p(ap(ap(c_2Ebool_2EIN(A_27b),V3z),ap(c_2Epred__set_2EGSPEC(A_27b,A_27a),f3397(A_27b,A_27a,V2f,V1P)))) => p(ap(V0Q,V3z))))) <=> ( ! [V5x:$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:$i] : (mem(V8z,A_27b) => (p(ap(ap(c_2Ebool_2EIN(A_27b),V8z),ap(c_2Epred__set_2EGSPEC(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),ap(c_2Epair_2EUNCURRY(A_27c,A_27d,ty_2Epair_2Eprod(A_27b,bool)),f3399(A_27b,A_27c,A_27d,V7f,V6P))))) => p(ap(V0Q,V8z))))) <=> ( ! [V11x:$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)),f3402(A_27e,A_27g,A_27b,A_27f,V13P,V14f))))) => p(ap(V0Q,V15z))))) <=> ( ! [V19w:$i] : (mem(V19w,A_27e) => ( ! [V20x:$i] : (mem(V20x,A_27f) => ( ! [V21y:$i] : (mem(V21y,A_27g) => (p(ap(ap(ap(V13P,V19w),V20x),V21y)) => p(ap(V0Q,ap(ap(ap(V14f,V19w),V20x),V21y)))))))))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EEXISTS__IN__GSPEC,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [A_27d:del]: ( ! [A_27e:del]: ( ! [A_27f:del]: ( ! [A_27g:del]: ( ! [V0Q:$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:$i] : (mem(V3z,A_27b) & (p(ap(ap(c_2Ebool_2EIN(A_27b),V3z),ap(c_2Epred__set_2EGSPEC(A_27b,A_27a),f3397(A_27b,A_27a,V2f,V1P)))) & p(ap(V0Q,V3z))))) <=> (? [V5x:$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:$i] : (mem(V8z,A_27b) & (p(ap(ap(c_2Ebool_2EIN(A_27b),V8z),ap(c_2Epred__set_2EGSPEC(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),ap(c_2Epair_2EUNCURRY(A_27c,A_27d,ty_2Epair_2Eprod(A_27b,bool)),f3399(A_27b,A_27c,A_27d,V7f,V6P))))) & p(ap(V0Q,V8z))))) <=> (? [V11x:$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)),f3402(A_27e,A_27g,A_27b,A_27f,V13P,V14f))))) & p(ap(V0Q,V15z))))) <=> (? [V19w:$i] : (mem(V19w,A_27e) & (? [V20x:$i] : (mem(V20x,A_27f) & (? [V21y:$i] : (mem(V21y,A_27g) & (p(ap(ap(ap(V13P,V19w),V20x),V21y)) & p(ap(V0Q,ap(ap(ap(V14f,V19w),V20x),V21y)))))))))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EEMPTY__BIGUNION,lemma,(( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,arr(arr(A_27a,bool),bool)) => ((ap(c_2Epred__set_2EBIGUNION(A_27a),V0s) = c_2Epred__set_2EEMPTY(A_27a)) <=> ( ! [V1t:$i] : (mem(V1t,arr(A_27a,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1t),V0s)) => (V1t = c_2Epred__set_2EEMPTY(A_27a))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__LT__RNEG,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1y)))) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))))).
tff(conj_thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET,lemma,(( ! [A_27a:del]: ( ! [V0f:$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))))))))))))))).
tff(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)))))))).
tff(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),f3595(A_27a,A_27b,V0f,V1s))))))))))).
tff(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),f3597(A_27a,A_27b,V0f,V1s))))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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))))))))).
tff(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)))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2EWLOG__LE,lemma,(( ! [V0P:$i] : (mem(V0P,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => ((( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : (p(ap(ap(V0P,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))) <=> p(ap(ap(V0P,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1m)))))) & ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n))) => p(ap(ap(V0P,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n))))))) => ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : p(ap(ap(V0P,inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V6n)))))))))).
tff(conj_thm_2Ereal__topology_2EFINITE__POWERSET,lemma,(( ! [A_27a:del]: ( ! [V0s:$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)),f1883(A_27a,V0s)))))))))).
tff(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)))))))).
tff(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)))))))).
tff(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))))).
tff(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))))).
tff(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))))).
tff(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE__EQ,lemma,(( ! [V0R:$i] : (mem(V0R,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => ((( ! [V1x:tp__ty_2Enum_2Enum] : p(ap(ap(V0R,inj__ty_2Enum_2Enum(V1x)),inj__ty_2Enum_2Enum(V1x)))) & ( ! [V2x:tp__ty_2Enum_2Enum] : ( ! [V3y:tp__ty_2Enum_2Enum] : ( ! [V4z:tp__ty_2Enum_2Enum] : ((p(ap(ap(V0R,inj__ty_2Enum_2Enum(V2x)),inj__ty_2Enum_2Enum(V3y))) & p(ap(ap(V0R,inj__ty_2Enum_2Enum(V3y)),inj__ty_2Enum_2Enum(V4z)))) => p(ap(ap(V0R,inj__ty_2Enum_2Enum(V2x)),inj__ty_2Enum_2Enum(V4z)))))))) => (( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V6n))) => p(ap(ap(V0R,inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V6n)))))) <=> ( ! [V7n:tp__ty_2Enum_2Enum] : p(ap(ap(V0R,inj__ty_2Enum_2Enum(V7n)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V7n))))))))))).
tff(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE,lemma,(( ! [V0R:$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))))))))))).
tff(conj_thm_2Ereal__topology_2ELAMBDA__PAIR,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0P:$i] : (mem(V0P,arr(A_27a,arr(A_27b,A_27c))) => (ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),f3599(A_27a,A_27c,A_27b,V0P)) = f3600(A_27c,A_27a,A_27b,V0P))))))))).
tff(conj_thm_2Ereal__topology_2ENOT__EQ,lemma,(( ! [A_27a:del]: ( ! [V0a:$i] : (mem(V0a,A_27a) => ( ! [V1b:$i] : (mem(V1b,A_27a) => ((~~ (V0a = V1b)) <=> (~~ (V0a = V1b)))))))))).
tff(conj_thm_2Ereal__topology_2EABS__LE__0,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V0x))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) <=> (V0x = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))).
tff(conj_thm_2Ereal__topology_2EREAL__OF__NUM__GE,lemma,(( ! [V0m:tp__ty_2Enum_2Enum] : ( ! [V1n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(V0m))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(V1n)))) <=> p(ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))))))).
tff(conj_thm_2Ereal__topology_2EPOWERSET__CLAUSES,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ((ap(c_2Epred__set_2EGSPEC(arr(A_27b,bool),arr(A_27b,bool)),f3601(A_27b)) = ap(ap(c_2Epred__set_2EINSERT(arr(A_27b,bool)),c_2Epred__set_2EEMPTY(A_27b)),c_2Epred__set_2EEMPTY(arr(A_27b,bool)))) & ( ! [V1a:$i] : (mem(V1a,A_27a) => ( ! [V2t:$i] : (mem(V2t,arr(A_27a,bool)) => (ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3602(A_27a,V1a,V2t)) = ap(ap(c_2Epred__set_2EUNION(arr(A_27a,bool)),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3603(A_27a,V2t))),ap(ap(c_2Epred__set_2EIMAGE(arr(A_27a,bool),arr(A_27a,bool)),f3604(A_27a,V1a)),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3605(A_27a,V2t)))))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__LT__LCANCEL__IMP,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V2z))))) => p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1y)),inj__ty_2Erealax_2Ereal(V2z))))))))).
tff(conj_thm_2Ereal__topology_2ESIMPLE__IMAGE__GEN,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,A_27b)) => ( ! [V1P:$i] : (mem(V1P,arr(A_27a,bool)) => (ap(c_2Epred__set_2EGSPEC(A_27b,A_27a),f3606(A_27b,A_27a,V0f,V1P)) = ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3607(A_27a,V1P)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSET__BIGUNION,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(arr(A_27a,bool),bool)) => ( ! [V1g:$i] : (mem(V1g,arr(arr(A_27a,bool),bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(arr(A_27a,bool)),V0f),V1g)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0f)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1g))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__LT__POW2,lemma,(( ! [V0n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V0n))))))).
tff(conj_thm_2Ereal__topology_2EFUN__IN__IMAGE,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$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)))))))))))))).
tff(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))))))))).
tff(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)),f3608(A_27a,V0u,V1s))))))))))).
tff(conj_thm_2Ereal__topology_2EBIGINTER__BIGUNION,lemma,(( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,arr(arr(A_27a,bool),bool)) => (ap(c_2Epred__set_2EBIGINTER(A_27a),V0s) = ap(ap(c_2Epred__set_2EDIFF(A_27a),c_2Epred__set_2EUNIV(A_27a)),ap(c_2Epred__set_2EBIGUNION(A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3609(A_27a,V0s)))))))))).
tff(conj_thm_2Ereal__topology_2EBIGUNION__BIGINTER,lemma,(( ! [A_27a:del]: ( ! [V0s:$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)),f3609(A_27a,V0s)))))))))).
tff(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))))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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)))))))).
tff(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))),f3396(V0x)))) = 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)))))))))).
tff(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))),f3610(V0x)))) = 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)))))))))))).
tff(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))),f3610(V0x))) = 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)))))))))))).
tff(stp_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,type,(tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o : $tType)).
tff(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)).
tff(stp_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,type,(surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o : $i > tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o)).
tff(stp_inj_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,( ! [X:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o] : (surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o(inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o(X)) = X))).
tff(stp_inj_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,( ! [X:tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o] : (mem(inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o(X),ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool))))).
tff(stp_iso_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_o,axiom,( ! [X:$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)))))).
tff(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)),f3612(V1t,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)),f3614(V0s,V1t)))))))))).
tff(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),f3615(A_27a,V1f))),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)))))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__WLOG__LE,lemma,(( ! [V0P:$i] : (mem(V0P,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ((( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y))) <=> p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V2y)),inj__ty_2Erealax_2Ereal(V1x)))))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V4y))) => p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V4y))))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6y:tp__ty_2Erealax_2Ereal] : p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V6y)))))))))).
tff(ax_thm_2Ereal__topology_2Epairwise,axiom,( ! [A_27a:del]: ( ! [V0r:$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:$i] : (mem(V3y,A_27a) => ((p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) & (p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V1s)) & (~~ (V2x = V3y)))) => p(ap(ap(V0r,V2x),V3y)))))))))))))).
tff(conj_thm_2Ereal__topology_2EPAIRWISE__EMPTY,lemma,(( ! [A_27a:del]: ( ! [V0r:$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))) <=> $true)))))).
tff(conj_thm_2Ereal__topology_2EPAIRWISE__SING,lemma,(( ! [A_27a:del]: ( ! [V0r:$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)))) <=> $true)))))))).
tff(conj_thm_2Ereal__topology_2EPAIRWISE__MONO,lemma,(( ! [A_27a:del]: ( ! [V0r:$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)))))))))))).
tff(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))))))))))))).
tff(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:$i] : (mem(V2f,arr(A_27b,A_27a)) => (p(ap(ap(c_2Ereal__topology_2Epairwise(A_27a),V1r),ap(ap(c_2Epred__set_2EIMAGE(A_27b,A_27a),V2f),V0s))) <=> p(ap(ap(c_2Ereal__topology_2Epairwise(A_27b),f3617(A_27a,A_27b,V1r,V2f)),V0s))))))))))))).
tff(ax_thm_2Ereal__topology_2Epermutes,axiom,( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V3y,A_27a) => p(ap(c_2Ebool_2E_3F_21(A_27a),f3618(A_27a,V0p,V3y))))))))))))).
tff(conj_thm_2Ereal__topology_2EPERMUTES__IMAGE,lemma,(( ! [A_27a:del]: ( ! [V0p:$i] : (mem(V0p,arr(A_27a,A_27a)) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(ap(c_2Ereal__topology_2Epermutes(A_27a),V0p),V1s)) => (ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),V0p),V1s) = V1s))))))))).
tff(conj_thm_2Ereal__topology_2EPERMUTES__INJECTIVE,lemma,(( ! [A_27a:del]: ( ! [V0p:$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:$i] : (mem(V3y,A_27a) => ((ap(V0p,V2x) = ap(V0p,V3y)) <=> (V2x = V3y)))))))))))))).
tff(conj_thm_2Ereal__topology_2EEXISTS__IN__INSERT,lemma,(( ! [A_27a:del]: ( ! [V0P:$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)))))))))))))))).
tff(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)))))))))))))))))))).
tff(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))))))))))))))))).
tff(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))))))))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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))))))).
tff(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))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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)))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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)))))).
tff(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))))))))).
tff(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))))))))))))).
tff(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)))))))).
tff(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))))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__SCALING,lemma,(( ! [V0c:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2Elinear,f3619(V0c)))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__CMUL,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(c_2Ereal__topology_2Elinear,f3620(V0f,V1c))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__NEG,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => (p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(c_2Ereal__topology_2Elinear,f3621(V0f)))))))).
tff(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,f3622(V0f,V1g)))))))))).
tff(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,f3623(V0f,V1g)))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__ID,lemma,(p(ap(c_2Ereal__topology_2Elinear,i(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__ZERO,lemma,(p(ap(c_2Ereal__topology_2Elinear,k(ty_2Erealax_2Ereal,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__NEGATION,lemma,(p(ap(c_2Ereal__topology_2Elinear,f3624)))).
tff(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:$i] : (mem(V2a,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)) => p(ap(c_2Ereal__topology_2Elinear,ap(V0f,V2a))))))) => p(ap(c_2Ereal__topology_2Elinear,f3626(A_27a,V0f,V1s))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__MUL__COMPONENT,lemma,(( ! [V0f:$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,f3627(V0f,V1v))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__0,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => (p(ap(c_2Ereal__topology_2Elinear,V0f)) => (surj__ty_2Erealax_2Ereal(ap(V0f,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__CMUL,lemma,(( ! [V0f:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__NEG,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Elinear,V0f)) => (surj__ty_2Erealax_2Ereal(ap(V0f,ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1x)))) = surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(V0f,inj__ty_2Erealax_2Ereal(V1x))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__ADD,lemma,(( ! [V0f:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__SUB,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Elinear,V0f)) => (surj__ty_2Erealax_2Ereal(ap(V0f,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Erealax_2Ereal(V1x))),ap(V0f,inj__ty_2Erealax_2Ereal(V2y)))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__SUM,lemma,(( ! [A_27a:del]: ( ! [V0f:$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)))))))))))))).
tff(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),f3628(A_27a,V2c,V3v)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3629(A_27a,V0f,V3v,V2c)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__BOUNDED,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => (p(ap(c_2Ereal__topology_2Elinear,V0f)) => (? [V1B:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Erealax_2Ereal(V2x)))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1B)),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x)))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__BOUNDED__POS,lemma,(( ! [V0f:$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))))))))))))).
tff(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)))))))))))).
tff(ax_thm_2Ereal__topology_2Ebilinear,axiom,( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => (p(ap(c_2Ereal__topology_2Ebilinear,V0f)) <=> (( ! [V1x:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2Elinear,f3630(V1x,V0f)))) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2Elinear,f3631(V0f,V3y))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__SWAP,lemma,(( ! [V0op:$i] : (mem(V0op,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => (p(ap(c_2Ereal__topology_2Ebilinear,f3633(V0op))) <=> p(ap(c_2Ereal__topology_2Ebilinear,V0op))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__LADD,lemma,(( ! [V0h:$i] : (mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3z:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => (surj__ty_2Erealax_2Ereal(ap(ap(V0h,ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y))),inj__ty_2Erealax_2Ereal(V3z))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(V0h,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V3z))),ap(ap(V0h,inj__ty_2Erealax_2Ereal(V2y)),inj__ty_2Erealax_2Ereal(V3z))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__RADD,lemma,(( ! [V0h:$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))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__LMUL,lemma,(( ! [V0h:$i] : (mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => (surj__ty_2Erealax_2Ereal(ap(ap(V0h,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V3y))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),ap(ap(V0h,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V3y))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__RMUL,lemma,(( ! [V0h:$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))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__LNEG,lemma,(( ! [V0h:$i] : (mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => (surj__ty_2Erealax_2Ereal(ap(ap(V0h,ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1x))),inj__ty_2Erealax_2Ereal(V2y))) = surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(ap(V0h,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y)))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__RNEG,lemma,(( ! [V0h:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__LZERO,lemma,(( ! [V0h:$i] : (mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => (surj__ty_2Erealax_2Ereal(ap(ap(V0h,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1x))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__RZERO,lemma,(( ! [V0h:$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)))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__LSUB,lemma,(( ! [V0h:$i] : (mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3z:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => (surj__ty_2Erealax_2Ereal(ap(ap(V0h,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y))),inj__ty_2Erealax_2Ereal(V3z))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(V0h,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V3z))),ap(ap(V0h,inj__ty_2Erealax_2Ereal(V2y)),inj__ty_2Erealax_2Ereal(V3z))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__RSUB,lemma,(( ! [V0h:$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))))))))))))).
tff(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:$i] : (mem(V4h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ((p(ap(c_2Ereal__topology_2Ebilinear,V4h)) & (p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) & p(ap(c_2Epred__set_2EFINITE(A_27b),V1t)))) => (surj__ty_2Erealax_2Ereal(ap(ap(V4h,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f)),ap(ap(c_2Eiterate_2ESum(A_27b),V1t),V3g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1t)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Erealax_2Ereal),f3635(A_27a,A_27b,V4h,V2f,V3g)))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__BOUNDED,lemma,(( ! [V0h:$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))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__BOUNDED__POS,lemma,(( ! [V0h:$i] : (mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => (p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => (? [V1B:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1B))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(V0h,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V3y)))),ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1B)),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x)))),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V3y)))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__SUC,lemma,(( ! [V0f:$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))),f3636(V2h,V0f,V1g))) = 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))),f3637(V2h,V0f,V1g)))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))))))))).
tff(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:$i] : (mem(V2h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : (p(ap(c_2Ereal__topology_2Ebilinear,V2h)) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n))),f3638(V2h,V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n))),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(V2h,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V4n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V1g,inj__ty_2Enum_2Enum(V4n)))),ap(ap(V2h,ap(V0f,inj__ty_2Enum_2Enum(V3m))),ap(V1g,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V3m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n))),f3639(V2h,V0f,V1g)))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))))))))).
tff(ax_thm_2Ereal__topology_2Esubspace,axiom,( ! [V0s:$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))))))))))).
tff(ax_thm_2Ereal__topology_2Espan,axiom,( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Espan,V0s) = ap(ap(c_2Etopology_2Ehull(ty_2Erealax_2Ereal),c_2Ereal__topology_2Esubspace),V0s))))).
tff(ax_thm_2Ereal__topology_2Edependent,axiom,( ! [V0s:$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))))))))))).
tff(ax_thm_2Ereal__topology_2Eindependent,axiom,( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Eindependent,V0s)) <=> (~~ p(ap(c_2Ereal__topology_2Edependent,V0s))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__UNIV,lemma,(p(ap(c_2Ereal__topology_2Esubspace,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__IMP__NONEMPTY,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Esubspace,V0s)) => (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__0,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Esubspace,V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0s))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__ADD,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$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))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__MUL,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Esubspace,V2s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),V2s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),inj__ty_2Erealax_2Ereal(V0x))),V2s))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__NEG,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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)))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__SUB,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Esubspace,V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),V2s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1y)),V2s)))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),V2s))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__SUM,lemma,(( ! [A_27a:del]: ( ! [V0s:$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:$i] : (mem(V3x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2t)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V1f,V3x)),V0s))))))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V1f)),V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__LINEAR__IMAGE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & p(ap(c_2Ereal__topology_2Esubspace,V1s))) => p(ap(c_2Ereal__topology_2Esubspace,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__LINEAR__PREIMAGE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & p(ap(c_2Ereal__topology_2Esubspace,V1s))) => p(ap(c_2Ereal__topology_2Esubspace,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3640(V1s,V0f))))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__TRIVIAL,lemma,(p(ap(c_2Ereal__topology_2Esubspace,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__INTER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Esubspace,V0s)) & p(ap(c_2Ereal__topology_2Esubspace,V1t))) => p(ap(c_2Ereal__topology_2Esubspace,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__BIGINTER,lemma,(( ! [V0f:$i] : (mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => (( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => p(ap(c_2Ereal__topology_2Esubspace,V1s))))) => p(ap(c_2Ereal__topology_2Esubspace,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f)))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__0__SUBSPACE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & p(ap(c_2Ereal__topology_2Esubspace,V1s))) => (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V1s)) & (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V2x))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V3y)))))) => (V2x = V3y)))) <=> ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V1s)) & (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) => (V4x = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__UNION__CHAIN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Esubspace,V0s)) & (p(ap(c_2Ereal__topology_2Esubspace,V1t)) & p(ap(c_2Ereal__topology_2Esubspace,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))))) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) | p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__SPAN,lemma,(( ! [V0s:$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)))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__SPAN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2Esubspace,ap(c_2Ereal__topology_2Espan,V0s))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__CLAUSES,lemma,(( ! [V0s:$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:$i] : (mem(V8s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),ap(c_2Ereal__topology_2Espan,V8s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V7c)),inj__ty_2Erealax_2Ereal(V6x))),ap(c_2Ereal__topology_2Espan,V8s))))))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__INDUCT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1h:$i] : (mem(V1h,arr(ty_2Erealax_2Ereal,bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1h)))) & p(ap(c_2Ereal__topology_2Esubspace,V1h))) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),ap(c_2Ereal__topology_2Espan,V0s))) => p(ap(V1h,inj__ty_2Erealax_2Ereal(V3x)))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__EMPTY,lemma,((ap(c_2Ereal__topology_2Espan,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2EINDEPENDENT__NONZERO,lemma,(( ! [V0s:$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)))))))).
tff(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))))))))).
tff(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)))))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__SUPERSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),ap(c_2Ereal__topology_2Espan,V0s))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__INC,lemma,(( ! [V0s:$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))))))).
tff(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)))))))))).
tff(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)))).
tff(conj_thm_2Ereal__topology_2ESPAN__0,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Ereal__topology_2Espan,V0s))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__ADD,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$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)))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__MUL,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal__topology_2Espan,V2s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),inj__ty_2Erealax_2Ereal(V0x))),ap(c_2Ereal__topology_2Espan,V2s)))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__MUL__EQ,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$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))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__NEG,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal__topology_2Espan,V1s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V0x))),ap(c_2Ereal__topology_2Espan,V1s))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__NEG__EQ,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__SUB,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal__topology_2Espan,V2s))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1y)),ap(c_2Ereal__topology_2Espan,V2s)))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(c_2Ereal__topology_2Espan,V2s)))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__SUM,lemma,(( ! [A_27a:del]: ( ! [V0s:$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:$i] : (mem(V3x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2t)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V1f,V3x)),ap(c_2Ereal__topology_2Espan,V0s))))))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V1f)),ap(c_2Ereal__topology_2Espan,V0s))))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__ADD__EQ,lemma,(( ! [V0s:$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))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__EQ__SELF,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((ap(c_2Ereal__topology_2Espan,V0s) = V0s) <=> p(ap(c_2Ereal__topology_2Esubspace,V0s))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__SUBSET__SUBSPACE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2Esubspace,V1t))) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Espan,V0s)),V1t))))))))).
tff(conj_thm_2Ereal__topology_2ESURJECTIVE__IMAGE__EQ,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0f:$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:$i] : (mem(V5x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V5x)),V2t)) <=> p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V1s)))))) => (ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V1s) = V2t)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF,lemma,(( ! [V0s:$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),f3641(V1a)),V0s) = V0s))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Esubspace,V0s)) => ((ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3641(V1a)),V0s) = V0s) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__SUMS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Esubspace,V0s)) & p(ap(c_2Ereal__topology_2Esubspace,V1t))) => p(ap(c_2Ereal__topology_2Esubspace,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__UNION,lemma,(( ! [V0s:$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)),f3643(V1t,V0s)))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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)))))))).
tff(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))))))))).
tff(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))))))))))).
tff(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))))))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2ECOLLINEAR__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Ecollinear,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(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))))))).
tff(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))))))))).
tff(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))))))).
tff(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)))))))))))).
tff(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)))))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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))))))))))))).
tff(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)))))))))))))))).
tff(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))))))))))))))).
tff(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))))))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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)))))).
tff(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)))))))))).
tff(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))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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)))))))))))).
tff(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)))))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ECOLLINEAR__1,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2Ecollinear,V0s)))))).
tff(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)))))))).
tff(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)))).
tff(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))))))))).
tff(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)))))))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))))))))).
tff(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)))))))))).
tff(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))))))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__BREAKDOWN,lemma,(( ! [V0b:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0b)),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),ap(c_2Ereal__topology_2Espan,V1s)))) => (? [V3k:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V2a)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V3k)),inj__ty_2Erealax_2Ereal(V0b)))),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V1s),inj__ty_2Erealax_2Ereal(V0b))))))))))))).
tff(conj_thm_2Ereal__topology_2EIN__SPAN__INSERT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$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))))))))))).
tff(conj_thm_2Ereal__topology_2EINDEPENDENT__INSERT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Eindependent,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),V1s))) <=> p(ap(ap(ap(c_2Ebool_2ECOND(bool),ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),V1s)),ap(c_2Ereal__topology_2Eindependent,V1s)),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ereal__topology_2Eindependent,V1s)),ap(c_2Ebool_2E_7E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),ap(c_2Ereal__topology_2Espan,V1s)))))))))))).
tff(conj_thm_2Ereal__topology_2EINDEPENDENT__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Eindependent,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2EINDEPENDENT__SING,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Eindependent,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) <=> (~~ (V0x = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))))).
tff(conj_thm_2Ereal__topology_2EINDEPENDENT__STDBASIS,lemma,(p(ap(c_2Ereal__topology_2Eindependent,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3644))))).
tff(conj_thm_2Ereal__topology_2ESPANNING__SUBSET__INDEPENDENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & (p(ap(c_2Ereal__topology_2Eindependent,V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Espan,V1t))))) => (V0s = V1t)))))))).
tff(conj_thm_2Ereal__topology_2EIN__SPAN__DELETE,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$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))))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__TRANS,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal__topology_2Espan,V2s))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1y)),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),V2s))))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1y)),ap(c_2Ereal__topology_2Espan,V2s)))))))))).
tff(conj_thm_2Ereal__topology_2EEXCHANGE__LEMMA,lemma,(( ! [V0s:$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))))))))))))))).
tff(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))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2EINDEPENDENT__BOUND,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Eindependent,V0s)) => (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V0s)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))))))).
tff(conj_thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET__EXTEND,lemma,(( ! [V0s:$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))))))))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__BREAKDOWN__EQ,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),V2s)))) <=> (? [V3k:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V0x)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V3k)),inj__ty_2Erealax_2Ereal(V1a)))),ap(c_2Ereal__topology_2Espan,V2s))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND__LEMMA,lemma,(( ! [V0f:$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)))))))))))))))))).
tff(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:$i] : (mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & (p(ap(c_2Ereal__topology_2Elinear,V2g)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V1b)) => (surj__ty_2Erealax_2Ereal(ap(V2g,inj__ty_2Erealax_2Ereal(V3x))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V3x)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__KERNEL,lemma,(( ! [V0f:$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),f3645(V0f))))))))).
tff(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))))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2s),ap(c_2Ereal__topology_2Espan,V1b))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V1b)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V3x))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V2s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__EQ,lemma,(( ! [V0f:$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:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & (p(ap(c_2Ereal__topology_2Elinear,V1g)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3s),ap(c_2Ereal__topology_2Espan,V2b))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V2b)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V4x))))))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V3s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V5x))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__EQ__STDBASIS,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : (mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & (p(ap(c_2Ereal__topology_2Elinear,V1g)) & ( ! [V2i:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V2i))) & p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2i)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V2i))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V2i)))))))) => (V0f = V1g)))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__LEFT__INVERSE,lemma,(( ! [V0f:$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:$i] : (mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & (p(ap(c_2Ereal__topology_2Elinear,V3g)) & (ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3g),V0f) = i(ty_2Erealax_2Ereal)))))))))).
tff(ax_thm_2Ereal__topology_2Edim,axiom,( ! [V0v:$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),f3647(V0v))))))).
tff(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)))))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__EXPLICIT,lemma,(( ! [V0p:$i] : (mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Espan,V0p) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3651(V0p))))))).
tff(conj_thm_2Ereal__topology_2EDEPENDENT__EXPLICIT,lemma,(( ! [V0p:$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),f3652(V2u))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__IMP__SURJECTIVE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V1x))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V2y)))) => (V1x = V2y))))) => ( ! [V3y:tp__ty_2Erealax_2Ereal] : (? [V4x:tp__ty_2Erealax_2Ereal] : (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = V3y)))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW__POS,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : (mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & (p(ap(c_2Ereal__topology_2Elinear,V1g)) & (ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f) = c_2Ecombin_2EI(ty_2Erealax_2Ereal)))) => (? [V2B:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2B))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2B)),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V3x)))),ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Erealax_2Ereal(V3x))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : (mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & (p(ap(c_2Ereal__topology_2Elinear,V1g)) & (ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f) = c_2Ecombin_2EI(ty_2Erealax_2Ereal)))) => (? [V2B:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2B)),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V3x)))),ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Erealax_2Ereal(V3x)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__BOUNDED__BELOW__POS,lemma,(( ! [V0f:$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))))))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2EDIM__LE__CARD,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Ereal__topology_2Edim,V0s)),ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V0s)))))))).
tff(conj_thm_2Ereal__topology_2ESPAN__STDBASIS,lemma,((ap(c_2Ereal__topology_2Espan,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3644)) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))).
tff(conj_thm_2Ereal__topology_2EHAS__SIZE__STDBASIS,lemma,(p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3644)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))).
tff(conj_thm_2Ereal__topology_2EDIM__UNIV,lemma,((surj__ty_2Enum_2Enum(ap(c_2Ereal__topology_2Edim,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))).
tff(conj_thm_2Ereal__topology_2EDIM__SUBSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Ereal__topology_2Edim,V0s)),ap(c_2Ereal__topology_2Edim,V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EDIM__SUBSET__UNIV,lemma,(( ! [V0s:$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))))))))).
tff(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)),f3653(A_27a,V1u,V0top)))))))))).
tff(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)))))))))))))))).
tff(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)),f3654(A_27a,V1t,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)),f3655(A_27b,V4t,V3s))))))))))))).
tff(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)),f3653(A_27a,V1u,V0top))))))))))).
tff(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))))))))))))))).
tff(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))))))))).
tff(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))))))))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__TOPSPACE,lemma,(( ! [A_27a:del]: ( ! [V0top:$i] : (mem(V0top,ty_2Etopology_2Etopology(A_27a)) => (ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),ap(c_2Etopology_2Etopspace(A_27a),V0top)) = V0top)))))).
tff(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__UNIV,lemma,(( ! [A_27a:del]: ( ! [V0top:$i] : (mem(V0top,ty_2Etopology_2Etopology(A_27a)) => (ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),c_2Epred__set_2EUNIV(A_27a)) = V0top)))))).
tff(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:$i] : (mem(V2t,arr(A_27a,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V1s)),V2t)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__IMP__SUBSET,lemma,(( ! [A_27a:del]: ( ! [V0top:$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)))))))))))).
tff(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)))))))))))))).
tff(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)))))))))))))).
tff(ax_thm_2Ereal__topology_2Eopen__def,axiom,( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => (? [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) & ( ! [V3x_27:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x_27)),inj__ty_2Erealax_2Ereal(V1x)))),inj__ty_2Erealax_2Ereal(V2e))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x_27)),V0s)))))))))))).
tff(ax_thm_2Ereal__topology_2Eclosed__def,axiom,( ! [V0s:$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))))))).
tff(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)))).
tff(conj_thm_2Ereal__topology_2EOPEN__EMPTY,lemma,(p(ap(c_2Ereal__topology_2EOpen,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2EOPEN__UNIV,lemma,(p(ap(c_2Ereal__topology_2EOpen,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
tff(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)))))))))).
tff(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)))))))).
tff(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:$i] : (mem(V2a,A_27a) => (p(ap(V0P,V2a)) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3656(A_27a,V1Q,V2a))))))) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3658(A_27a,V1Q,V0P)))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__EXISTS,lemma,(( ! [A_27a:del]: ( ! [V0Q:$i] : (mem(V0Q,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => (( ! [V1a:$i] : (mem(V1a,A_27a) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3659(A_27a,V0Q,V1a)))))) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3661(A_27a,V0Q)))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN,lemma,(( ! [V0s:$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))))))).
tff(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)))).
tff(conj_thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN__SUBTOPOLOGY,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Etopology_2Etopspace(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)) = V0s))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__REFL,lemma,(( ! [V0s:$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)))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__REFL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V0s)))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN,lemma,(( ! [V0s:$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))))))).
tff(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)))))))))).
tff(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))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__EMPTY,lemma,(p(ap(c_2Ereal__topology_2EClosed,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__UNIV,lemma,(p(ap(c_2Ereal__topology_2EClosed,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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),f3662(A_27b,A_27a,V1f,V0P))) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3664(A_27a,A_27b,V1f,V0P))))))) & (( ! [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)),f3666(A_27e,A_27c,A_27d,V6f,V5P)))) = ap(c_2Epred__set_2EGSPEC(A_27e,A_27e),f3669(A_27c,A_27d,A_27e,V5P,V6f))))))) & ( ! [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)),f3672(A_27f,A_27h,A_27i,A_27g,V12P,V13f)))) = ap(c_2Epred__set_2EGSPEC(A_27i,A_27i),f3676(A_27f,A_27h,A_27g,A_27i,V13f,V12P)))))))))))))))))))).
tff(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:$i] : (mem(V2a,A_27a) => (p(ap(V0P,V2a)) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3656(A_27a,V1Q,V2a))))))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3678(A_27a,V1Q,V0P)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__FORALL,lemma,(( ! [A_27a:del]: ( ! [V0Q:$i] : (mem(V0Q,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => (( ! [V1a:$i] : (mem(V1a,A_27a) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3659(A_27a,V0Q,V1a)))))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3679(A_27a,V0Q)))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__CLOSED,lemma,(( ! [V0s:$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)))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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)))))))).
tff(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),f3680(V1e,V0x)))))).
tff(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),f3681(V1e,V0x)))))).
tff(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),f3682(V1e,V0x)))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))).
tff(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)))))))).
tff(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)))))))).
tff(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)))))).
tff(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))))).
tff(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)))))).
tff(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)))))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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),f3683(V0a)),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)))))))))).
tff(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),f3683(V0a)),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)))))))))).
tff(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),f3683(V0a)),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)))))))))).
tff(conj_thm_2Ereal__topology_2EBALL__LINEAR__IMAGE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2r:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & (( ! [V3y:tp__ty_2Erealax_2Ereal] : (? [V4x:tp__ty_2Erealax_2Ereal] : (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = V3y))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Erealax_2Ereal(V5x)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V5x))))))) => (ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V1x))),inj__ty_2Erealax_2Ereal(V2r))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2r)))))))))))).
tff(conj_thm_2Ereal__topology_2ECBALL__LINEAR__IMAGE,lemma,(( ! [V0f:$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)))))))))))).
tff(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)))))))))))).
tff(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),f3684(V0c)),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))))))))))).
tff(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),f3684(V0c)),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))))))))))).
tff(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)))))))).
tff(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)))))))).
tff(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)))))))).
tff(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)))))))).
tff(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))))))))).
tff(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)))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__BALL__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) <=> (? [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2e)))),V0s))))))))))).
tff(conj_thm_2Ereal__topology_2EBALL__EQ__EMPTY,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1e))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) <=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1e)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))))).
tff(conj_thm_2Ereal__topology_2EBALL__EMPTY,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1e)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) => (ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1e))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL,lemma,(( ! [V0s:$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))))))))))).
tff(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))))))))))).
tff(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))))))))).
tff(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))))))).
tff(conj_thm_2Ereal__topology_2ENEGATIONS__BALL,lemma,(( ! [V0r:tp__ty_2Erealax_2Ereal] : (ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),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))))))).
tff(conj_thm_2Ereal__topology_2ENEGATIONS__CBALL,lemma,(( ! [V0r:tp__ty_2Erealax_2Ereal] : (ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),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))))))).
tff(conj_thm_2Ereal__topology_2ENEGATIONS__SPHERE,lemma,(( ! [V0r:tp__ty_2Erealax_2Ereal] : (ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),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))))))).
tff(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)))))))))))).
tff(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:$i] : (mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t))) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__INTER,lemma,(( ! [V0u:$i] : (mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V1s)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0u),V1s)))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__OPEN__IN__TRANS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EOpen,V0s)) & (p(ap(c_2Ereal__topology_2EOpen,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)))) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__SUBSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2EOpen,V0s))) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED,lemma,(( ! [V0s:$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)))))))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__INTER,lemma,(( ! [V0u:$i] : (mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V1s)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0u),V1s)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__SUBSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2EClosed,V0s))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__SUBSET__TRANS,lemma,(( ! [V0s:$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))))))))))).
tff(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:$i] : (mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s))))))))))).
tff(conj_thm_2Ereal__topology_2Eopen__in,lemma,(( ! [V0u:$i] : (mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) <=> (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V0u)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x_27)),V0u)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x_27)),inj__ty_2Erealax_2Ereal(V2x)))),inj__ty_2Erealax_2Ereal(V3e)))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x_27)),V1s)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__CONTAINS__BALL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) <=> (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V3e)))),V1t)),V0s)))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__CONTAINS__CBALL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) <=> (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V3e)))),V1t)),V0s)))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__TRANS,lemma,(( ! [V0s:$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))))))))))).
tff(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:$i] : (mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V2u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t))))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__TRANS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t))) => p(ap(c_2Ereal__topology_2EOpen,V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__TRANS,lemma,(( ! [V0s:$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))))))))))).
tff(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:$i] : (mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V2u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t))))) <=> p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__TRANS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t))) => p(ap(c_2Ereal__topology_2EClosed,V0s))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__INTER__SUBSET,lemma,(( ! [V0s:$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)))))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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),f3686(V0l))))).
tff(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)))))))).
tff(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),f3687(V0a,V1b)))))))).
tff(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),f3688(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(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)))))))))).
tff(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)))))).
tff(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)))))))))))))))).
tff(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))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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)))))))))))).
tff(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))))))))))).
tff(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)))))))))))))).
tff(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),f3689(V0c)),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),f3690(V4c)),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))))))))))).
tff(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))))))))))))).
tff(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)))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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)))))))))))).
tff(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))))))))))).
tff(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))))))))))))).
tff(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)))))))))))).
tff(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)))))))))))).
tff(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))))))))))))))))).
tff(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:$i] : (mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(c_2Ereal__topology_2EClosed,V1e1)) & (p(ap(c_2Ereal__topology_2EClosed,V2e2)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2))) & ((ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2)),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) & ((~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) & (~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2e2),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__IN,lemma,(( ! [V0s:$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)))))))))))))))))).
tff(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:$i] : (mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1e1)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V2e2)) & ((ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2) = V0s) & ((ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) & ((~~ (V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) & (~~ (V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__IN,lemma,(( ! [V0s:$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)))))))))))))))))).
tff(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:$i] : (mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1e1)) & (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V2e2)) & ((ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2) = V0s) & ((ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) & ((~~ (V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) & (~~ (V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EEXISTS__DIFF,lemma,(( ! [A_27a:del]: ( ! [V0P:$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)))))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(c_2Ereal__topology_2EClosed,V1e1)) & (p(ap(c_2Ereal__topology_2EClosed,V2e2)) & ((~~ (V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) & ((~~ (V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) & ((ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2) = V0s) & (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__SET,lemma,(( ! [V0s:$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)))))))))))))))))).
tff(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))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Econnected,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(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))))))).
tff(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)))))))))))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__UNIV,lemma,(p(ap(c_2Ereal__topology_2Econnected,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
tff(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)))))))).
tff(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)))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)) & (p(ap(c_2Ereal__topology_2EOpen,V0s)) & (p(ap(c_2Ereal__topology_2EClosed,V1t)) & (p(ap(c_2Ereal__topology_2Econnected,V2u)) & p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V1t),V0s)))))))) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V2u),V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__DISJOINT__BIGUNION__OPEN__UNIQUE,lemma,(( ! [V0f:$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:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3s),V1f_27)) => (p(ap(c_2Ereal__topology_2EOpen,V3s)) & (p(ap(c_2Ereal__topology_2Econnected,V3s)) & (~~ (V3s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))) & (ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1f_27)))))) => (V0f = V1f_27)))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__FROM__CLOSED__UNION__AND__INTER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V0s)) & (p(ap(c_2Ereal__topology_2EClosed,V1t)) & (p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) & p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))))) => (p(ap(c_2Ereal__topology_2Econnected,V0s)) & p(ap(c_2Ereal__topology_2Econnected,V1t)))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__FROM__OPEN__UNION__AND__INTER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EOpen,V0s)) & (p(ap(c_2Ereal__topology_2EOpen,V1t)) & (p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) & p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))))) => (p(ap(c_2Ereal__topology_2Econnected,V0s)) & p(ap(c_2Ereal__topology_2Econnected,V1t)))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__INDUCTION,lemma,(( ! [V0P:$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))))))))))))))).
tff(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:$i] : (mem(V12t,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V12t)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V11a)),V12t)) & ( ! [V13x:tp__ty_2Erealax_2Ereal] : ( ! [V14y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V13x)),V12t)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V14y)),V12t)) & (p(ap(V0P,inj__ty_2Erealax_2Ereal(V13x))) & p(ap(V0P,inj__ty_2Erealax_2Ereal(V14y)))))) => p(ap(ap(V1R,inj__ty_2Erealax_2Ereal(V13x)),inj__ty_2Erealax_2Ereal(V14y)))))))))))))))) => ( ! [V15a:tp__ty_2Erealax_2Ereal] : ( ! [V16b:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V15a)),V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V16b)),V2s)) & (p(ap(V0P,inj__ty_2Erealax_2Ereal(V15a))) & p(ap(V0P,inj__ty_2Erealax_2Ereal(V16b)))))) => p(ap(ap(V1R,inj__ty_2Erealax_2Ereal(V15a)),inj__ty_2Erealax_2Ereal(V16b))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__INDUCTION__SIMPLE,lemma,(( ! [V0P:$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))))))))))))).
tff(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:$i] : (mem(V8t,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V8t)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V7a)),V8t)) & ( ! [V9x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V9x)),V8t)) => p(ap(ap(V0R,inj__ty_2Erealax_2Ereal(V7a)),inj__ty_2Erealax_2Ereal(V9x)))))))))))))) => ( ! [V10a:tp__ty_2Erealax_2Ereal] : ( ! [V11b:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V10a)),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V11b)),V1s))) => p(ap(ap(V0R,inj__ty_2Erealax_2Ereal(V10a)),inj__ty_2Erealax_2Ereal(V11b))))))))))))).
tff(ax_thm_2Ereal__topology_2Elimit__point__of,axiom,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),V1s)) <=> ( ! [V2t:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),V2t)) & p(ap(c_2Ereal__topology_2EOpen,V2t))) => (? [V3y:tp__ty_2Erealax_2Ereal] : ((~~ (V3y = V0x)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V2t))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__SUBSET,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),V1s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V2t))) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),V2t)))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__APPROACHABLE,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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)))))))))))))).
tff(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))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__APPROACHABLE__LE,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),V1s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (? [V3x_27:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x_27)),V1s)) & ((~~ (V3x_27 = V0x)) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x_27)),inj__ty_2Erealax_2Ereal(V0x)))),inj__ty_2Erealax_2Ereal(V2e)))))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__CHOOSE__SIZE,lemma,(( ! [V0c:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0c))) => (? [V1x:tp__ty_2Erealax_2Ereal] : (surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V1x))) = V0c)))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__UNIV,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__LIMPT,lemma,(( ! [V0s:$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))))))))).
tff(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))))))).
tff(conj_thm_2Ereal__topology_2ENO__LIMIT__POINT__IMP__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((~~ (? [V1x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V1x)),V0s)))) => p(ap(c_2Ereal__topology_2EClosed,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__POSITIVE__ORTHANT,lemma,(p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3691))))).
tff(conj_thm_2Ereal__topology_2EFINITE__SET__AVOID,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIMIT__POINT__FINITE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => (~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V1a)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__SING,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1y)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))).
tff(conj_thm_2Ereal__topology_2ELIMIT__POINT__UNION,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2x)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) <=> (p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2x)),V0s)) | p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2x)),V1t))))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__INSERT,lemma,(( ! [V0s:$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))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__OF__LIMPTS,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3692(V1s)))) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),V1s)))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__LIMPTS,lemma,(( ! [V0s:$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),f3693(V0s)))))))).
tff(conj_thm_2Ereal__topology_2EDISCRETE__IMP__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1e))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V3y)),inj__ty_2Erealax_2Ereal(V2x)))),inj__ty_2Erealax_2Ereal(V1e))))) => (V3y = V2x))))) => p(ap(c_2Ereal__topology_2EClosed,V0s)))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__OF__UNIV,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__OF__OPEN__IN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) & (p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2x)),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1t)))) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2x)),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__OF__OPEN,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__SING,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) <=> (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),V0s)) & (~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V1a)),V0s)))))))))).
tff(ax_thm_2Ereal__topology_2Einterior,axiom,( ! [V0s:$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),f3695(V0s)))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((ap(c_2Ereal__topology_2Einterior,V0s) = V0s) <=> p(ap(c_2Ereal__topology_2EOpen,V0s))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__OPEN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) => (ap(c_2Ereal__topology_2Einterior,V0s) = V0s)))))).
tff(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)))).
tff(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)))).
tff(conj_thm_2Ereal__topology_2EOPEN__INTERIOR,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Ereal__topology_2Einterior,V0s))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__INTERIOR,lemma,(( ! [V0s:$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)))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__SUBSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,V0s)),V0s)))))).
tff(conj_thm_2Ereal__topology_2ESUBSET__INTERIOR__EQ,lemma,(( ! [V0s:$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))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V2t_27,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t_27),V0s)) & p(ap(c_2Ereal__topology_2EOpen,V2t_27))) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t_27),V1t))))))) => (ap(c_2Ereal__topology_2Einterior,V0s) = V1t)))))))).
tff(conj_thm_2Ereal__topology_2EIN__INTERIOR,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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)))))))))).
tff(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)))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__FINITE__BIGINTER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => (p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0s)) => (ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0s)) = ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),c_2Ereal__topology_2Einterior),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__BIGINTER__SUBSET,lemma,(( ! [V0f:$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)))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__LIMIT__POINT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),ap(c_2Ereal__topology_2Einterior,V0s))) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V1x)),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__SING,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : (ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__UNION__EMPTY__INTERIOR,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V0s)) & (ap(c_2Ereal__topology_2Einterior,V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) => (ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)) = ap(c_2Ereal__topology_2Einterior,V0s))))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__UNION__EQ__EMPTY,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V0s)) | p(ap(c_2Ereal__topology_2EClosed,V1t))) => ((ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) <=> ((ap(c_2Ereal__topology_2Einterior,V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) & (ap(c_2Ereal__topology_2Einterior,V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__UNIONS__OPEN__SUBSETS,lemma,(( ! [V0s:$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)),f3696(V0s))) = ap(c_2Ereal__topology_2Einterior,V0s)))))).
tff(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)))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2EFORALL__POS__MONO,lemma,(( ! [V0P:$i] : (mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ((( ! [V1d:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1d)),inj__ty_2Erealax_2Ereal(V2e))) & p(ap(V0P,inj__ty_2Erealax_2Ereal(V1d)))) => p(ap(V0P,inj__ty_2Erealax_2Ereal(V2e)))))) & ( ! [V3n:tp__ty_2Enum_2Enum] : ((~~ (V3n = fo__c_2Enum_2E0)) => p(ap(V0P,ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(V3n)))))))) => ( ! [V4e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4e))) => p(ap(V0P,inj__ty_2Erealax_2Ereal(V4e)))))))))).
tff(conj_thm_2Ereal__topology_2EFORALL__SUC,lemma,(( ! [V0P:$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)))))))))).
tff(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)))))).
tff(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))))))))).
tff(ax_thm_2Ereal__topology_2Eclosure,axiom,( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Eclosure,V0s) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3693(V0s))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__APPROACHABLE,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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))))))))))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Eclosure,V0s) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__CLOSURE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2Eclosure,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__HULL,lemma,(( ! [V0s:$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)))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((ap(c_2Ereal__topology_2Eclosure,V0s) = V0s) <=> p(ap(c_2Ereal__topology_2EClosed,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V0s)) => (ap(c_2Ereal__topology_2Eclosure,V0s) = V0s)))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__CLOSURE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Eclosure,V0s)) = ap(c_2Ereal__topology_2Eclosure,V0s)))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__SUBSET,lemma,(( ! [V0s:$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))))))).
tff(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)))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__BIGINTER__SUBSET,lemma,(( ! [V0f:$i] : (mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f))),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),c_2Ereal__topology_2Eclosure),V0f)))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__MINIMAL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2EClosed,V1t))) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),V1t))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__MINIMAL__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V1t)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),V1t)) <=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__UNIQUE,lemma,(( ! [V0s:$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)))))))).
tff(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)))).
tff(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)))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__BIGUNION,lemma,(( ! [V0f:$i] : (mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => (p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0f)) => (ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f)) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),f3697(V0f))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__EQ__EMPTY,lemma,(( ! [V0s:$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))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__SUBSET__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),V0s)) <=> p(ap(c_2Ereal__topology_2EClosed,V0s))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__EQ__EMPTY,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) => ((ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) <=> (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__IN__INTER__CLOSURE,lemma,(( ! [V0s:$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)))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__COMPLEMENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Einterior,V0s))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__COMPLEMENT,lemma,(( ! [V0s:$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))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__CLOSURE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Econnected,V0s)) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Ereal__topology_2Eclosure,V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__UNION__STRONG,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Econnected,V0s)) & (p(ap(c_2Ereal__topology_2Econnected,V1t)) & (~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__DIFF,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__OF__CLOSURE,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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)))))))).
tff(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)))))))))))).
tff(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)))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE__IDEMP,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s)))) = ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR__IDEMP,lemma,(( ! [V0s:$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))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(c_2Ereal__topology_2EOpen,V2u)) & ((~~ (V2u = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2u),V1t)) & (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2u),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE__INTER__OPEN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EOpen,V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t))) => (ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s))),ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V1t))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR__UNION__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t))) => (ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V0s))),ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V1t))))))))))).
tff(conj_thm_2Ereal__topology_2EREGULAR__OPEN__INTER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (((ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s)) = V0s) & (ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V1t)) = V1t)) => (ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))))))))).
tff(conj_thm_2Ereal__topology_2EREGULAR__CLOSED__UNION,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (((ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V0s)) = V0s) & (ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V1t)) = V1t)) => (ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))))))))).
tff(conj_thm_2Ereal__topology_2EREGULAR__CLOSED__BIGUNION,lemma,(( ! [V0f:$i] : (mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ((p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0f)) & ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1t),V0f)) => (ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V1t)) = V1t))))) => (ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f))) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f))))))).
tff(conj_thm_2Ereal__topology_2EDIFF__CLOSURE__SUBSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t))),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EDENSE__OPEN__INTER,lemma,(( ! [V0s:$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)))))))))))))).
tff(ax_thm_2Ereal__topology_2Efrontier,axiom,( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Efrontier,V0s) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Einterior,V0s)))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2Efrontier,V0s))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__CLOSURES,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Efrontier,V0s) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__STRADDLE,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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)))))))))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V0s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),V0s))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__EMPTY,lemma,((ap(c_2Ereal__topology_2Efrontier,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__UNIV,lemma,((ap(c_2Ereal__topology_2Efrontier,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__EQ,lemma,(( ! [V0s:$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))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__COMPLEMENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s)) = ap(c_2Ereal__topology_2Efrontier,V0s)))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__DISJOINT__EQ,lemma,(( ! [V0s:$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))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__INTERIORS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Efrontier,V0s) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Einterior,V0s))),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER__SUBSET,lemma,(( ! [V0s:$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))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__FRONTIER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Efrontier,V0s)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s))),ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V0s)))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__FRONTIER__EMPTY,lemma,(( ! [V0s:$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))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EOpen,V0s)) | p(ap(c_2Ereal__topology_2EClosed,V0s))) => (ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Efrontier,V0s)) = ap(c_2Ereal__topology_2Efrontier,V0s))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER__FRONTIER,lemma,(( ! [V0s:$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))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__UNION__FRONTIER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Eclosure,V0s) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Efrontier,V0s))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__INTERIOR__SUBSET,lemma,(( ! [V0s:$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))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__CLOSURE__SUBSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Eclosure,V0s))),ap(c_2Ereal__topology_2Efrontier,V0s))))))).
tff(conj_thm_2Ereal__topology_2ESET__DIFF__FRONTIER,lemma,(( ! [V0s:$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)))))).
tff(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))))))))))).
tff(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:$i] : (mem(V4z,A_27a) => (p(ap(ap(V0g,V4z),V2y)) => p(ap(ap(V0g,V4z),V1x)))))))))))))))).
tff(ax_thm_2Ereal__topology_2Enet__TY__DEF,axiom,( ! [A_27a:del]: (? [V0rep:$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)))))).
tff(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))))))))).
tff(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:$i] : (mem(V4z,A_27a) => (p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V4z),V2y)) => p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V4z),V1x))))))))))))))).
tff(conj_thm_2Ereal__topology_2EOLDNET,lemma,(( ! [A_27a:del]: ( ! [V0n:$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))))))))))))))))))).
tff(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))))))))))))))))))).
tff(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),f3699(V0a)))))).
tff(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),f3701)))).
tff(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),f3703)))).
tff(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),f3705)))).
tff(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),f3707)))).
tff(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),f3709(A_27a,V0net,V1s))))))))).
tff(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),f3711(V1v,V0a)))))))).
tff(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)))))))))))).
tff(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))))))))).
tff(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)))))))).
tff(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)))))))).
tff(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)))))))).
tff(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))))))))))))))).
tff(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))))))))))))))).
tff(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))))))).
tff(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:$i] : (mem(V2t,arr(A_27a,bool)) => (ap(ap(c_2Ereal__topology_2Ewithin(A_27a),ap(ap(c_2Ereal__topology_2Ewithin(A_27a),V0net),V1s)),V2t) = ap(ap(c_2Ereal__topology_2Ewithin(A_27a),V0net),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)))))))))))).
tff(ax_thm_2Ereal__topology_2Etrivial__limit,axiom,( ! [A_27a:del]: ( ! [V0net:$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:$i] : (mem(V5x,A_27a) => ((~~ p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0net),V5x),V3a))) & (~~ p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0net),V5x),V4b))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__WITHIN,lemma,(( ! [V0s:$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))))))))).
tff(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)))))))).
tff(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__INFINITY,lemma,((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__infinity))))).
tff(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__POSINFINITY,lemma,((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__posinfinity))))).
tff(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__NEGINFINITY,lemma,((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__neginfinity))))).
tff(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__SEQUENTIALLY,lemma,((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Enum_2Enum),c_2Ereal__topology_2Esequentially))))).
tff(conj_thm_2Ereal__topology_2ELIM__WITHIN__CLOSED__TRIVIAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V1s)) & (~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),V1s)))) => p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0a))),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ENONTRIVIAL__LIMIT__WITHIN,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1s:$i] : (mem(V1s,arr(A_27a,bool)) => (p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) => p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),ap(ap(c_2Ereal__topology_2Ewithin(A_27a),V0net),V1s))))))))))).
tff(ax_thm_2Ereal__topology_2Eeventually,axiom,( ! [A_27a:del]: ( ! [V0p:$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)))))))))))))))).
tff(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:$i] : (mem(V2x,A_27a) & p(ap(V1p,V2x))))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN__LE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2p:$i] : (mem(V2p,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V2p),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1a))),V0s))) <=> (? [V3d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3d))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s)) & (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V1a))))) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V1a)))),inj__ty_2Erealax_2Ereal(V3d))))) => p(ap(V2p,inj__ty_2Erealax_2Ereal(V4x))))))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2p:$i] : (mem(V2p,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V2p),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1a))),V0s))) <=> (? [V3d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3d))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s)) & (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V1a))))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V1a)))),inj__ty_2Erealax_2Ereal(V3d))))) => p(ap(V2p,inj__ty_2Erealax_2Ereal(V4x))))))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__AT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1p:$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))))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__SEQUENTIALLY,lemma,(( ! [V0p:$i] : (mem(V0p,arr(ty_2Enum_2Enum,bool)) => (p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Enum_2Enum),V0p),c_2Ereal__topology_2Esequentially)) <=> (? [V1N:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1N)),inj__ty_2Enum_2Enum(V2n))) => p(ap(V0p,inj__ty_2Enum_2Enum(V2n))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY,lemma,(( ! [V0p:$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))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__POSINFINITY,lemma,(( ! [V0p:$i] : (mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),c_2Ereal__topology_2Eat__posinfinity)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal_2Ereal__ge,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1b))) => p(ap(V0p,inj__ty_2Erealax_2Ereal(V2x))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__NEGINFINITY,lemma,(( ! [V0p:$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))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY__POS,lemma,(( ! [V0p:$i] : (mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),c_2Ereal__topology_2Eat__infinity)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1b))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V1b))) => p(ap(V0p,inj__ty_2Erealax_2Ereal(V2x)))))))))))).
tff(conj_thm_2Ereal__topology_2EALWAYS__EVENTUALLY,lemma,(( ! [A_27a:del]: ( ! [V0p:$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)))))))))).
tff(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:$i] : (mem(V2q,arr(A_27a,bool)) => (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3712(A_27a,V1p,V2q)),V0net)) <=> (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V1p),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V2q),V0net))))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__MONO,lemma,(( ! [A_27a:del]: ( ! [V0net:$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:$i] : (mem(V3x,A_27a) => (p(ap(V1p,V3x)) => p(ap(V2q,V3x))))) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V1p),V0net))) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V2q),V0net)))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__MP,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3713(A_27a,V1p,V2q)),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V1p),V0net))) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V2q),V0net)))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__FALSE,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),k(A_27a,c_2Ebool_2EF)),V0net)) <=> p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__TRUE,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),k(A_27a,c_2Ebool_2ET)),V0net)) <=> $true)))))).
tff(conj_thm_2Ereal__topology_2ENOT__EVENTUALLY,lemma,(( ! [A_27a:del]: ( ! [V0net:$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))))))))))).
tff(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:$i] : (mem(V2s,arr(A_27b,bool)) => ((p(ap(c_2Epred__set_2EFINITE(A_27b),V2s)) & (~~ (V2s = c_2Epred__set_2EEMPTY(A_27b)))) => (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3715(A_27a,A_27b,V2s,V1p)),V0net)) <=> ( ! [V5a:$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))))))))))))))))).
tff(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),f3717(A_27a,A_27b,V2s,V1p)),V0net)))))))))))))).
tff(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),f3718(A_27a,V1l,V0f,V3e)),V2net)))))))))))).
tff(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),f3719(A_27a,V1f,V0net)))))))))).
tff(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:$i] : (mem(V6x,A_27a) => (p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V6x),V4y)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V6x)),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V3e))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__WITHIN__LE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a))),V3s))) <=> ( ! [V4e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4e))) => (? [V5d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5d))) & ( ! [V6x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),V3s)) & (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V2a))))) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V2a)))),inj__ty_2Erealax_2Ereal(V5d))))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V6x))),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V4e)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__WITHIN,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a))),V3s))) <=> ( ! [V4e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4e))) => (? [V5d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5d))) & ( ! [V6x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),V3s)) & (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V2a))))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V2a)))),inj__ty_2Erealax_2Ereal(V5d))))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V6x))),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V4e)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__AT__LE,lemma,(( ! [V0f:$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)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__AT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a)))) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4d))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V2a))))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V2a)))),inj__ty_2Erealax_2Ereal(V4d)))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V5x))),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V3e)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__AT__INFINITY,lemma,(( ! [V0f:$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)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__AT__INFINITY__POS,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Eat__infinity)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (? [V3b:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3b))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V4x))),inj__ty_2Erealax_2Ereal(V3b))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V4x))),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V2e))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__AT__POSINFINITY,lemma,(( ! [V0f:$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)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__AT__NEGINFINITY,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Eat__neginfinity)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (? [V3b:tp__ty_2Erealax_2Ereal] : ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V3b))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V4x))),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V2e)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__SEQUENTIALLY,lemma,(( ! [V0s:$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)))))))))))))).
tff(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),f3720(A_27a,V1f,V2l)),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__POSINFINITY__SEQUENTIALLY,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Eat__posinfinity)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3721(V0f)),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially)))))))).
tff(conj_thm_2Ereal__topology_2ELIM__INFINITY__POSINFINITY,lemma,(( ! [V0f:$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)))))))).
tff(conj_thm_2Ereal__topology_2ELIM__WITHIN__EMPTY,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__WITHIN__SUBSET,lemma,(( ! [V0t:$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)))))))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V2l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x))),V3s))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V2l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x))),V4t)))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V2l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x))),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__UNION__UNIV,lemma,(( ! [V0f:$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))))))))))))))).
tff(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),f3722(A_27a,V5z,V4y,V2g,V3s,V1f)),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)))))))))))))))).
tff(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:$i] : (mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V3y:tp__ty_2Erealax_2Ereal] : ( ! [V4z:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3y)),V0net)) & (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3723(A_27a,V1f,V2g,V3y,V4z)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),inj__ty_2Erealax_2Ereal(V4z)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3y)))))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2g),V1f)),inj__ty_2Erealax_2Ereal(V4z)),V0net)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__AT__WITHIN,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a)))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a))),V3s)))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__WITHIN__OPEN,lemma,(( ! [V0f:$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)))))))))))))).
tff(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),f3724(V0n))))).
tff(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)))).
tff(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)))))))).
tff(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))))))))).
tff(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)))))))).
tff(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))))))).
tff(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)))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT__EQ,lemma,(( ! [V0R:$i] : (mem(V0R,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => (( ! [V1x:tp__ty_2Enum_2Enum] : ( ! [V2y:tp__ty_2Enum_2Enum] : ( ! [V3z:tp__ty_2Enum_2Enum] : ((p(ap(ap(V0R,inj__ty_2Enum_2Enum(V1x)),inj__ty_2Enum_2Enum(V2y))) & p(ap(ap(V0R,inj__ty_2Enum_2Enum(V2y)),inj__ty_2Enum_2Enum(V3z)))) => p(ap(ap(V0R,inj__ty_2Enum_2Enum(V1x)),inj__ty_2Enum_2Enum(V3z))))))) => (( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V4m)),inj__ty_2Enum_2Enum(V5n))) => p(ap(ap(V0R,inj__ty_2Enum_2Enum(V4m)),inj__ty_2Enum_2Enum(V5n)))))) <=> ( ! [V6n:tp__ty_2Enum_2Enum] : p(ap(ap(V0R,inj__ty_2Enum_2Enum(V6n)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V6n))))))))))).
tff(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT,lemma,(( ! [V0R:$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))))))))))).
tff(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)))))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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))))))))))))))))).
tff(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))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__BALL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V1x)),V0s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2e))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__CBALL,lemma,(( ! [V0s:$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))))))))))))))).
tff(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)))))))))).
tff(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)))))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__IMP__LIMPT,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,inj__ty_2Erealax_2Ereal(V0x)),V1s)) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V0x)),V1s)))))))).
tff(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL__CBALL,lemma,((( ! [V0s:$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:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,inj__ty_2Erealax_2Ereal(V4x)),V3s)) <=> ( ! [V5e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5e))) => (~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V5e)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL,lemma,(( ! [V0s:$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))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__CBALL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,inj__ty_2Erealax_2Ereal(V1x)),V0s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2e))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__LINEAR,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3725(A_27a,V2f,V1h)),ap(V1h,inj__ty_2Erealax_2Ereal(V3l))),V0net))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__CONST,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),k(A_27a,inj__ty_2Erealax_2Ereal(V1a))),inj__ty_2Erealax_2Ereal(V1a)),V0net)))))))).
tff(conj_thm_2Ereal__topology_2ELIM__CMUL,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3c:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3726(A_27a,V1f,V3c)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V2l))),V0net)))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__CMUL__EQ,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3c:tp__ty_2Erealax_2Ereal] : ((~~ (V3c = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3726(A_27a,V1f,V3c)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V2l))),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__NEG,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3727(A_27a,V1f)),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V2l))),V0net))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__NEG__EQ,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3727(A_27a,V1f)),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V2l))),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__ADD,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3728(A_27a,V1f,V2g)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V3l)),inj__ty_2Erealax_2Ereal(V4m))),V0net)))))))))))))).
tff(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),f3729(A_27a,V1f)),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2l))),V0net))))))))))).
tff(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:$i] : (mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),inj__ty_2Erealax_2Ereal(V4m)),V0net))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3730(A_27a,V1f,V2g)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V3l)),inj__ty_2Erealax_2Ereal(V4m))),V0net)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__MAX,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3731(A_27a,V1f,V2g)),ap(ap(c_2Ereal_2Emax,inj__ty_2Erealax_2Ereal(V3l)),inj__ty_2Erealax_2Ereal(V4m))),V0net)))))))))))))).
tff(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:$i] : (mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),inj__ty_2Erealax_2Ereal(V4m)),V0net))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3732(A_27a,V1f,V2g)),ap(ap(c_2Ereal_2Emin,inj__ty_2Erealax_2Ereal(V3l)),inj__ty_2Erealax_2Ereal(V4m))),V0net)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__NULL,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3733(A_27a,V1f,V2l)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__NULL__ABS,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3734(A_27a,V1f)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL__EQ,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2c:tp__ty_2Erealax_2Ereal] : ((~~ (V2c = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3172(A_27a,V1f,V2c)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2c:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3172(A_27a,V1f,V2c)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__NULL__ADD,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3735(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)))))))))))).
tff(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:$i] : (mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3736(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__NULL__COMPARISON,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3737(A_27a,V1f,V2g)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3738(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)))))))))))).
tff(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:$i] : (mem(V2i,A_27b) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l)),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3739(A_27a,V1f)),inj__ty_2Erealax_2Ereal(V3l)),V0net)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__BOUND,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3740(A_27a,V1f,V2g)),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)))))))))))).
tff(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:$i] : (mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V3B:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3741(A_27a,V3B,V1f,V2g)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3742(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__SUM,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$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),f3744(A_27b,A_27a,V1f,V3s)),ap(ap(c_2Eiterate_2ESum(A_27a),V3s),V2l)),V0net))))))))))))))).
tff(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),f3745(A_27a,A_27b,V1f,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),f3746(A_27a,A_27b,V1f,V2s)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net))))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2EClosed,V2s)) & (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3747(A_27a,V1f,V2s)),V0net)) & ((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l)),V0net))))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3l)),V2s))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__ABS__UBOUND,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3748(A_27a,V1f,V3b)),V0net)))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2l))),inj__ty_2Erealax_2Ereal(V3b))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__ABS__LBOUND,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : (((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3749(A_27a,V1f,V3b)),V0net)))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3b)),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2l)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__UNIQUE,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3l_27:tp__ty_2Erealax_2Ereal] : (((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l_27)),V0net)))) => (V2l = V3l_27))))))))))).
tff(conj_thm_2Ereal__topology_2ETENDSTO__LIM,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : (((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__topology_2Elim__def(A_27a),V0net),V1f)) = V2l)))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__CONST__EQ,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),k(A_27a,inj__ty_2Erealax_2Ereal(V1c))),inj__ty_2Erealax_2Ereal(V2d)),V0net)) <=> (p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) | (V1c = V2d)))))))))).
tff(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),f3751(A_27a,A_27b,V1P,V4l,V2f,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),f3753(A_27a,A_27b,V1P,V5m,V3g,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),f3755(A_27a,A_27b,V1P,V5m,V4l,V3g,V2f,V12e)),V0net))))))))))))))))))))).
tff(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),f3751(A_27a,A_27b,V1P,V4l,V2f,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),f3753(A_27a,A_27b,V1P,V5m,V3g,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),f3757(A_27a,A_27b,V1P,V5m,V4l,V3g,V2f,V12e)),V0net))))))))))))))))))))).
tff(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:$i] : (mem(V6m,arr(A_27b,ty_2Erealax_2Ereal)) => ( ! [V7b1:tp__ty_2Erealax_2Ereal] : ( ! [V8b2:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Ebilinear,V2h)) & (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),k(A_27a,ap(c_2Ebool_2E_21(A_27b),f3758(A_27b,V7b1,V5l,V1P)))),V0net)) & (p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),k(A_27a,ap(c_2Ebool_2E_21(A_27b),f3759(A_27b,V8b2,V6m,V1P)))),V0net)) & (( ! [V13e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V13e))) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3761(A_27a,A_27b,V1P,V5l,V3f,V13e)),V0net)))) & ( ! [V16e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V16e))) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3763(A_27a,A_27b,V1P,V6m,V4g,V16e)),V0net)))))))) => ( ! [V19e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V19e))) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3765(A_27a,A_27b,V1P,V6m,V5l,V2h,V4g,V3f,V19e)),V0net))))))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__BILINEAR,lemma,(( ! [A_27a:del]: ( ! [V0net:$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:$i] : (mem(V3g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V4l:tp__ty_2Erealax_2Ereal] : ( ! [V5m:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2f),inj__ty_2Erealax_2Ereal(V4l)),V0net)) & (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V3g),inj__ty_2Erealax_2Ereal(V5m)),V0net)) & p(ap(c_2Ereal__topology_2Ebilinear,V1h)))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3766(A_27a,V1h,V2f,V3g)),ap(ap(V1h,inj__ty_2Erealax_2Ereal(V4l)),inj__ty_2Erealax_2Ereal(V5m))),V0net)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__WITHIN__ID,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),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)))))))).
tff(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),i(ty_2Erealax_2Ereal)),inj__ty_2Erealax_2Ereal(V0a)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0a))))))).
tff(conj_thm_2Ereal__topology_2ELIM__AT__ZERO,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a)))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),f3767(V2a,V0f)),inj__ty_2Erealax_2Ereal(V1l)),ap(c_2Ereal__topology_2Eat,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))))))))).
tff(ax_thm_2Ereal__topology_2Enetlimit,axiom,( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => (ap(c_2Ereal__topology_2Enetlimit(A_27a),V0net) = ap(c_2Emin_2E_40(A_27a),f3769(A_27a,V0net))))))).
tff(conj_thm_2Ereal__topology_2ENETLIMIT__WITHIN,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0a))),V1s)))) => (surj__ty_2Erealax_2Ereal(ap(c_2Ereal__topology_2Enetlimit(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0a))),V1s))) = V0a))))))).
tff(conj_thm_2Ereal__topology_2ENETLIMIT__AT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : (surj__ty_2Erealax_2Ereal(ap(c_2Ereal__topology_2Enetlimit(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0a)))) = V0a)))).
tff(conj_thm_2Ereal__topology_2ELIM__TRANSFORM,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3770(A_27a,V1f,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),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))))))))))))).
tff(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:$i] : (mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3771(A_27a,V1f,V2g)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l)),V0net))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),inj__ty_2Erealax_2Ereal(V3l)),V0net))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN,lemma,(( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$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))))))))))))))).
tff(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)))))))))))))).
tff(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:$i] : (mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3770(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0net)) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l)),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),inj__ty_2Erealax_2Ereal(V3l)),V0net)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET,lemma,(( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$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),f3772(V3s,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))))))))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),f3773(V4t,V3s)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a)))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a))),V3s)))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a))),V4t)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__WITHIN,lemma,(( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$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))))))))))))))).
tff(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)))))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4l:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2EOpen,V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3a)),V2s)) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V2s)) & (~~ (V5x = V3a))) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V5x)))))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V4l)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3a))))))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1g),inj__ty_2Erealax_2Ereal(V4l)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3a))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN__IN,lemma,(( ! [V0f:$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:$i] : (mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5l:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),V2s)) & (( ! [V6x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),V2s)) & (~~ (V6x = V4a))) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V6x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V6x)))))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V5l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V4a))),V3t)))))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1g),inj__ty_2Erealax_2Ereal(V5l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V4a))),V3t)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__CASES__FINITE__SEQUENTIALLY,lemma,(( ! [V0P:$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),f3774(V0P)))) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3775(V1f,V0P,V2g)),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))))))))))))).
tff(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:$i] : (mem(V2g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3776(V0P)))) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3775(V1f,V0P,V2g)),inj__ty_2Erealax_2Ereal(V3l)),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1f),inj__ty_2Erealax_2Ereal(V3l)),c_2Ereal__topology_2Esequentially))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__CASES__SEQUENTIALLY,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : (mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3m:tp__ty_2Enum_2Enum] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3777(V0f,V3m,V1g)),inj__ty_2Erealax_2Ereal(V2l)),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),inj__ty_2Erealax_2Ereal(V2l)),c_2Ereal__topology_2Esequentially))) & ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3778(V0f,V3m,V1g)),inj__ty_2Erealax_2Ereal(V2l)),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),inj__ty_2Erealax_2Ereal(V2l)),c_2Ereal__topology_2Esequentially))) & ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3779(V0f,V3m,V1g)),inj__ty_2Erealax_2Ereal(V2l)),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1g),inj__ty_2Erealax_2Ereal(V2l)),c_2Ereal__topology_2Esequentially))) & (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3780(V0f,V3m,V1g)),inj__ty_2Erealax_2Ereal(V2l)),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1g),inj__ty_2Erealax_2Ereal(V2l)),c_2Ereal__topology_2Esequentially)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__CONG__WITHIN,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$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),f3781(V1f)),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))))))))))))))).
tff(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),f3782(V1f)),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)))))))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__APPROACHABLE,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V1s)) => (( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (? [V3y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),inj__ty_2Erealax_2Ereal(V0x)))),inj__ty_2Erealax_2Ereal(V2e))))))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2EIN__CLOSURE__DELETE,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2EDENSE__IMP__PERFECT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((ap(c_2Ereal__topology_2Eclosure,V0s) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V1x)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EDENSE__LIMIT__POINTS,lemma,(( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:$i] : (mem(V1x,A_27a) => ((ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3783(V0s)) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) <=> (ap(c_2Ereal__topology_2Eclosure,V0s) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))))))))).
tff(conj_thm_2Ereal__topology_2ESEQ__OFFSET,lemma,(( ! [V0f:$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),f3784(V2k,V0f)),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially))))))))).
tff(conj_thm_2Ereal__topology_2ESEQ__OFFSET__NEG,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2k:tp__ty_2Enum_2Enum] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3785(V2k,V0f)),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially))))))))).
tff(conj_thm_2Ereal__topology_2ESEQ__OFFSET__REV,lemma,(( ! [V0f:$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),f3784(V2k,V0f)),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))))))))).
tff(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),f3786(V0a)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Ereal__topology_2Esequentially))))).
tff(conj_thm_2Ereal__topology_2ESEQ__HARMONIC,lemma,(p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3787),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Ereal__topology_2Esequentially)))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))))))).
tff(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))))))))).
tff(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)))))))).
tff(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)))))))).
tff(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))))))))).
tff(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)))))))).
tff(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))))))))).
tff(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))))))).
tff(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))))))))).
tff(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)))))))).
tff(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)))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN__INTERIOR,lemma,(( ! [V0p:$i] : (mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),ap(c_2Ereal__topology_2Einterior,V1s))) => (p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V1s))) <=> p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__WITHIN__INTERIOR,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),ap(c_2Ereal__topology_2Einterior,V2s))) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3x))),V2s))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3x)))))))))))))).
tff(conj_thm_2Ereal__topology_2ENETLIMIT__WITHIN__INTERIOR,lemma,(( ! [V0s:$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))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Econnected,V0s)) & ((~~ (? [V2a:tp__ty_2Erealax_2Ereal] : (V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)))) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V1x)),V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT__CLOSED,lemma,(( ! [V0s:$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))))))))).
tff(ax_thm_2Ereal__topology_2Ebounded__def,axiom,( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> (? [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V1a)))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Ebounded__def,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t))) => p(ap(c_2Ereal__topology_2Ebounded__def,V0s))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__INTERIOR,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__CLOSURE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Eclosure,V0s)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__CLOSURE__EQ,lemma,(( ! [V0s:$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))))))).
tff(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))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2EFINITE__IMP__BOUNDED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,V0s))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__UNION,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) <=> (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__BIGUNION,lemma,(( ! [V0f:$i] : (mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ((p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0f)) & ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => p(ap(c_2Ereal__topology_2Ebounded__def,V1s)))))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__POS,lemma,(( ! [V0s:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__POS__LT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> (? [V1b:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1b))) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V1b)))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__INTER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) | p(ap(c_2Ereal__topology_2Ebounded__def,V1t))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__DIFF,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__INSERT,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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)))))))).
tff(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))))))).
tff(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)))))))).
tff(conj_thm_2Ereal__topology_2ENOT__BOUNDED__UNIV,lemma,((~~ p(ap(c_2Ereal__topology_2Ebounded__def,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))))).
tff(conj_thm_2Ereal__topology_2ECOBOUNDED__IMP__UNBOUNDED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) => (~~ p(ap(c_2Ereal__topology_2Ebounded__def,V0s)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__LINEAR__IMAGE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) & p(ap(c_2Ereal__topology_2Elinear,V0f))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SCALING,lemma,(( ! [V0c:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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),f3788(V0c)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__NEGATIONS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__TRANSLATION,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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),f3789(V0a)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__TRANSLATION__EQ,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))) <=> p(ap(c_2Ereal__topology_2Ebounded__def,V1s)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__DIFFS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3791(V1t,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUMS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGE,lemma,(( ! [A_27a:del]: ( ! [V0f:$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),f3792(A_27a,V0f,V2t)))) & p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,A_27a),f3793(A_27a,V1g,V2t))))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,A_27a),f3794(A_27a,V0f,V1g,V2t)))))))))))))).
tff(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),f3795(A_27b,A_27a,V0f,V3a,V1t)))))))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,A_27a),f3796(A_27b,A_27a,V0f,V2s,V1t))))))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__BALL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => (? [V2r:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2r))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2r))))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CBALL,lemma,(( ! [V0s:$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))))))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__BOUNDED__EQ__TRIVIAL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Esubspace,V0s)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> (V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__COMPONENTWISE,lemma,(( ! [V0s:$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),i(ty_2Erealax_2Ereal)),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__HAS__SUP,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1x)),ap(c_2Ereal_2Esup,V0s))))) & ( ! [V2b:tp__ty_2Erealax_2Ereal] : (( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),inj__ty_2Erealax_2Ereal(V2b))))))))))).
tff(conj_thm_2Ereal__topology_2ESUP__INSERT,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__HAS__INF,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),inj__ty_2Erealax_2Ereal(V1x))))) & ( ! [V2b:tp__ty_2Erealax_2Ereal] : (( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3x))))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2b)),ap(c_2Eiterate_2Einf,V0s))))))))))).
tff(conj_thm_2Ereal__topology_2EINF__INSERT,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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))))))))))).
tff(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))))))))))))))).
tff(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))))))))))))))))).
tff(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:$i] : (mem(V4r,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3l)),V0s)) & (( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V6n))) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(V4r,inj__ty_2Enum_2Enum(V5m))),ap(V4r,inj__ty_2Enum_2Enum(V6n))))))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Enum_2Enum),V1f),V4r)),inj__ty_2Erealax_2Ereal(V3l)),c_2Ereal__topology_2Esequentially)))))))))))))).
tff(conj_thm_2Ereal__topology_2EMONOTONE__BIGGER,lemma,(( ! [V0r:$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)))))))))).
tff(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)))))))))).
tff(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)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONVERGENT__BOUNDED__INCREASING,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1b:tp__ty_2Erealax_2Ereal] : ((( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,inj__ty_2Enum_2Enum(V2m))),ap(V0s,inj__ty_2Enum_2Enum(V3n))))))) & ( ! [V4n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0s,inj__ty_2Enum_2Enum(V4n)))),inj__ty_2Erealax_2Ereal(V1b))))) => (? [V5l:tp__ty_2Erealax_2Ereal] : ( ! [V6e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V6e))) => (? [V7N:tp__ty_2Enum_2Enum] : ( ! [V8n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V7N)),inj__ty_2Enum_2Enum(V8n))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0s,inj__ty_2Enum_2Enum(V8n))),inj__ty_2Erealax_2Ereal(V5l)))),inj__ty_2Erealax_2Ereal(V6e))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONVERGENT__BOUNDED__MONOTONE,lemma,(( ! [V0s:$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))))))))))))))).
tff(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)))))))))))))))))).
tff(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)))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__IMP__COMPACT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V0s))) => p(ap(c_2Ereal__topology_2Ecompact,V0s))))))).
tff(ax_thm_2Ereal__topology_2Ecauchy,axiom,( ! [V0s:$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))))))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2ECAUCHY,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => (p(ap(c_2Ereal__topology_2Ecauchy,V0s)) <=> ( ! [V1e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1e))) => (? [V2N:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V2N))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0s,inj__ty_2Enum_2Enum(V3n))),ap(V0s,inj__ty_2Enum_2Enum(V2N))))),inj__ty_2Erealax_2Ereal(V1e))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONVERGENT__IMP__CAUCHY,lemma,(( ! [V0s:$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)))))))).
tff(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)))))).
tff(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:$i] : (mem(V3x,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3x)),inj__ty_2Erealax_2Ereal(V2a))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECAUCHY__IMP__BOUNDED,lemma,(( ! [V0s:$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),f3798(V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__IMP__COMPLETE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecomplete,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECOMPLETE__UNIV,lemma,(p(ap(c_2Ereal__topology_2Ecomplete,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2ECOMPLETE__EQ__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecomplete,V0s)) <=> p(ap(c_2Ereal__topology_2EClosed,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECONVERGENT__EQ__CAUCHY,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ((? [V1l:tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially))) <=> p(ap(c_2Ereal__topology_2Ecauchy,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECONVERGENT__IMP__BOUNDED,lemma,(( ! [V0s:$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)))))))))).
tff(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)),f3799(V1e)),V2k))))))))))))))).
tff(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))))))))))))))))).
tff(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))))))))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => (((~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V4t))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V0s))) => (? [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V0s)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V5x)),V4t)))))))))))).
tff(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__BOUNDED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (((~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s))) => (? [V2x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2x)),V1t)))))) => p(ap(c_2Ereal__topology_2Ebounded__def,V0s))))))).
tff(conj_thm_2Ereal__topology_2EINF__FINITE__LEMMA,lemma,(( ! [V0s:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2EINF__FINITE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Eiterate_2Einf,V0s)),V0s)) & ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),inj__ty_2Erealax_2Ereal(V1x))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__LE__INF__FINITE,lemma,(( ! [V0s:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__INF__LE__FINITE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),inj__ty_2Erealax_2Ereal(V1a))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) & p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1a)))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__LT__INF__FINITE,lemma,(( ! [V0s:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2EREAL__INF__LT__FINITE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Eiterate_2Einf,V0s)),inj__ty_2Erealax_2Ereal(V1a))) <=> (? [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1a)))))))))))).
tff(conj_thm_2Ereal__topology_2ESEQUENCE__INFINITE__LEMMA,lemma,(( ! [V0f:$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),f3801(V0f))))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2ESEQUENCE__UNIQUE__LIMPT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2l_27:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2l_27)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3801(V0f))))) => (V2l_27 = V1l)))))))).
tff(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (((~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s))) => (? [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2x)),V1t))))))) => p(ap(c_2Ereal__topology_2EClosed,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL,lemma,(( ! [V0s:$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:$i] : (mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f)) & (p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V3f_27))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__EQ__BOLZANO__WEIERSTRASS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecompact,V0s)) <=> ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (((~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s))) => (? [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,inj__ty_2Erealax_2Ereal(V2x)),V1t)))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__EQ__BOUNDED__CLOSED,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__IMP__BOUNDED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__IMP__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2EClosed,V0s))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__SEQUENCE__WITH__LIMIT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Erealax_2Ereal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPACT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t))) => p(ap(c_2Ereal__topology_2Ecompact,V1t))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPACT__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) <=> (p(ap(c_2Ereal__topology_2Ecompact,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL__SUBTOPOLOGY,lemma,(( ! [V0s:$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:$i] : (mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & (p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f)) & (p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V3f_27))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__CLOSURE,lemma,(( ! [V0s:$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))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2EDISCRETE__BOUNDED__IMP__FINITE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1e:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1e))) & (( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V3y)),inj__ty_2Erealax_2Ereal(V2x)))),inj__ty_2Erealax_2Ereal(V1e))))) => (V3y = V2x)))) & p(ap(c_2Ereal__topology_2Ebounded__def,V0s)))) => p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS,lemma,(( ! [V0s:$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)))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Ecompact,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(conj_thm_2Ereal__topology_2EFINITE__IMP__CLOSED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => p(ap(c_2Ereal__topology_2EClosed,V0s))))))).
tff(conj_thm_2Ereal__topology_2EFINITE__IMP__CLOSED__IN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EFINITE__IMP__COMPACT,lemma,(( ! [V0s:$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))))))).
tff(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))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__INSERT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecompact,V1s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__SING,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__SING,lemma,(( ! [V0u:$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)))))))).
tff(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)))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__INSERT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V1s)) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__CBALL,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1e:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1e))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__FRONTIER__BOUNDED,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__FRONTIER,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Efrontier,V0s)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__FRONTIER,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__COMPACT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),V0s))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__DELETE,lemma,(( ! [V0s:$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)))))))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__BIGINTER__COMPACT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V0s)) <=> ( ! [V1e:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1e)))),V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__BIGUNION,lemma,(( ! [V0s:$i] : (mem(V0s,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ((p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0s)) & ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1t),V0s)) => p(ap(c_2Ereal__topology_2Ecompact,V1t)))))) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__DIFF,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t))) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__SPHERE,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1r))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SPHERE,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1r))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__SPHERE,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1r))))))))).
tff(conj_thm_2Ereal__topology_2EFRONTIER__SING,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : (ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__IMP__FIP,lemma,(( ! [V0s:$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:$i] : (mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ((p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f))) => (~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V3f_27)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))) => (~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V1f)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IMP__FIP,lemma,(( ! [V0s:$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)))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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:$i] : (mem(V2f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ((p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V2f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V2f_27),V0f))) => (~~ (ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V2f_27) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))) => (~~ (ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__NEST,lemma,(( ! [V0s:$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))))))))))).
tff(conj_thm_2Ereal__topology_2EDECREASING__CLOSED__NEST,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) => ((( ! [V1n:tp__ty_2Enum_2Enum] : p(ap(c_2Ereal__topology_2EClosed,ap(V0s,inj__ty_2Enum_2Enum(V1n))))) & (( ! [V2n:tp__ty_2Enum_2Enum] : (~~ (ap(V0s,inj__ty_2Enum_2Enum(V2n)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) & (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n))) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0s,inj__ty_2Enum_2Enum(V4n))),ap(V0s,inj__ty_2Enum_2Enum(V3m))))))) & ( ! [V5e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5e))) => (? [V6n:tp__ty_2Enum_2Enum] : ( ! [V7x:tp__ty_2Erealax_2Ereal] : ( ! [V8y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V7x)),ap(V0s,inj__ty_2Enum_2Enum(V6n)))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V8y)),ap(V0s,inj__ty_2Enum_2Enum(V6n))))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V7x)),inj__ty_2Erealax_2Ereal(V8y)))),inj__ty_2Erealax_2Ereal(V5e)))))))))))) => (? [V9a:tp__ty_2Erealax_2Ereal] : ( ! [V10n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V9a)),ap(V0s,inj__ty_2Enum_2Enum(V10n))))))))))).
tff(conj_thm_2Ereal__topology_2EDECREASING__CLOSED__NEST__SING,lemma,(( ! [V0s:$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)),f3803(V0s))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V9a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3s),V0f)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V4t),V0f))) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3s),V4t)) | p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V3s)))))))) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1b),V0f)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1b))))) => (~~ (ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__CHAIN,lemma,(( ! [V0f:$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:$i] : (mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2s),V0f)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3t),V0f))) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2s),V3t)) | p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),V2s))))))))) => (~~ (ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__NEST,lemma,(( ! [V0s:$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),f3804(V0s))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))).
tff(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:$i] : (mem(V11x,A_27a) => ((p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V8N)),inj__ty_2Enum_2Enum(V9m))) & (p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V8N)),inj__ty_2Enum_2Enum(V10n))) & p(ap(V0P,V11x)))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(V1s,inj__ty_2Enum_2Enum(V9m)),V11x)),ap(ap(V1s,inj__ty_2Enum_2Enum(V10n)),V11x)))),inj__ty_2Erealax_2Ereal(V7e))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY__ALT,lemma,(( ! [A_27a:del]: ( ! [V0P:$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))))))))))))))))))).
tff(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)))))))))))))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x)))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),ap(V0f,inj__ty_2Erealax_2Ereal(V1x))),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x)))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x)))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x))),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__CLOSED__NONTRIVIAL,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V2s)) & (~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),V2s)))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1a))),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN,lemma,(( ! [V0f:$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)))))))))))))).
tff(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))))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2EOpen,V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3a)),V2s)) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V2s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V4x)))))) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3a))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3a)))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN__IN,lemma,(( ! [V0f:$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:$i] : (mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V4a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),V2s)) & (( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V2s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V5x)))))) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V4a))),V3t)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V4a))),V3t))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__SET__IMP,lemma,(( ! [V0f:$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),f3805(V3t,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))))))))))))).
tff(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))))))))))))))))).
tff(conj_thm_2Ereal__topology_2Econtinuous__at,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x)))) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (? [V3d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3d))) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x_27)),inj__ty_2Erealax_2Ereal(V1x)))),inj__ty_2Erealax_2Ereal(V3d))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V4x_27))),ap(V0f,inj__ty_2Erealax_2Ereal(V1x))))),inj__ty_2Erealax_2Ereal(V2e))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__BALL,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V1s))) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4d))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V4d)))),V1s))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V3e))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__BALL,lemma,(( ! [V0f:$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))))))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3a))),V2s))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V3a))),ap(V0f,inj__ty_2Erealax_2Ereal(V4x))))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V1g,inj__ty_2Erealax_2Ereal(V3a))),ap(V1g,inj__ty_2Erealax_2Ereal(V4x))))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V3a))),V2s))))))))))))).
tff(ax_thm_2Ereal__topology_2Econtinuous__on,axiom,( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) => ( ! [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4d))) & ( ! [V5x_27:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x_27)),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x_27)),inj__ty_2Erealax_2Ereal(V2x)))),inj__ty_2Erealax_2Ereal(V4d)))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V5x_27))),ap(V0f,inj__ty_2Erealax_2Ereal(V2x))))),inj__ty_2Erealax_2Ereal(V3e))))))))))))))))).
tff(ax_thm_2Ereal__topology_2Euniformly__continuous__on,axiom,( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (? [V3d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3d))) & ( ! [V4x:tp__ty_2Erealax_2Ereal] : ( ! [V5x_27:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V1s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x_27)),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x_27)),inj__ty_2Erealax_2Ereal(V4x)))),inj__ty_2Erealax_2Ereal(V3d))))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V5x_27))),ap(V0f,inj__ty_2Erealax_2Ereal(V4x))))),inj__ty_2Erealax_2Ereal(V2e)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CONTINUOUS,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__IMP__CONTINUOUS__ON,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__WITHIN,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V1s)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),ap(V0f,inj__ty_2Erealax_2Ereal(V2x))),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V1s)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__AT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V1s)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x)))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SUBSET,lemma,(( ! [V0f:$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))))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t),V1s))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2t))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUBSET,lemma,(( ! [V0f:$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))))))))))).
tff(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)))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V2s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V3x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V3x)))))) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__EQ,lemma,(( ! [V0f:$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))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SING,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EMPTY,lemma,(( ! [V0f:$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))))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2EFORALL__POS__MONO__1,lemma,(( ! [V0P:$i] : (mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ((( ! [V1d:tp__ty_2Erealax_2Ereal] : ( ! [V2e:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1d)),inj__ty_2Erealax_2Ereal(V2e))) & p(ap(V0P,inj__ty_2Erealax_2Ereal(V1d)))) => p(ap(V0P,inj__ty_2Erealax_2Ereal(V2e)))))) & ( ! [V3n:tp__ty_2Enum_2Enum] : p(ap(V0P,ap(c_2Erealax_2Einv,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(V3n))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))))) => ( ! [V4e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4e))) => p(ap(V0P,inj__ty_2Erealax_2Ereal(V4e)))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SEQUENTIALLY,lemma,(( ! [V0f:$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))))))))))))).
tff(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))))))))))).
tff(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:$i] : (mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3a)),V1s)) & (( ! [V4n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,inj__ty_2Enum_2Enum(V4n))),V1s))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V2x),inj__ty_2Erealax_2Ereal(V3a)),c_2Ereal__topology_2Esequentially)))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x)),ap(V0f,inj__ty_2Erealax_2Ereal(V3a))),c_2Ereal__topology_2Esequentially))))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SEQUENTIALLY,lemma,(( ! [V0f:$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:$i] : (mem(V3y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ((( ! [V4n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,inj__ty_2Enum_2Enum(V4n))),V1s))) & (( ! [V5n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V3y,inj__ty_2Enum_2Enum(V5n))),V1s))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3806(V2x,V3y)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Ereal__topology_2Esequentially)))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3807(V2x,V3y,V0f)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Ereal__topology_2Esequentially)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__CONTINUOUS__FUNCTION,lemma,(( ! [A_27a:del]: ( ! [V0f:$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),f3808(A_27a,V2g,V0f)),ap(V0f,inj__ty_2Erealax_2Ereal(V3l))),V1net))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__CONST,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),k(A_27a,inj__ty_2Erealax_2Ereal(V1c))),V0net)))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__CMUL,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2net:$i] : (mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V2net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3356(A_27a,V0f,V1c)),V2net))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__NEG,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V1net:$i] : (mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V1net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3358(A_27a,V0f)),V1net)))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ADD,lemma,(( ! [A_27a:del]: ( ! [V0f:$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),f3352(A_27a,V0f,V1g)),V2net)))))))))))).
tff(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:$i] : (mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V2net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1g),V2net))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3359(A_27a,V0f,V1g)),V2net)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ABS,lemma,(( ! [A_27a:del]: ( ! [V0f:$i] : (mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V1net:$i] : (mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V1net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3360(A_27a,V0f)),V1net)))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__MAX,lemma,(( ! [A_27a:del]: ( ! [V0f:$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),f3809(A_27a,V0f,V1g)),V2net)))))))))))).
tff(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:$i] : (mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V2net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1g),V2net))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3810(A_27a,V0f,V1g)),V2net)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__SUM,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$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:$i] : (mem(V3a,A_27b) => (p(ap(ap(c_2Ebool_2EIN(A_27b),V3a),V2s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),ap(V1f,V3a)),V0net)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3812(A_27a,A_27b,V1f,V2s)),V0net))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CONST,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Econtinuous__on,k(ty_2Erealax_2Ereal,inj__ty_2Erealax_2Ereal(V1c))),V0s))))))).
tff(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,f3813(V0f,V1c)),V2s)))))))))).
tff(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,f2906(V0f)),V1s))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f2904(V0f,V1g)),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUB,lemma,(( ! [V0f:$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,f2908(V0f,V1g)),V2s))))))))))).
tff(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,f3814(V0f)),V1s))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3815(V0f,V1g)),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MIN,lemma,(( ! [V0f:$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,f3816(V0f,V1g)),V2s))))))))))).
tff(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,f3818(A_27a,V1f,V2s)),V0t)))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CONST,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,k(ty_2Erealax_2Ereal,inj__ty_2Erealax_2Ereal(V1c))),V0s))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__UNIFORMLY__CONTINUOUS__ON,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__COMPOSE,lemma,(( ! [V0f:$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))))))))))).
tff(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,f3819(V2h,V0f,V1g)),V3s))))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) & (p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1g),V2s)) & (p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))) & p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V2s)))))) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f2905(V0f,V1g)),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CMUL,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f3813(V0f,V1c)),V2s)))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__VMUL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:$i] : (mem(V1c,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V2v:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1c),V0s)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f3820(V1c,V2v)),V0s)))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__NEG,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f2906(V0f)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ADD,lemma,(( ! [V0f:$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,f2904(V0f,V1g)),V2s))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1g),V2s))) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f2908(V0f,V1g)),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUM,lemma,(( ! [A_27a:del]: ( ! [V0t:$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:$i] : (mem(V3a,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V3a),V2s)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,ap(V1f,V3a)),V0t)))))) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f3818(A_27a,V1f,V2s)),V0t)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__ID,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0a))),V1s)))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__ID,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0a))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ID,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,i(ty_2Erealax_2Ereal)),V0s)))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ID,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,i(ty_2Erealax_2Ereal)),V0s)))))).
tff(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:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V3s))) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,ap(V0f,inj__ty_2Erealax_2Ereal(V2x)))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3s))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V3s))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1g:$i] : (mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x)))) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(c_2Ereal__topology_2Eat,ap(V0f,inj__ty_2Erealax_2Ereal(V2x)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE,lemma,(( ! [V0f:$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))))))))))).
tff(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))))))))))))))))))).
tff(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:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(c_2Ereal__topology_2EOpen,V3s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V3s)) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x_27)),V3s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V4x_27))),V2t))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__GEN,lemma,(( ! [V0f:$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:$i] : (mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN,lemma,(( ! [V0f:$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),f3822(V1s,V0f,V2t)))))))))))))).
tff(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),f3821(V1s,V0f,V3u))))))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V2t))) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED__GEN,lemma,(( ! [V0f:$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:$i] : (mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED,lemma,(( ! [V0f:$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),f3822(V1s,V0f,V2t)))))))))))))).
tff(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),f3821(V1s,V0f,V3u))))))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V2t))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE,lemma,(( ! [V0f:$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),f3822(V1s,V0f,V2t))))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(c_2Ereal__topology_2EClosed,V2t))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE,lemma,(( ! [V0f:$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),f3822(V1s,V0f,V2t))))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & (p(ap(c_2Ereal__topology_2EClosed,V1s)) & p(ap(c_2Ereal__topology_2EClosed,V2t)))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE__UNIV,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))))) & p(ap(c_2Ereal__topology_2EOpen,V1s))) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3823(V1s,V0f))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__UNIV,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((( ! [V2x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))))) & p(ap(c_2Ereal__topology_2EClosed,V1s))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3823(V1s,V0f))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__EQ,lemma,(( ! [V0f:$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),f3822(V1s,V0f,V2t)))))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V2t)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__LIM__0,lemma,(( ! [V0f:$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)))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__AT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1a)))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__WITHIN,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V1s))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__COMPOSE,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3824(A_27a,V1f,V2g)),V0net)))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(c_2Ereal__topology_2Elinear,V1g))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f2910(V0f,V1g)),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__COMPONENT__COMPOSE,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$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),f3825(A_27a,V1f)),V0net))))))))))))).
tff(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,f3826(V0f)),V1s))))))))).
tff(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),f3827(A_27a,V3h,V1f,V2g)),V0net)))))))))))))).
tff(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,f3819(V2h,V0f,V1g)),V3s))))))))))))).
tff(conj_thm_2Ereal__topology_2EBILINEAR__DOT,lemma,(p(ap(c_2Ereal__topology_2Ebilinear,f3829)))).
tff(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:$i] : (mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V2g),V0net))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3830(A_27a,V1f,V2g)),V0net)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DOT2,lemma,(( ! [V0f:$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,f2905(V0f,V1g)),V2s))))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__TRANSLATION,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3641(V1a)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__TRANSLATION__EQ,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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),f3789(V0a)),V1s))) <=> p(ap(c_2Ereal__topology_2Ecompact,V1s)))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__TRANSLATION,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Econnected,V1s)) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__TRANSLATION__EQ,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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),f3789(V0a)),V1s))) <=> p(ap(c_2Ereal__topology_2Econnected,V1s)))))))).
tff(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)))))))))).
tff(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),f3821(V1s,V0f,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:$i] : (mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5u),V2t)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3831(V1s,V0f,V5u)))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V5u)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__OPEN,lemma,(( ! [V0f:$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:$i] : (mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s))))))))))).
tff(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__CLOSED,lemma,(( ! [V0f:$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:$i] : (mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__MAP__IMP__QUOTIENT__MAP,lemma,(( ! [V0f:$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:$i] : (mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3832(V1s,V0f,V3t)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V3t))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__MAP__IMP__QUOTIENT__MAP,lemma,(( ! [V0f:$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:$i] : (mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3832(V1s,V0f,V3t)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V3t))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__RIGHT__INVERSE__IMP__QUOTIENT__MAP,lemma,(( ! [V0f:$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),f3833(V2s,V0f,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))))))))))))))))).
tff(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),f3834(V2s,V0f,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))))))))))))))).
tff(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),f3821(V1s,V0f,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:$i] : (mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5u),V2t)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3831(V1s,V0f,V5u)))) <=> p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V5u)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE__QUOTIENT,lemma,(( ! [V0f:$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:$i] : (mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V3t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3835(V2s,V0f,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V5v)))))) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V2s))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t))))))))))))))).
tff(conj_thm_2Ereal__topology_2EFUNCTION__FACTORS__LEFT__GEN,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [V0P:$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)))))))))))))))))))).
tff(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),f3835(V2s,V0f,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:$i] : (mem(V9g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V9g),V3t)) & ((ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V9g),V3t) = V4u) & ( ! [V10x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V10x)),V2s)) => (surj__ty_2Erealax_2Ereal(ap(V1h,inj__ty_2Erealax_2Ereal(V10x))) = surj__ty_2Erealax_2Ereal(ap(V9g,ap(V0f,inj__ty_2Erealax_2Ereal(V10x)))))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__COMPOSE,lemma,(( ! [V0f:$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:$i] : (mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V3t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3835(V2s,V0f,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V5v)))))) & ( ! [V7v:$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),f3836(V3t,V1g,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:$i] : (mem(V9v,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V9v),V4u)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3837(V2s,V0f,V1g,V9v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),V9v))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__COMPOSITION,lemma,(( ! [V0f:$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:$i] : (mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V4u)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3838(V2s,V0f,V1g,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),V5v)))))))))) => ( ! [V7v:$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),f3836(V3t,V1g,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))))))))))))))))))).
tff(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:$i] : (mem(V4v,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4v),V3u)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3839(V1s,V0f,V4v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3u)),V4v)))))))))) => ( ! [V6v:$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),f3840(V2t,V0f,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))))))))))))))))).
tff(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:$i] : (mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4u),V2t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3841(V1s,V0f,V4u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V4u)))))) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3c)) | p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3c))))) => ( ! [V6u:$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),f3842(V1s,V0f,V3c)))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3844(V6u,V0f,V3c,V1s)))) <=> 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))))))))))))))))).
tff(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),f3821(V1s,V0f,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),f3845(V1s,V0f,V5y)))))) & p(ap(c_2Ereal__topology_2Econnected,V2t)))))) => p(ap(c_2Ereal__topology_2Econnected,V1s))))))))))).
tff(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:$i] : (mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4u),V2t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3841(V1s,V0f,V4u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V4u)))))) & (( ! [V6y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6y)),V2t)) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3846(V1s,V0f,V6y)))))) & ((p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3c)) | p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3c))) & p(ap(c_2Ereal__topology_2Econnected,V3c)))))) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3847(V1s,V0f,V3c))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__MAP__CLOSURES,lemma,(( ! [V0f:$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))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Einterior,V2s))),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__MAP__RESTRICT,lemma,(( ! [V0f:$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:$i] : (mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3848(V1s,V0f,V3t_27)))),V5u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t_27)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__MAP__RESTRICT,lemma,(( ! [V0f:$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:$i] : (mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3848(V1s,V0f,V3t_27)))),V5u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t_27)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__MAP__EQ,lemma,(( ! [V0f:$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:$i] : (mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u))))))) => (( ! [V5k:$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:$i] : (mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3849(V1s,V0f,V6k))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__CLOSED__MAP__EQ,lemma,(( ! [V0f:$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:$i] : (mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u))))))) => (( ! [V5k:$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:$i] : (mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3849(V1s,V0f,V6k))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__MAP__IMP__OPEN__MAP,lemma,(( ! [V0f:$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),f3850(V1s,V0f,V4u))))))))) => ( ! [V6u:$i] : (mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u))))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__MAP__IMP__CLOSED__MAP,lemma,(( ! [V0f:$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),f3850(V1s,V0f,V4u))))))))) => ( ! [V6u:$i] : (mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u))))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__SURJECTIVE,lemma,(( ! [V0f:$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))))))))))))))))))).
tff(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:$i] : (mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V6k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V6k))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__INJECTIVE,lemma,(( ! [V0f:$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))))))))))))))))))).
tff(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:$i] : (mem(V8k,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V8k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V8k))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE,lemma,(( ! [V0f:$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:$i] : (mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5k)))))) & (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4w),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3851(V1s,V0f,V4w))),V3u))))) => (? [V7v:$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),f3852(V1s,V0f,V7v))),V3u))))))))))))))))))).
tff(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),f3853(V1s,V0f,V5w))),V4u)))) => (? [V7v:$i] : (mem(V7v,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V7v)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5w),V7v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3852(V1s,V0f,V7v))),V4u))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE,lemma,(( ! [V0f:$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:$i] : (mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5k)))))) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4w),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3851(V1s,V0f,V4w))),V3u))))) => (? [V7v:$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),f3852(V1s,V0f,V7v))),V3u))))))))))))))))))).
tff(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),f3853(V1s,V0f,V5w))),V4u)))) => (? [V7v:$i] : (mem(V7v,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V7v)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5w),V7v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3852(V1s,V0f,V7v))),V4u))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EBIGUNION__GSPEC,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [A_27c:del]: ( ! [A_27d:del]: ( ! [A_27e:del]: ( ! [A_27f:del]: ( ! [A_27g:del]: ( ! [A_27h:del]: ( ! [A_27i:del]: (( ! [V0P:$i] : (mem(V0P,arr(A_27a,bool)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,arr(A_27b,bool))) => (ap(c_2Epred__set_2EBIGUNION(A_27b),ap(c_2Epred__set_2EGSPEC(arr(A_27b,bool),A_27a),f3662(A_27b,A_27a,V1f,V0P))) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3855(A_27a,A_27b,V1f,V0P))))))) & (( ! [V5P:$i] : (mem(V5P,arr(A_27c,arr(A_27d,bool))) => ( ! [V6f:$i] : (mem(V6f,arr(A_27c,arr(A_27d,arr(A_27e,bool)))) => (ap(c_2Epred__set_2EBIGUNION(A_27e),ap(c_2Epred__set_2EGSPEC(arr(A_27e,bool),ty_2Epair_2Eprod(A_27c,A_27d)),ap(c_2Epair_2EUNCURRY(A_27c,A_27d,ty_2Epair_2Eprod(arr(A_27e,bool),bool)),f3666(A_27e,A_27c,A_27d,V6f,V5P)))) = ap(c_2Epred__set_2EGSPEC(A_27e,A_27e),f3858(A_27c,A_27d,A_27e,V5P,V6f))))))) & ( ! [V12P:$i] : (mem(V12P,arr(A_27f,arr(A_27g,arr(A_27h,bool)))) => ( ! [V13f:$i] : (mem(V13f,arr(A_27f,arr(A_27g,arr(A_27h,arr(A_27i,bool))))) => (ap(c_2Epred__set_2EBIGUNION(A_27i),ap(c_2Epred__set_2EGSPEC(arr(A_27i,bool),ty_2Epair_2Eprod(A_27f,ty_2Epair_2Eprod(A_27g,A_27h))),ap(c_2Epair_2EUNCURRY(A_27f,ty_2Epair_2Eprod(A_27g,A_27h),ty_2Epair_2Eprod(arr(A_27i,bool),bool)),f3672(A_27f,A_27h,A_27i,A_27g,V12P,V13f)))) = ap(c_2Epred__set_2EGSPEC(A_27i,A_27i),f3862(A_27f,A_27h,A_27g,A_27i,V13f,V12P)))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__POINT,lemma,(( ! [V0f:$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),f3845(V1s,V0f,V5y))),V4u)))) => (? [V7v:$i] : (mem(V7v,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V7v)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5y)),V7v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3852(V1s,V0f,V7v))),V4u)))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__MONOTONE__PREIMAGE,lemma,(( ! [V0f:$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:$i] : (mem(V3c,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3c)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3c)))))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4y)),V2t)) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3863(V1s,V0f,V4y))))))))) => ( ! [V6c:$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),f3864(V1s,V0f,V6c)))))))))))))))).
tff(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),f3863(V1s,V0f,V4y))))))))) => ( ! [V6c:$i] : (mem(V6c,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Econnected,V6c)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6c),V2t))) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3864(V1s,V0f,V6c)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EPROPER__MAP,lemma,(( ! [V0f:$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:$i] : (mem(V3k,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3k),V2t)) & p(ap(c_2Ereal__topology_2Ecompact,V3k))) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3865(V1s,V0f,V3k))))))) <=> (( ! [V5k:$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),f3866(V1s,V0f,V6a))))))))))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V4t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V1s))) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4t)))))))))))))).
tff(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPACT,lemma,(( ! [V0t:$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:$i] : (mem(V3k,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V2s)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V2s)),V0t)) & (p(ap(c_2Ereal__topology_2Ecompact,V2s)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0t)),V3k))))) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3867(V2s,V1f,V3k))))))))))))))).
tff(conj_thm_2Ereal__topology_2EPROPER__MAP__COMPOSE,lemma,(( ! [V0f:$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:$i] : (mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5k),V3t)) & p(ap(c_2Ereal__topology_2Ecompact,V5k))) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3868(V2s,V0f,V5k))))))) & ( ! [V7k:$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),f3869(V3t,V1g,V7k))))))))) => ( ! [V9k:$i] : (mem(V9k,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V9k),V4u)) & p(ap(c_2Ereal__topology_2Ecompact,V9k))) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3870(V2s,V0f,V1g,V9k)))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__LEFT,lemma,(( ! [V0f:$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:$i] : (mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5k),V4u)) & p(ap(c_2Ereal__topology_2Ecompact,V5k))) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3871(V2s,V0f,V1g,V5k))))))))))) => ( ! [V7k:$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),f3869(V3t,V1g,V7k)))))))))))))))))))).
tff(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),f3871(V2s,V0f,V1g,V5k))))))))))) => ( ! [V7k:$i] : (mem(V7k,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V7k),V3t)) & p(ap(c_2Ereal__topology_2Ecompact,V7k))) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3872(V2s,V0f,V7k)))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EPASTING__LEMMA,lemma,(( ! [A_27a:del]: ( ! [V0f:$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)))))))))))))))).
tff(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),f3873(A_27a,V1t,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:$i] : (mem(V11i,A_27a) => ((p(ap(ap(c_2Ebool_2EIN(A_27a),V11i),V3k)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V10x)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2s),ap(V1t,V11i))))) => (surj__ty_2Erealax_2Ereal(ap(V9g,inj__ty_2Erealax_2Ereal(V10x))) = surj__ty_2Erealax_2Ereal(ap(ap(V0f,V11i),inj__ty_2Erealax_2Ereal(V10x))))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL__OPEN,lemma,(( ! [V0t:$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)))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EOpen,V1s)) & (p(ap(c_2Ereal__topology_2EOpen,V2t)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2t))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1s),V2t)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL__OPEN,lemma,(( ! [V0P:$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,f3874(V1f,V0P,V2g)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t)))))))))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EOpen,V3s)) & (p(ap(c_2Ereal__topology_2EOpen,V4t)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V3s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V4t)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V3s)) & (~~ p(ap(V0P,inj__ty_2Erealax_2Ereal(V5x))))) | (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V4t)) & p(ap(V0P,inj__ty_2Erealax_2Ereal(V5x))))) => (surj__ty_2Erealax_2Ereal(ap(V1f,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V2g,inj__ty_2Erealax_2Ereal(V5x)))))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3874(V1f,V0P,V2g)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EPASTING__LEMMA__CLOSED,lemma,(( ! [A_27a:del]: ( ! [V0f:$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)))))))))))))))).
tff(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),f3873(A_27a,V1t,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:$i] : (mem(V11i,A_27a) => ((p(ap(ap(c_2Ebool_2EIN(A_27a),V11i),V3k)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V10x)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2s),ap(V1t,V11i))))) => (surj__ty_2Erealax_2Ereal(ap(V9g,inj__ty_2Erealax_2Ereal(V10x))) = surj__ty_2Erealax_2Ereal(ap(ap(V0f,V11i),inj__ty_2Erealax_2Ereal(V10x))))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__LIFT__DOT,lemma,(( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1net:tp__c_ty_2Ereal__topology_2Enet_ty_2Erealax_2Ereal] : ( ! [V2f:$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),f3875(V2f,V3a)),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))))))))))).
tff(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),f3876(V0a)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x)))))))).
tff(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,f3876(V0a)),V1s))))))).
tff(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),f3877(V0b))))))).
tff(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),f3878(V0a))))))).
tff(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),f3879(V1b,V0a)))))))).
tff(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),f3880(V1b,V0a)))))))).
tff(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),f3881(V1b,V0a)))))))).
tff(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),f3881(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a))))))).
tff(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),f3883(V0a))))))).
tff(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),f3884(V0a))))))).
tff(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),f3885(V0a))))))).
tff(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),f3886(V1b,V0a)))))))).
tff(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),f3887(V0a))))))).
tff(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),f3888(V1b,V0a)))))))).
tff(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),f3889(V0a))))))).
tff(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)),f3891(V0s)))))))))).
tff(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),f3892(V0b))))))).
tff(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),f3263(V0a))))))).
tff(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),f3893))))).
tff(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),f3879(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3894(V1b,V0a)))))))).
tff(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),f3880(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3895(V1b,V0a)))))))).
tff(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),f3884(V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3896(V0a)))))).
tff(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),f3885(V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3897(V0a)))))).
tff(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),f3886(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3898(V1b,V0a)))))))).
tff(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),f3888(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3899(V1b,V0a)))))))).
tff(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),f3887(V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3900(V0a)))))).
tff(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),f3889(V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3901(V0a)))))).
tff(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),f3881(V1b,V0a))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))).
tff(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),f3879(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)))))))).
tff(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),f3880(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)))))))).
tff(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),f3886(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)))))))).
tff(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),f3888(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)))))))).
tff(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),f3883(V0a))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(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),f3884(V0a)))))))).
tff(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),f3885(V0a)))))))).
tff(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),f3887(V0a)))))))).
tff(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),f3889(V0a)))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V2t)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V1s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V3x))),V2t)))))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),ap(c_2Ereal__topology_2Eclosure,V1s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V4x))),V2t))))))))))))).
tff(conj_thm_2Ereal__topology_2EFORALL__IN__CLOSURE__EQ,lemma,(( ! [V0f:$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)))))))))))))).
tff(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))))))))))))).
tff(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)))))))))))))).
tff(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))))))))))).
tff(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:$i] : (mem(V2h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(c_2Ereal__topology_2Eclosure,V0s))) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2h),ap(c_2Ereal__topology_2Eclosure,V0s))) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s)) => (surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V3x))) = surj__ty_2Erealax_2Ereal(ap(V2h,inj__ty_2Erealax_2Ereal(V3x)))))))) => ( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),ap(c_2Ereal__topology_2Eclosure,V0s))) => (surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V4x))) = surj__ty_2Erealax_2Ereal(ap(V2h,inj__ty_2Erealax_2Ereal(V4x))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__CONSTANT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3902(V1s,V0f,V2a)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__CONSTANT,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$i] : (mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V2s)) & p(ap(c_2Ereal__topology_2EClosed,V2s))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3903(V2s,V1f,V0a)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3e:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),ap(c_2Ereal__topology_2Eclosure,V1s))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e)))) => (? [V4d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4d))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5y)),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5y)),inj__ty_2Erealax_2Ereal(V2x)))),inj__ty_2Erealax_2Ereal(V4d)))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V5y))),ap(V0f,inj__ty_2Erealax_2Ereal(V2x))))),inj__ty_2Erealax_2Ereal(V3e))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__SEQUENTIALLY,lemma,(( ! [V0f:$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))))))))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ((p(ap(c_2Ereal__topology_2Ecauchy,V2x)) & ( ! [V3n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,inj__ty_2Enum_2Enum(V3n))),V1s)))) => p(ap(c_2Ereal__topology_2Ecauchy,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IMP__CAUCHY__CONTINUOUS,lemma,(( ! [V0f:$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))))))))))))).
tff(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:$i] : (mem(V8y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ((( ! [V9n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V8y,inj__ty_2Enum_2Enum(V9n))),V1s))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V8y),inj__ty_2Erealax_2Ereal(V4a)),c_2Ereal__topology_2Esequentially))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V8y)),inj__ty_2Erealax_2Ereal(V7l)),c_2Ereal__topology_2Esequentially)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__EXTENDS__TO__CLOSURE,lemma,(( ! [V0f:$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:$i] : (mem(V4g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4g),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V1s)) => (surj__ty_2Erealax_2Ereal(ap(V4g,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__EXTENDS__TO__CLOSURE,lemma,(( ! [V0f:$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:$i] : (mem(V4h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4h),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V1s)) => (surj__ty_2Erealax_2Ereal(ap(V4h,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))))))) => ( ! [V6x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),ap(c_2Ereal__topology_2Eclosure,V1s))) => (surj__ty_2Erealax_2Ereal(ap(V4h,inj__ty_2Erealax_2Ereal(V6x))) = surj__ty_2Erealax_2Ereal(ap(V2g,inj__ty_2Erealax_2Ereal(V6x)))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__IMP__CONTINUOUS,lemma,(( ! [V0f:$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))))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V3h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V2g),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0x)))) & (p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V3h),ap(c_2Ereal__topology_2Eat,ap(V2g,inj__ty_2Erealax_2Ereal(V0x))))) & (( ! [V4y:tp__ty_2Erealax_2Ereal] : (surj__ty_2Erealax_2Ereal(ap(V2g,ap(V3h,inj__ty_2Erealax_2Ereal(V4y)))) = V4y)) & (surj__ty_2Erealax_2Ereal(ap(V3h,ap(V2g,inj__ty_2Erealax_2Ereal(V0x)))) = V0x)))) => (p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1f),ap(c_2Ereal__topology_2Eat,ap(V2g,inj__ty_2Erealax_2Ereal(V0x))))) <=> p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),f3904(V2g,V1f)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0x))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__TRANSLATION,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1z:tp__ty_2Erealax_2Ereal] : ( ! [V2f:$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),f3905(V0a,V2f)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1z))))))))))).
tff(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))))))))))).
tff(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))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__AVOID,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x)))) & (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V1x))) = V2a))) => (? [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V4y)))),inj__ty_2Erealax_2Ereal(V3e))) => (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4y))) = V2a))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__AVOID,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V2s)) & (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V1x))) = V3a)))) => (? [V4e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4e))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5y)),V2s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V5y)))),inj__ty_2Erealax_2Ereal(V4e)))) => (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5y))) = V3a))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__AVOID,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & (p(ap(c_2Ereal__topology_2EOpen,V2s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V2s)) & (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V1x))) = V3a))))) => (? [V4e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4e))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V5y)))),inj__ty_2Erealax_2Ereal(V4e))) => (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5y))) = V3a))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN__CASES,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Econnected,V1s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3902(V1s,V0f,V2a)))))) => (( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V1s)) => (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = V2a)))) | ( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V1s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) = V2a)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Econnected,V1s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3902(V1s,V0f,V2a)))) & (? [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V1s)) & (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = V2a)))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V1s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) = V2a))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Econnected,V1s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & (p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3902(V1s,V0f,V2a)))) & (? [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V1s)) & (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) = V2a)))))) => ( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V1s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) = V2a))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__SCALING,lemma,(( ! [V0s:$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),f3906(V1c)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__NEGATIONS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__TRANSLATION,lemma,(( ! [V0s:$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),f3641(V1a)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__TRANSLATION__EQ,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))) <=> p(ap(c_2Ereal__topology_2EOpen,V1s)))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__AFFINITY,lemma,(( ! [V0s:$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),f3907(V2c,V1a)),V0s)))))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__TRANSLATION,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3908(V0a)),ap(c_2Ereal__topology_2Einterior,V1s)))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__SUMS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EOpen,V0s)) | p(ap(c_2Ereal__topology_2EOpen,V1t))) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2EUPPER__HEMICONTINUOUS,lemma,(( ! [V0f:$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:$i] : (mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3909(V2s,V0f,V4u))))))) <=> ( ! [V6u:$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),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),f3910(V2s,V6u,V0f))))))))))))))))).
tff(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),f3909(V2s,V0f,V4u))))))) <=> ( ! [V6u:$i] : (mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V6u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3910(V2s,V6u,V0f))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__MAP__IFF__LOWER__HEMICONTINUOUS__PREIMAGE,lemma,(( ! [V0f:$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),f3911(V2t,V0f,V1s,V4u))))))))))))))))).
tff(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:$i] : (mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3911(V2t,V0f,V1s,V4u))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS__EXPLICIT,lemma,(( ! [V0f:$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:$i] : (mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3909(V2s,V0f,V4u))))))) & ( ! [V6u:$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),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),f3912(V2s,V0f,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)))))))))))))))))))))))).
tff(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))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__IN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y))) => (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),V0s)))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL,lemma,(( ! [V0s:$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))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V1x))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM,lemma,(( ! [V0s:$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)))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__TRANS,lemma,(( ! [V0z:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V3y))) & p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),inj__ty_2Erealax_2Ereal(V3y)),inj__ty_2Erealax_2Ereal(V0z)))) => p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V0z))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__OF__SUBSET,lemma,(( ! [V0y:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V2t)) & p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V0y)))) => p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V2t),inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V0y)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SET,lemma,(( ! [V0s:$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),f3914(V0s,V1x)))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__BIGUNION,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),f3915(V1x,V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SUBSET,lemma,(( ! [V0s:$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))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT__SET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => (ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)) = V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIV,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : (ap(ap(c_2Ereal__topology_2Econnected__component,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),inj__ty_2Erealax_2Ereal(V0x)) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__UNIV,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__SELF,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Econnected,V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s))) => (ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)) = V0s))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTED__COMPONENT,lemma,(( ! [V0s:$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))))))))))).
tff(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)))))))))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EMPTY,lemma,(( ! [V0s:$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))))))))).
tff(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))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)))) => (ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V2y)) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__CONNECTED__COMPONENT,lemma,(( ! [V0s:$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)))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__DISJOINT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1a))),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V2b)))) <=> (~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V2b)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__NONOVERLAP,lemma,(( ! [V0s:$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))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__OVERLAP,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ((~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1a))),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V2b))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) <=> (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2b)),V0s)) & (ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1a)) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V2b)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM__EQ,lemma,(( ! [V0s:$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)))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V2y))) <=> (((~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s))) & (~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),V0s)))) | (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),V0s)) & p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENT__EQ,lemma,(( ! [V0s:$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))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__IDEMP,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (ap(ap(c_2Ereal__topology_2Econnected__component,ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x))),inj__ty_2Erealax_2Ereal(V1x)) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIQUE,lemma,(( ! [V0s:$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))))))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2EBIGUNION__CONNECTED__COMPONENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal),f3916(V0s))) = V0s))))).
tff(conj_thm_2Ereal__topology_2ECOMPLEMENT__CONNECTED__COMPONENT__BIGUNION,lemma,(( ! [V0s:$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),f3917(V0s))),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__CONNECTED__COMPONENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),inj__ty_2Erealax_2Ereal(V1x))))))))).
tff(conj_thm_2Ereal__topology_2EBIGUNION__DIFF,lemma,(( ! [A_27a:del]: ( ! [V0s:$i] : (mem(V0s,arr(arr(A_27a,bool),bool)) => ( ! [V1t:$i] : (mem(V1t,arr(A_27a,bool)) => (ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s)),V1t) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3918(A_27a,V1t,V0s))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IN__CONNECTED__COMPONENT,lemma,(( ! [V0s:$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),f3919(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)))))))))).
tff(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:$i] : (mem(V8t,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V8t)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V7a)),V8t)) & ( ! [V9x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V9x)),V8t)) => p(ap(ap(V0R,inj__ty_2Erealax_2Ereal(V7a)),inj__ty_2Erealax_2Ereal(V9x))))))))))))) => ( ! [V10a:tp__ty_2Erealax_2Ereal] : ( ! [V11b:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),inj__ty_2Erealax_2Ereal(V10a)),inj__ty_2Erealax_2Ereal(V11b))) => p(ap(ap(V0R,inj__ty_2Erealax_2Ereal(V10a)),inj__ty_2Erealax_2Ereal(V11b))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__INTERMEDIATE__SUBSET,lemma,(( ! [V0t:$i] : (mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1u:$i] : (mem(V1u,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V1u),inj__ty_2Erealax_2Ereal(V2a))),V0t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0t),V1u))) => (ap(ap(c_2Ereal__topology_2Econnected__component,V0t),inj__ty_2Erealax_2Ereal(V2a)) = ap(ap(c_2Ereal__topology_2Econnected__component,V1u),inj__ty_2Erealax_2Ereal(V2a))))))))))).
tff(ax_thm_2Ereal__topology_2Ecomponents,axiom,( ! [V0s:$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),f3916(V0s)))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2EBIGUNION__COMPONENTS,lemma,(( ! [V0u:$i] : (mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => (V0u = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecomponents,V0u))))))).
tff(conj_thm_2Ereal__topology_2EPAIRWISE__DISJOINT__COMPONENTS,lemma,(( ! [V0u:$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))))))).
tff(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)))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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:$i] : (mem(V2c_27,arr(ty_2Erealax_2Ereal,bool)) => (((~~ (V2c_27 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1c),V2c_27)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2c_27),V0s)) & p(ap(c_2Ereal__topology_2Econnected,V2c_27))))) => (V2c_27 = V1c)))))))))))))).
tff(conj_thm_2Ereal__topology_2EJOINABLE__COMPONENTS__EQ,lemma,(( ! [V0s:$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:$i] : (mem(V3c2,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Econnected,V1t)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c1),ap(c_2Ereal__topology_2Ecomponents,V0s))) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3c2),ap(c_2Ereal__topology_2Ecomponents,V0s))) & ((~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2c1),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) & (~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3c2),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))) => (V2c1 = V3c2)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPONENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:$i] : (mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1c))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__COMPONENTS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:$i] : (mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s)))) => p(ap(c_2Ereal__topology_2EClosed,V1c))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__COMPONENTS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:$i] : (mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s)))) => p(ap(c_2Ereal__topology_2Ecompact,V1c))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__GEN,lemma,(( ! [V0f:$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))))))))).
tff(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:$i] : (mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),ap(c_2Ereal__topology_2Ecomponents,V1s))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2c)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPONENTS__NONOVERLAP,lemma,(( ! [V0s:$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)))))))))))).
tff(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:$i] : (mem(V2c_27,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c_27),ap(c_2Ereal__topology_2Ecomponents,V0s)))) => ((V1c = V2c_27) <=> (~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1c),V2c_27) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__EMPTY,lemma,(( ! [V0s:$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))))))).
tff(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))))).
tff(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:$i] : (mem(V2c_27,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c_27),ap(c_2Ereal__topology_2Ecomponents,V0s)))) => (V1c = V2c_27))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__N__EXISTS,lemma,((( ! [V0s:$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)))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((ap(c_2Ereal__topology_2Ecomponents,V0s) = ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),V0s),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool)))) <=> (p(ap(c_2Ereal__topology_2Econnected,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__EXISTS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((? [V1a:$i] : (mem(V1a,arr(ty_2Erealax_2Ereal,bool)) & (ap(c_2Ereal__topology_2Ecomponents,V0s) = ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),V1a),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool)))))) <=> (p(ap(c_2Ereal__topology_2Econnected,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIV,lemma,((ap(c_2Ereal__topology_2Ecomponents,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) = ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool)))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING,lemma,(( ! [V0s:$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)))))))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2EIN__COMPONENTS__SELF,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V0s),ap(c_2Ereal__topology_2Ecomponents,V0s))) <=> (p(ap(c_2Ereal__topology_2Econnected,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPONENTS__MAXIMAL,lemma,(( ! [V0s:$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))))))))))).
tff(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)))))))).
tff(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))))))))))))))))).
tff(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:$i] : (mem(V2c,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),ap(c_2Ereal__topology_2Ecomponents,V0s))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2c)))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPONENTS__INTERMEDIATE__SUBSET,lemma,(( ! [V0s:$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)))))))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & (p(ap(c_2Ereal__topology_2Econnected,V2c)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2c),V0u)) & (~~ (ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2c),V1s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2c),V1s))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOPEN__BIGUNION__COMPONENTS,lemma,(( ! [V0u:$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)))))))))))).
tff(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)))))))))).
tff(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))))))))))))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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),f3920(A_27a,V1f,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),f3922(A_27a,V3s,V2g,V1f,V5e)),V0net)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V3s)))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => (? [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) & ( ! [V3x_27:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V3x_27)),inj__ty_2Erealax_2Ereal(V1x)))),inj__ty_2Erealax_2Ereal(V2e))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x_27)),V0s))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED,lemma,(( ! [V0s:$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))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__RANGE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x)))) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (? [V3d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3d))) & ( ! [V4x_27:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V4x_27)),inj__ty_2Erealax_2Ereal(V1x)))),inj__ty_2Erealax_2Ereal(V3d))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Erealax_2Ereal(V4x_27))),ap(V0f,inj__ty_2Erealax_2Ereal(V1x))))),inj__ty_2Erealax_2Ereal(V2e))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__RANGE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) => ( ! [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) => (? [V4d:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4d))) & ( ! [V5x_27:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x_27)),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V5x_27)),inj__ty_2Erealax_2Ereal(V2x)))),inj__ty_2Erealax_2Ereal(V4d)))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Erealax_2Ereal(V5x_27))),ap(V0f,inj__ty_2Erealax_2Ereal(V2x))))),inj__ty_2Erealax_2Ereal(V3e)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ABS__COMPOSE,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3734(A_27a,V1f)),V0net)))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ABS__COMPOSE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3814(V0f)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__ABS,lemma,(( ! [V0x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),c_2Ereal_2Eabs),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V0x))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__DIST,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),f3923(V0a)),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V1x)))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DIST,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3923(V0a)),V1s))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__ATTAINS__SUP,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (? [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) & ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2y)),inj__ty_2Erealax_2Ereal(V1x)))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__ATTAINS__INF,lemma,(( ! [V0s:$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)))))))))))).
tff(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))))))))))))))).
tff(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))))))))))))))).
tff(conj_thm_2Ereal__topology_2EDISTANCE__ATTAINS__SUP,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Ecompact,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (? [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3y)))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2x))))))))))))))).
tff(conj_thm_2Ereal__topology_2EDISTANCE__ATTAINS__INF,lemma,(( ! [V0s:$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))))))))))))))).
tff(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:$i] : (mem(V3c,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V4d:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V3c),inj__ty_2Erealax_2Ereal(V4d)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3924(A_27a,V3c,V1f)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V4d)),inj__ty_2Erealax_2Ereal(V2l))),V0net)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__VMUL,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1c:$i] : (mem(V1c,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2d:tp__ty_2Erealax_2Ereal] : ( ! [V3v:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1c),inj__ty_2Erealax_2Ereal(V2d)),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3925(A_27a,V1c,V3v)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2d)),inj__ty_2Erealax_2Ereal(V3v))),V0net)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__VMUL,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1c:$i] : (mem(V1c,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2v:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1c),V0net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3926(A_27a,V1c,V2v)),V0net))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__MUL,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3927(A_27a,V2c,V1f)),V0net)))))))))))).
tff(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,f3820(V1c,V2v)),V0s)))))))))).
tff(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:$i] : (mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1c),V0s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V0s))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3928(V1c,V2f)),V0s))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__POW,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3825(A_27a,V1f)),V0net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3929(A_27a,V1f,V2n)),V0net))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__POW,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3930(V0f)),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3931(V0f,V2n)),V1s)))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__PRODUCT,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$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:$i] : (mem(V3i,A_27b) => (p(ap(ap(c_2Ebool_2EIN(A_27b),V3i),V2t)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3932(A_27a,A_27b,V1f,V3i)),V0net)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3933(A_27a,A_27b,V1f,V2t)),V0net))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__PRODUCT,lemma,(( ! [A_27a:del]: ( ! [V0f:$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:$i] : (mem(V3i,A_27a) => (p(ap(ap(c_2Ebool_2EIN(A_27a),V3i),V2t)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3934(A_27a,V0f,V3i)),V1s)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3935(A_27a,V0f,V2t)),V1s)))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__INV,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) & (~~ (V2l = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Erealax_2Einv),V1f)),ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V2l))),V0net))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__INV,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) & (~~ (surj__ty_2Erealax_2Ereal(ap(V1f,ap(c_2Ereal__topology_2Enetlimit(A_27a),V0net))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Erealax_2Einv),V1f)),V0net)))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN__INV,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a))),V1s))) & (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V2a))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Erealax_2Einv),V0f)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2a))),V1s))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__INV,lemma,(( ! [V0f:$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)))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__SCALING,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Econnected,V0s)) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__NEGATIONS,lemma,(( ! [V0s:$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),f3685),V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__SCALING,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__NEGATIONS,lemma,(( ! [V0s:$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),f3685),V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__AFFINITY,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3907(V2c,V1a)),V0s)))))))))).
tff(ax_thm_2Ereal__topology_2Ediameter,axiom,( ! [V0s:$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)),f3937(V0s)))))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__BOUNDED,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => (( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),V0s))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y)))),ap(c_2Ereal__topology_2Ediameter,V0s)))))) & ( ! [V3d:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3d))) & p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V3d)),ap(c_2Ereal__topology_2Ediameter,V0s)))) => (? [V4x:tp__ty_2Erealax_2Ereal] : (? [V5y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5y)),V0s)) & p(ap(ap(c_2Ereal_2Ereal__gt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V5y)))),inj__ty_2Erealax_2Ereal(V3d))))))))))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__BOUNDED__BOUND,lemma,(( ! [V0s:$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)))))))))).
tff(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)))))))))).
tff(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)))))).
tff(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))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__POS__LE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Ereal__topology_2Ediameter,V0s)))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ediameter,V0s)),ap(c_2Ereal__topology_2Ediameter,V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__CLOSURE,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL__NONEMPTY,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (? [V1z:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1z)),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1z)),ap(c_2Ereal__topology_2Ediameter,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL,lemma,(( ! [V0s:$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))))))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__EQ__0,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ((surj__ty_2Erealax_2Ereal(ap(c_2Ereal__topology_2Ediameter,V0s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) <=> ((V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) | (? [V1a:tp__ty_2Erealax_2Ereal] : (V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))))))))).
tff(conj_thm_2Ereal__topology_2EDIAMETER__LE,lemma,(( ! [V0s:$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))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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)),f3612(V1t,V0s))))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2Ediameter,V0s)),ap(c_2Ereal__topology_2Ediameter,V1t))))))))))).
tff(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:$i] : (mem(V5b,arr(ty_2Erealax_2Ereal,bool)) & (p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V5b),V1c)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V5b))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__SCALING,lemma,(( ! [V0s:$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),f3906(V1c)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__NEGATIONS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EClosed,V0s)) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__CLOSED__SUMS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__COMPACT__SUMS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V1t))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__SUMS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) | p(ap(c_2Ereal__topology_2Ebounded__def,V1t))) => (ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s)))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3939(V1t,V0s))))))))))).
tff(conj_thm_2Ereal__topology_2ECOMPACT__CLOSED__DIFFERENCES,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3791(V1t,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__COMPACT__DIFFERENCES,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V1t))) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3791(V1t,V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2ETRANSLATION__DIFF,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => (ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3908(V0a)),ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V1s),V2t)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3940(V0a)),V1s)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3941(V0a)),V2t)))))))))).
tff(conj_thm_2Ereal__topology_2ESEPARATE__POINT__CLOSED,lemma,(( ! [V0s:$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))))))))))))))).
tff(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))))))))))))))))).
tff(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))))))))))))))))).
tff(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),f3942(V1f))) = V0s) & ( ! [V6k:$i] : (mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V6k)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6k),V0s))) => (? [V7N:tp__ty_2Enum_2Enum] : ( ! [V8n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V8n)),inj__ty_2Enum_2Enum(V7N))) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6k),ap(V1f,inj__ty_2Enum_2Enum(V8n))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__UNION__COMPACT__SUBSETS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) => (? [V1f:$i] : (mem(V1f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) & (( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(c_2Ereal__topology_2Ecompact,ap(V1f,inj__ty_2Enum_2Enum(V2n))))) & (( ! [V3n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V1f,inj__ty_2Enum_2Enum(V3n))),V0s))) & (( ! [V4n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V1f,inj__ty_2Enum_2Enum(V4n))),ap(c_2Ereal__topology_2Einterior,ap(V1f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V4n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))))) & ((ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum),f3942(V1f))) = V0s) & ( ! [V6k:$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))))))))))))))))))))).
tff(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),f3943(V1b,V0a)))))).
tff(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),f3944(V0l))))).
tff(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),f3943(V1b,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_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3945(V1b,V0a)))))))).
tff(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))))))))))).
tff(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)))))))))))).
tff(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),f3946),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),f3947),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)))))))))).
tff(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)))))))).
tff(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))))))))).
tff(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))))))))))))))).
tff(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)))))).
tff(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))))))))))))))).
tff(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)))))))))))))))).
tff(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))))))))))))).
tff(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)))))))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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))))))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL__SYMMETRIC,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => (? [V1a:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1a))),inj__ty_2Erealax_2Ereal(V1a))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL,lemma,(( ! [V0s:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL__SYMMETRIC,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => (? [V1a:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1a))),inj__ty_2Erealax_2Ereal(V1a))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL,lemma,(( ! [V0s:$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))))))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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),f3689(V0c)),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),f3690(V4c)),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))))))))))).
tff(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))))))).
tff(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)))))).
tff(conj_thm_2Ereal__topology_2EIMAGE__STRETCH__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2m:$i] : (mem(V2m,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => (ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3949(V2m)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3950(V1b,V2m,V0a))),ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3951(V1b,V2m,V0a)))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))))))))).
tff(conj_thm_2Ereal__topology_2EINTERVAL__IMAGE__STRETCH__INTERVAL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : ( ! [V2m:$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),f3953(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),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))))))))))))).
tff(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),f3789(V0a)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3955(V1b,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),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)))))))))))).
tff(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)),f3957(V3d,V2c,V1b,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),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)),f3959(V9d,V8c,V7b,V6a))) = 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))))))))))))).
tff(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))))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) => (? [V2a:tp__ty_2Erealax_2Ereal] : (? [V3b:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__OPEN__INTERVAL,lemma,(( ! [V0s:$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)))))))))))).
tff(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))))))))))).
tff(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),i(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))))) = 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)))))))))))).
tff(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))))))))))))).
tff(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)))))))).
tff(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)))))))))).
tff(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)))))))).
tff(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)))))))))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))))).
tff(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))))))).
tff(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)))))))).
tff(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))))))))).
tff(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)))))))).
tff(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)))))))))).
tff(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))))))))).
tff(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),f3960(V0a,V1b,V2v)))))) & ( ! [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),f3961(V4a,V5b,V6v)))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2EIS__INTERVAL__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Eis__interval,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2EIS__INTERVAL__UNIV,lemma,(p(ap(c_2Ereal__topology_2Eis__interval,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
tff(conj_thm_2Ereal__topology_2EIS__INTERVAL__POINTWISE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Eis__interval,V0s)) => ((? [V2a:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),V0s)) & (V2a = V1x))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EIS__INTERVAL__COMPACT,lemma,(( ! [V0s:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2EIS__INTERVAL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Eis__interval,V0s)) <=> ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2b)),V0s)) & (p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3x))) & p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b)))))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))))))))))).
tff(conj_thm_2Ereal__topology_2EIS__INTERVAL__CASES,lemma,(( ! [V0s:$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),f3962(V1a)))) | ((? [V3a:tp__ty_2Erealax_2Ereal] : (V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3963(V3a)))) | ((? [V5b:tp__ty_2Erealax_2Ereal] : (V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3964(V5b)))) | ((? [V7b:tp__ty_2Erealax_2Ereal] : (V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3965(V7b)))) | ((? [V9a:tp__ty_2Erealax_2Ereal] : (? [V10b:tp__ty_2Erealax_2Ereal] : (V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3966(V10b,V9a))))) | ((? [V12a:tp__ty_2Erealax_2Ereal] : (? [V13b:tp__ty_2Erealax_2Ereal] : (V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3967(V13b,V12a))))) | ((? [V15a:tp__ty_2Erealax_2Ereal] : (? [V16b:tp__ty_2Erealax_2Ereal] : (V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3968(V16b,V15a))))) | (? [V18a:tp__ty_2Erealax_2Ereal] : (? [V19b:tp__ty_2Erealax_2Ereal] : (V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3969(V19b,V18a))))))))))))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2EINTERVAL__SUBSET__IS__INTERVAL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Eis__interval,V0s)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),V0s)) <=> ((ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) | (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2b)),V0s)))))))))))).
tff(conj_thm_2Ereal__topology_2EINTERVAL__CONTAINS__COMPACT__NEIGHBOURHOOD,lemma,(( ! [V0s:$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)))))))))))))))))).
tff(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))))))).
tff(conj_thm_2Ereal__topology_2EIS__INTERVAL__SCALING,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1c:tp__ty_2Erealax_2Ereal] : (p(ap(c_2Ereal__topology_2Eis__interval,V0s)) => p(ap(c_2Ereal__topology_2Eis__interval,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EIS__INTERVAL__SCALING__EQ,lemma,(( ! [V0s:$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),f3906(V1c)),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))))))))).
tff(conj_thm_2Ereal__topology_2ECARD__FRONTIER__INTERVAL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2Eis__interval,V0s)) => (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s))) & p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__COMPONENT__UBOUND,lemma,(( ! [A_27a:del]: ( ! [A_27b:del]: ( ! [V0net:$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),f3970(A_27a,V1f,V3b)),V0net)))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2l)),inj__ty_2Erealax_2Ereal(V3b)))))))))))))))).
tff(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),f3971(A_27a,V1f,V3b)),V0net)))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3b)),inj__ty_2Erealax_2Ereal(V2l))))))))))))).
tff(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:$i] : (mem(V2i,A_27b) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l)),V0net)) & ((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3972(A_27a,V1f,V4b)),V0net)))) => (V3l = V4b)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__COMPONENT__LE,lemma,(( ! [A_27a:del]: ( ! [V0net:$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),f3973(A_27a,V1f,V2g)),V0net))))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3l)),inj__ty_2Erealax_2Ereal(V4m))))))))))))))).
tff(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:$i] : (mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ( ! [V4m:tp__ty_2Erealax_2Ereal] : (((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V3l)),V0net)) & (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),inj__ty_2Erealax_2Ereal(V4m)),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3973(A_27a,V1f,V2g)),V0net))))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3l)),inj__ty_2Erealax_2Ereal(V4m))))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__DROP__UBOUND,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) & ((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3974(A_27a,V1f,V3b)),V0net)))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2l)),inj__ty_2Erealax_2Ereal(V3b))))))))))))).
tff(conj_thm_2Ereal__topology_2ELIM__DROP__LBOUND,lemma,(( ! [A_27a:del]: ( ! [V0net:$i] : (mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ! [V1f:$i] : (mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ! [V2l:tp__ty_2Erealax_2Ereal] : ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),inj__ty_2Erealax_2Ereal(V2l)),V0net)) & ((~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net))) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3971(A_27a,V1f,V3b)),V0net)))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3b)),inj__ty_2Erealax_2Ereal(V2l))))))))))))).
tff(conj_thm_2Ereal__topology_2EIMAGE__CLOSURE__SUBSET,lemma,(( ! [V0f:$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))))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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)))))))))))).
tff(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)))))))))))).
tff(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))))))))))))).
tff(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))))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t)))) <=> (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V3s))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,inj__ty_2Erealax_2Ereal(V2x))),V4t))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION,lemma,(( ! [V0f:$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)))))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EClosed,V3s)) & (p(ap(c_2Ereal__topology_2EClosed,V4t)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V3s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V4t)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V3s)) & (~~ p(ap(V0P,inj__ty_2Erealax_2Ereal(V5x))))) | (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V4t)) & p(ap(V0P,inj__ty_2Erealax_2Ereal(V5x))))) => (surj__ty_2Erealax_2Ereal(ap(V1f,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V2g,inj__ty_2Erealax_2Ereal(V5x)))))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3874(V1f,V0P,V2g)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL,lemma,(( ! [V0t:$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)))))))))))).
tff(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:$i] : (mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))),V3s)) & (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))),V4t)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V3s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V4t)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V3s)) & (~~ p(ap(V0P,inj__ty_2Erealax_2Ereal(V5x))))) | (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V4t)) & p(ap(V0P,inj__ty_2Erealax_2Ereal(V5x))))) => (surj__ty_2Erealax_2Ereal(ap(V1f,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V2g,inj__ty_2Erealax_2Ereal(V5x)))))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3874(V1f,V0P,V2g)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LE,lemma,(( ! [V0f:$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:$i] : (mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V4a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3975(V3s,V2h,V4a)))) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3976(V3s,V2h,V4a)))) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2h),V3s)) & ( ! [V7t:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V7t)),V3s)) & (surj__ty_2Erealax_2Ereal(ap(V2h,inj__ty_2Erealax_2Ereal(V7t))) = V4a)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V7t))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V7t))))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3977(V0f,V4a,V2h,V1g)),V3s)))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__1,lemma,(( ! [V0f:$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),f3978(V3a,V2s)))) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3979(V3a,V2s)))) & (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,f3980(V0f,V3a,V1g)),V2s)))))))))))).
tff(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:$i] : (mem(V4g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4g),V1s)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4g),V1s)),V3u)) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V2t)) => (surj__ty_2Erealax_2Ereal(ap(V4g,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__IVT__HYPERPLANE,lemma,(( ! [V0s:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2ECONNECTED__IVT__COMPONENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : ( ! [V3a:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Econnected,V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),V0s)) & (p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V3a))) & p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3a)),inj__ty_2Erealax_2Ereal(V2y))))))) => (? [V4z:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4z)),V0s)) & (V4z = V3a))))))))))).
tff(conj_thm_2Ereal__topology_2EMAPPING__CONNECTED__ONTO__SEGMENT,lemma,(( ! [V0s:$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:$i] : (mem(V4f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4f),V0s)) & (ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4f),V0s) = ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__INCREASING__CONVERGENT,lemma,(( ! [V0s:$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),f3981(V0s)))) & ( ! [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)))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__DECREASING__CONVERGENT,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ((p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f3981(V0s)))) & ( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n)))),ap(V0s,inj__ty_2Enum_2Enum(V2n)))))) => (? [V3l:tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),inj__ty_2Erealax_2Ereal(V3l)),c_2Ereal__topology_2Esequentially)))))))).
tff(ax_thm_2Ereal__topology_2Ehomeomorphism,axiom,( ! [V0s:$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:$i] : (mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2f),V3g))) <=> (( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s)) => (surj__ty_2Erealax_2Ereal(ap(V3g,ap(V2f,inj__ty_2Erealax_2Ereal(V4x)))) = V4x))) & ((ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V0s) = V1t) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V0s)) & (( ! [V5y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5y)),V1t)) => (surj__ty_2Erealax_2Ereal(ap(V2f,ap(V3g,inj__ty_2Erealax_2Ereal(V5y)))) = V5y))) & ((ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3g),V1t) = V0s) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V3g),V1t))))))))))))))))).
tff(ax_thm_2Ereal__topology_2Ehomeomorphic,axiom,( ! [V0s:$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:$i] : (mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2f),V3g))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM,lemma,(( ! [V0s:$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:$i] : (mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2f),V3g))) <=> (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V0s)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V0s)),V1t)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V3g),V1t)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3g),V1t)),V0s)) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s)) => (surj__ty_2Erealax_2Ereal(ap(V3g,ap(V2f,inj__ty_2Erealax_2Ereal(V4x)))) = V4x))) & ( ! [V5y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5y)),V1t)) => (surj__ty_2Erealax_2Ereal(ap(V2f,ap(V3g,inj__ty_2Erealax_2Ereal(V5y)))) = V5y))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__OF__SUBSETS,lemma,(( ! [V0f:$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:$i] : (mem(V5t_27,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4s_27),V2s)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5t_27),V3t)) & (ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4s_27) = V5t_27)))) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V4s_27),V5t_27)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g)))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__ID,lemma,(( ! [V0s:$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)),i(ty_2Erealax_2Ereal)),i(ty_2Erealax_2Ereal)))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__REFL,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V0s)))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__SYM,lemma,(( ! [V0f:$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:$i] : (mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V3t),V2s)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V1g),V0f)))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SYM,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1t),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__COMPOSE,lemma,(( ! [V0f:$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))))))))))))))))))))).
tff(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:$i] : (mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1t),V2u))) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V2u))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__IMP__CARD__EQ,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0s),V1t))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITENESS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) => (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) <=> p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__EMPTY,lemma,((( ! [V0s:$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:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),V1s)) <=> (V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__MINIMAL,lemma,(( ! [V0s:$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:$i] : (mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & (( ! [V4x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s)) => (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2f,inj__ty_2Erealax_2Ereal(V4x))),V1t)) & (surj__ty_2Erealax_2Ereal(ap(V3g,ap(V2f,inj__ty_2Erealax_2Ereal(V4x)))) = V4x)))) & (( ! [V5y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5y)),V1t)) => (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V3g,inj__ty_2Erealax_2Ereal(V5y))),V0s)) & (surj__ty_2Erealax_2Ereal(ap(V2f,ap(V3g,inj__ty_2Erealax_2Ereal(V5y)))) = V5y)))) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V0s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V3g),V1t)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__SELF,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V2x))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V3y)))) => (V2x = V3y))))) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__LEFT__EQ,lemma,(( ! [V0f:$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)))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V3x))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4y)))) => (V3x = V4y))))) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2t))) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__SELF,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1s:$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),f3789(V0a)),V1s)),V1s))))))).
tff(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),f3908(V0a)),V1s)),V2t)) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)))))))))).
tff(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),f3908(V0a)),V2t))) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)))))))))).
tff(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:$i] : (mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4u),V3t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3834(V2s,V0f,V4u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V4u))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__LEFT,lemma,(( ! [V0c:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0c))) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3684(V0c)),V1s)),V2t)) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__RIGHT,lemma,(( ! [V0c:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0c))) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V2t:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3684(V0c)),V2t))) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t))) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) <=> (surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V0s)) = surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V1t))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE__STRONG,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) | p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t))) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) <=> (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & (p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) & (surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V0s)) = surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V1t))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SING,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1b)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))).
tff(conj_thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE__UNIQUE,lemma,(( ! [V0f:$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:$i] : (mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V3t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3835(V2s,V0f,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V5v)))))) & (( ! [V7v:$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),f3982(V2s,V1g,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))))))))))))))).
tff(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:$i] : (mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V5u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))))))))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__CLOSED__MAP,lemma,(( ! [V0f:$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))))))))))))).
tff(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:$i] : (mem(V6g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V6g)))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP,lemma,(( ! [V0f:$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)))))))))))))).
tff(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:$i] : (mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V4u))) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4u)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__CLOSED__MAP,lemma,(( ! [V0f:$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)))))))))))))))).
tff(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:$i] : (mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u)))))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP__EQ,lemma,(( ! [V0f:$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)))))))))))))))).
tff(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:$i] : (mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__IMP__CLOSED__MAP,lemma,(( ! [V0f:$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:$i] : (mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3u))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__IMP__QUOTIENT__MAP,lemma,(( ! [V0f:$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:$i] : (mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => (p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u))))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE,lemma,(( ! [V0f:$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)))))))))))).
tff(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)))))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Ecompact,V0s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) & ((ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V0s) = V2t) & ( ! [V3x:tp__ty_2Erealax_2Ereal] : ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s)) & (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4y)),V0s)) & (surj__ty_2Erealax_2Ereal(ap(V1f,inj__ty_2Erealax_2Ereal(V3x))) = surj__ty_2Erealax_2Ereal(ap(V1f,inj__ty_2Erealax_2Ereal(V4y)))))) => (V3x = V4y))))))) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V2t))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__SURJECTIVE,lemma,(( ! [V0f:$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:$i] : (mem(V7g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V3t),V4u)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V1g),V7g_27))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__INJECTIVE,lemma,(( ! [V0f:$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:$i] : (mem(V9g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V3t),V4u)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V1g),V9g_27))))))))))))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACTNESS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) => (p(ap(c_2Ereal__topology_2Ecompact,V0s)) <=> p(ap(c_2Ereal__topology_2Ecompact,V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__CONNECTEDNESS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1t:$i] : (mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) => (p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> p(ap(c_2Ereal__topology_2Econnected,V1t)))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING,lemma,(( ! [V0s:$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),f3906(V1c)),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V1a:tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3641(V1a)),V0s)))))))).
tff(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__AFFINITY,lemma,(( ! [V0s:$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),f3907(V2c,V1a)),V0s)))))))))).
tff(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)))))))))))))).
tff(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)))))))))))).
tff(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)))))))))))).
tff(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)))))))))))).
tff(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))))))))))).
tff(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)))))))))))).
tff(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)))))))).
tff(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)))))).
tff(conj_thm_2Ereal__topology_2EUNCOUNTABLE__REAL,lemma,((~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))))).
tff(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)))))))).
tff(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))))).
tff(conj_thm_2Ereal__topology_2EUNCOUNTABLE__EUCLIDEAN,lemma,((~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))))).
tff(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))))))))).
tff(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)))))))))))).
tff(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))))))).
tff(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)))))))).
tff(conj_thm_2Ereal__topology_2EUNCOUNTABLE__OPEN,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2EOpen,V0s)) & (~~ (V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) => (~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),V0s)))))))).
tff(conj_thm_2Ereal__topology_2ECARD__EQ__BALL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1r))) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1r)))),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))))))).
tff(conj_thm_2Ereal__topology_2ECARD__EQ__CBALL,lemma,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1r:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1r))) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1r)))),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)))))))).
tff(conj_thm_2Ereal__topology_2EFINITE__IMP__NOT__OPEN,lemma,(( ! [V0s:$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)))))))).
tff(conj_thm_2Ereal__topology_2EOPEN__IMP__INFINITE,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (p(ap(c_2Ereal__topology_2EOpen,V0s)) => ((V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) | (~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))))))))).
tff(conj_thm_2Ereal__topology_2EEMPTY__INTERIOR__FINITE,lemma,(( ! [V0s:$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))))))).
tff(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))))))))).
tff(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))))))))).
tff(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:$i] : (mem(V14f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ((p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V14f),V13s)) & p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V14f),V13s)))) => (? [V15a:tp__ty_2Erealax_2Ereal] : ( ! [V16x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V16x)),V13s)) => (surj__ty_2Erealax_2Ereal(ap(V14f,inj__ty_2Erealax_2Ereal(V16x))) = V15a)))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT__EQ,lemma,(( ! [V0s:$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)))))))))))))).
tff(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)))))))))))).
tff(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)))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Econnected,V2s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V2s)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V2s)),V0t)) & ( ! [V3y:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V0t)) => (ap(ap(c_2Ereal__topology_2Econnected__component,V0t),inj__ty_2Erealax_2Ereal(V3y)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))))) => (? [V4a:tp__ty_2Erealax_2Ereal] : ( ! [V5x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V2s)) => (surj__ty_2Erealax_2Ereal(ap(V1f,inj__ty_2Erealax_2Ereal(V5x))) = V4a))))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ((p(ap(c_2Ereal__topology_2Econnected,V1s)) & (p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ! [V2x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V1s)) => (? [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) & ( ! [V4y:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4y)),V1s)) & (~~ (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V4y))) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V2x)))))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3e)),ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Erealax_2Ereal(V4y))),ap(V0f,inj__ty_2Erealax_2Ereal(V2x)))))))))))))) => (? [V5a:tp__ty_2Erealax_2Ereal] : ( ! [V6x:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),V1s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V6x))) = V5a))))))))))).
tff(conj_thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT,lemma,(( ! [V0f:$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))))))))))).
tff(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),f3983(V1b,V0a))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3984(V3d,V2c))))))))))).
tff(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),f3882(V1b,V0a))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3985(V2c)))))))))).
tff(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),f3986(V2c))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3983(V1b,V0a)))))))))).
tff(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)))))))))))).
tff(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))))))))))).
tff(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)))))))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ! [V3x:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Ereal__topology_2Elinear,V0f)) & (p(ap(c_2Ereal__topology_2Elinear,V1g)) & ((ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1g) = c_2Ecombin_2EI(ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),ap(c_2Ereal__topology_2Einterior,V2s)))))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,inj__ty_2Erealax_2Ereal(V3x))),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)))))))))))))).
tff(conj_thm_2Ereal__topology_2ELINEAR__OPEN__MAPPING,lemma,(( ! [V0f:$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))))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(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)))))))))).
tff(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))))))))))).
tff(conj_thm_2Ereal__topology_2EINTERIOR__NEGATIONS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3946),ap(c_2Ereal__topology_2Einterior,V0s))))))).
tff(conj_thm_2Ereal__topology_2ESYMMETRIC__INTERIOR,lemma,(( ! [V0s:$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)))))))))).
tff(conj_thm_2Ereal__topology_2ECLOSURE__NEGATIONS,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => (ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3946),ap(c_2Ereal__topology_2Eclosure,V0s))))))).
tff(conj_thm_2Ereal__topology_2ESYMMETRIC__CLOSURE,lemma,(( ! [V0s:$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)))))))))).
tff(conj_thm_2Ereal__topology_2ESUBSPACE__SUBSTANDARD,lemma,(p(ap(c_2Ereal__topology_2Esubspace,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3987))))).
tff(conj_thm_2Ereal__topology_2ECLOSED__SUBSTANDARD,lemma,(p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3987))))).
tff(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),f3987))) = fo__c_2Enum_2E0))).
tff(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),f3988(V0m,V1c)),f3989(V1c,V0m)) = i(ty_2Erealax_2Ereal)) & (ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3990(V1c,V0m)),f3991(V0m,V1c)) = i(ty_2Erealax_2Ereal)))))))).
tff(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)))))))))))))).
tff(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))))))))))).
tff(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)))))))))))))).
tff(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))))))))))).
tff(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)))))))))))))).
tff(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))))))))).
tff(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),f3992(V2m,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))))))))))))).
tff(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),f3993(V2s,V0f)),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially))))))))).
tff(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),f3994(V1f,V0s))))))))).
tff(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))))))))).
tff(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)))))))))).
tff(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))))))))).
tff(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),f3995(V1s,V0f))),V1s)) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f))))))))).
tff(conj_thm_2Ereal__topology_2EFINITE__INTER__NUMSEG,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V0s),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__FROM,lemma,(( ! [V0f:$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),f3996(V2k,V0f)),inj__ty_2Erealax_2Ereal(V1l)),c_2Ereal__topology_2Esequentially))))))))).
tff(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)))))))))).
tff(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))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__TERMS__TOZERO,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2n:tp__ty_2Enum_2Enum] : (p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(c_2Ereal__topology_2Efrom,inj__ty_2Enum_2Enum(V2n)))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Ereal__topology_2Esequentially))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__FINITE,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,bool)) => (p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),V1s),V0f)),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__LINEAR,lemma,(( ! [V0f:$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,f3997(V0f,V1h)),ap(V1h,inj__ty_2Erealax_2Ereal(V2l))),V3s)))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__0,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,bool)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,k(ty_2Enum_2Enum,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0s)))))).
tff(conj_thm_2Ereal__topology_2ESERIES__ADD,lemma,(( ! [V0x:$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,f3998(V0x,V2y)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V3y0))),V4s))))))))))))).
tff(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:$i] : (mem(V4s,arr(ty_2Enum_2Enum,bool)) => ((p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),inj__ty_2Erealax_2Ereal(V1x0)),V4s)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V2y),inj__ty_2Erealax_2Ereal(V3y0)),V4s))) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f3999(V0x,V2y)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V3y0))),V4s))))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__CMUL,lemma,(( ! [V0x:$i] : (mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1x0:tp__ty_2Erealax_2Ereal] : ( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3s:$i] : (mem(V3s,arr(ty_2Enum_2Enum,bool)) => (p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),inj__ty_2Erealax_2Ereal(V1x0)),V3s)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f4000(V0x,V2c)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2c)),inj__ty_2Erealax_2Ereal(V1x0))),V3s))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__NEG,lemma,(( ! [V0x:$i] : (mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1x0:tp__ty_2Erealax_2Ereal] : ( ! [V2s:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => (p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),inj__ty_2Erealax_2Ereal(V1x0)),V2s)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f4001(V0x)),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1x0))),V2s)))))))))).
tff(conj_thm_2Ereal__topology_2ESUMS__IFF,lemma,(( ! [V0l:tp__ty_2Erealax_2Ereal] : ( ! [V1f:$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))))))))))))).
tff(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:$i] : (mem(V3k,arr(ty_2Enum_2Enum,bool)) => ((( ! [V4x:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4x)),V3k)) => (surj__ty_2Erealax_2Ereal(ap(V1f,inj__ty_2Enum_2Enum(V4x))) = surj__ty_2Erealax_2Ereal(ap(V2g,inj__ty_2Enum_2Enum(V4x)))))) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V1f),inj__ty_2Erealax_2Ereal(V0l)),V3k))) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V2g),inj__ty_2Erealax_2Ereal(V0l)),V3k)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMS__0,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,bool)) => (( ! [V2n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V2n)),V1s)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V2n))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V1s))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__FINITE__SUPPORT,lemma,(( ! [V0f:$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))))))))))).
tff(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,f4002(V0f)),inj__ty_2Erealax_2Ereal(V2l)),V1s)))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__DIFFS,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1k:tp__ty_2Enum_2Enum] : (p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Ereal__topology_2Esequentially)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f4003(V0f)),ap(V0f,inj__ty_2Enum_2Enum(V1k))),ap(c_2Ereal__topology_2Efrom,inj__ty_2Enum_2Enum(V1k)))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__TRIVIAL,lemma,(( ! [V0f:$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))))))).
tff(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,f4004(V0f,V1k)),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)))))))))).
tff(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:$i] : (mem(V3s,arr(ty_2Enum_2Enum,bool)) => ((p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V3s)) & (p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V3s),V2k)) & (( ! [V4x:tp__ty_2Enum_2Enum] : ((~~ p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4x)),V3s))) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V4x))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))) & (surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),V3s),V0f)) = V1l)))) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),inj__ty_2Erealax_2Ereal(V1l)),V2k)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMS__REINDEX,lemma,(( ! [V0k:tp__ty_2Enum_2Enum] : ( ! [V1a:$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,f4005(V0k,V1a)),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))))))))))))).
tff(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,f4005(V0k,V1a)),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),f4006(V0k)),V3s)))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => ((p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0f)) & p(ap(c_2Ereal__topology_2Elinear,V1h))) => p(ap(ap(c_2Ereal__topology_2Esummable,V2s),f4007(V0f,V1h)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__0,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,bool)) => p(ap(ap(c_2Ereal__topology_2Esummable,V0s),k(ty_2Enum_2Enum,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => ((p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0x)) & p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V1y))) => p(ap(ap(c_2Ereal__topology_2Esummable,V2s),f4008(V0x,V1y)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__SUB,lemma,(( ! [V0x:$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),f4009(V0x,V1y)))))))))))).
tff(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),f4010(V1x,V2c))))))))))).
tff(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),f2866(V0x)))))))))).
tff(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:$i] : (mem(V2k,arr(ty_2Enum_2Enum,bool)) => (( ! [V3x:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3x)),V2k)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V3x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Enum_2Enum(V3x)))))) => (p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V0f)) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V1g)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__EQ,lemma,(( ! [V0f:$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))))))))))).
tff(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),f4011(V0f)))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Enum_2Enum,bool)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V1s),V2t)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,f4012(V0x,V1s)),inj__ty_2Erealax_2Ereal(V3l)),V2t))) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),inj__ty_2Erealax_2Ereal(V3l)),V1s)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__SUBSET,lemma,(( ! [V0x:$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),f4013(V0x,V1s)))) => p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0x))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__TRIVIAL,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => p(ap(ap(c_2Ereal__topology_2Esummable,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)),V0f)))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__RESTRICT,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : (mem(V1k,arr(ty_2Enum_2Enum,bool)) => (p(ap(ap(c_2Ereal__topology_2Esummable,c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),f4014(V0f,V1k))) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V1k),V0f))))))))).
tff(conj_thm_2Ereal__topology_2ESUMS__FINITE__DIFF,lemma,(( ! [V0f:$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))))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Enum_2Enum,bool)) => ( ! [V3l:tp__ty_2Erealax_2Ereal] : ((p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V2t)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),inj__ty_2Erealax_2Ereal(V3l)),V1s))) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V3l)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V2t),V1s)),V0f))),ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V1s),V2t))))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMS__OFFSET,lemma,(( ! [V0f:$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)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMS__OFFSET__REV,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1l:tp__ty_2Erealax_2Ereal] : ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ((p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),inj__ty_2Erealax_2Ereal(V1l)),ap(c_2Ereal__topology_2Efrom,inj__ty_2Enum_2Enum(V2m)))) & (p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2m))) & p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V2m))))) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1l)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V3n)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),V0f))),ap(c_2Ereal__topology_2Efrom,inj__ty_2Enum_2Enum(V3n)))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__REINDEX,lemma,(( ! [V0k:tp__ty_2Enum_2Enum] : ( ! [V1a:$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))),f4015(V0k,V1a))) <=> 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))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ( ! [V4b:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),inj__ty_2Erealax_2Ereal(V3a)),V2s)) & (p(ap(ap(ap(c_2Ereal__topology_2Esums,V1g),inj__ty_2Erealax_2Ereal(V4b)),V2s)) & ( ! [V5x:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V5x)),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,inj__ty_2Enum_2Enum(V5x))),ap(V1g,inj__ty_2Enum_2Enum(V5x)))))))) => p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3a)),inj__ty_2Erealax_2Ereal(V4b)))))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__DROP__POS,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( ! [V2a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),inj__ty_2Erealax_2Ereal(V2a)),V1s)) & ( ! [V3x:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3x)),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,inj__ty_2Enum_2Enum(V3x))))))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2a))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__BOUND,lemma,(( ! [V0f:$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)))))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => ((p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0f)) & p(ap(c_2Ereal__topology_2Elinear,V1h))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__topology_2Einfsum,V2s),f4007(V0f,V1h))) = surj__ty_2Erealax_2Ereal(ap(V1h,ap(ap(c_2Ereal__topology_2Einfsum,V2s),V0f))))))))))))).
tff(conj_thm_2Ereal__topology_2EINFSUM__0,lemma,(( ! [V0s:$i] : (mem(V0s,arr(ty_2Enum_2Enum,bool)) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__topology_2Einfsum,V0s),k(ty_2Enum_2Enum,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)))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => ((p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0x)) & p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V1y))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__topology_2Einfsum,V2s),f4016(V0x,V1y))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal__topology_2Einfsum,V2s),V0x)),ap(ap(c_2Ereal__topology_2Einfsum,V2s),V1y))))))))))))).
tff(conj_thm_2Ereal__topology_2EINFSUM__SUB,lemma,(( ! [V0x:$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),f4017(V0x,V1y))) = 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))))))))))))).
tff(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),f4010(V1x,V2c))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2c)),ap(ap(c_2Ereal__topology_2Einfsum,V0s),V1x)))))))))))).
tff(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),f4018(V1x))) = surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal__topology_2Einfsum,V0s),V1x))))))))))).
tff(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:$i] : (mem(V2k,arr(ty_2Enum_2Enum,bool)) => ((p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V0f)) & (p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V1g)) & ( ! [V3x:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3x)),V2k)) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V3x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Enum_2Enum(V3x)))))))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__topology_2Einfsum,V2k),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__topology_2Einfsum,V2k),V1g)))))))))))).
tff(conj_thm_2Ereal__topology_2EINFSUM__RESTRICT,lemma,(( ! [V0k:$i] : (mem(V0k,arr(ty_2Enum_2Enum,bool)) => ( ! [V1a:$i] : (mem(V1a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__topology_2Einfsum,c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),f4019(V1a,V0k))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal__topology_2Einfsum,V0k),V1a))))))))).
tff(conj_thm_2Ereal__topology_2EPARTIAL__SUMS__COMPONENT__LE__INFSUM,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( ! [V2n:tp__ty_2Enum_2Enum] : ((( ! [V3i:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3i)),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,inj__ty_2Enum_2Enum(V3i)))))) & p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2n)))),V0f)),ap(ap(c_2Ereal__topology_2Einfsum,V1s),V0f))))))))))).
tff(conj_thm_2Ereal__topology_2EPARTIAL__SUMS__DROP__LE__INFSUM,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( ! [V2n:tp__ty_2Enum_2Enum] : ((( ! [V3i:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3i)),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,inj__ty_2Enum_2Enum(V3i)))))) & p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2n)))),V0f)),ap(ap(c_2Ereal__topology_2Einfsum,V1s),V0f))))))))))).
tff(conj_thm_2Ereal__topology_2ESEQUENCE__CAUCHY__WLOG,lemma,(( ! [V0e:tp__ty_2Erealax_2Ereal] : ( ! [V1P:$i] : (mem(V1P,arr(ty_2Enum_2Enum,bool)) => ( ! [V2s:$i] : (mem(V2s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => (( ! [V3m:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p(ap(V1P,inj__ty_2Enum_2Enum(V3m))) & p(ap(V1P,inj__ty_2Enum_2Enum(V4n)))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V2s,inj__ty_2Enum_2Enum(V3m))),ap(V2s,inj__ty_2Enum_2Enum(V4n))))),inj__ty_2Erealax_2Ereal(V0e)))))) <=> ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : ((p(ap(V1P,inj__ty_2Enum_2Enum(V5m))) & (p(ap(V1P,inj__ty_2Enum_2Enum(V6n))) & p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V6n))))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V2s,inj__ty_2Enum_2Enum(V5m))),ap(V2s,inj__ty_2Enum_2Enum(V6n))))),inj__ty_2Erealax_2Ereal(V0e)))))))))))))).
tff(conj_thm_2Ereal__topology_2ESUM__DIFF__LEMMA,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : (mem(V1k,arr(ty_2Enum_2Enum,bool)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))) => (surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1k),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V3n)))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1k),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2m)))),V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1k),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V3n)))),V0f)))))))))))).
tff(conj_thm_2Ereal__topology_2EABS__SUM__TRIVIAL__LEMMA,lemma,(( ! [V0P:tp__o] : ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( ! [V2m:tp__ty_2Enum_2Enum] : ( ! [V3n:tp__ty_2Enum_2Enum] : ( ! [V4f:$i] : (mem(V4f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V5e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5e))) => ((p(inj__o(V0P)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n)))),V4f))),inj__ty_2Erealax_2Ereal(V5e)))) <=> (p(inj__o(V0P)) => (p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V2m))) | p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n)))),V4f))),inj__ty_2Erealax_2Ereal(V5e))))))))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__CAUCHY,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,bool)) => ((? [V2l:tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),inj__ty_2Erealax_2Ereal(V2l)),V1s))) <=> ( ! [V3e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e))) => (? [V4N:tp__ty_2Enum_2Enum] : ( ! [V5m:tp__ty_2Enum_2Enum] : ( ! [V6n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V4N))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V6n)))),V0f))),inj__ty_2Erealax_2Ereal(V3e)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__CAUCHY,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1s:$i] : (mem(V1s,arr(ty_2Enum_2Enum,bool)) => (p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) <=> ( ! [V2e:tp__ty_2Erealax_2Ereal] : (p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2e))) => (? [V3N:tp__ty_2Enum_2Enum] : ( ! [V4m:tp__ty_2Enum_2Enum] : ( ! [V5n:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V4m)),inj__ty_2Enum_2Enum(V3N))) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V4m)),inj__ty_2Enum_2Enum(V5n)))),V0f))),inj__ty_2Erealax_2Ereal(V2e)))))))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__IFF__EVENTUALLY,lemma,(( ! [V0f:$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)))))))))))).
tff(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:$i] : (mem(V2k,arr(ty_2Enum_2Enum,bool)) => (((? [V3N:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3N)),inj__ty_2Enum_2Enum(V4n))) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4n)),V2k))) => (surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V4n))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Enum_2Enum(V4n))))))) & p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V0f))) => p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V1g))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__IFF__COFINITE,lemma,(( ! [V0f:$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)))))))))))).
tff(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:$i] : (mem(V2t,arr(ty_2Enum_2Enum,bool)) => ((p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V1s),V2t)),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V2t),V1s)))) & p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f))) => p(ap(ap(c_2Ereal__topology_2Esummable,V2t),V0f))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__FROM__ELSEWHERE,lemma,(( ! [V0f:$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))))))))).
tff(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))))))))))))))))))))).
tff(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),f4020(V2e,V1x,V0s)),c_2Ereal__topology_2Esequentially))))))))))).
tff(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),f4014(V0f,V1k)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Ereal__topology_2Esequentially))))))))).
tff(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)))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__SUMS__BOUNDED,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1k:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,inj__ty_2Enum_2Enum(V1k))),V0f)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f4021(V1k,V0f)))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__COMPARISON,lemma,(( ! [V0f:$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)))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => ((p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V1g)) & (? [V3N:tp__ty_2Enum_2Enum] : ( ! [V4n:tp__ty_2Enum_2Enum] : ((p(ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V4n)),inj__ty_2Enum_2Enum(V3N))) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4n)),V2s))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Enum_2Enum(V4n)))),ap(V1g,inj__ty_2Enum_2Enum(V4n)))))))) => p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0f))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__ABSCONV__IMP__CONV,lemma,(( ! [V0x:$i] : (mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1k:$i] : (mem(V1k,arr(ty_2Enum_2Enum,bool)) => (p(ap(ap(c_2Ereal__topology_2Esummable,V1k),f4022(V0x))) => p(ap(ap(c_2Ereal__topology_2Esummable,V1k),V0x))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__SUBSET__ABSCONV,lemma,(( ! [V0x:$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),f4023(V0x))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V2t),V1s))) => p(ap(ap(c_2Ereal__topology_2Esummable,V2t),f4024(V0x)))))))))))).
tff(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:$i] : (mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ! [V3a:tp__ty_2Erealax_2Ereal] : ((p(ap(ap(ap(c_2Ereal__topology_2Esums,V1g),inj__ty_2Erealax_2Ereal(V3a)),V2s)) & ( ! [V4i:tp__ty_2Enum_2Enum] : (p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4i)),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Enum_2Enum(V4i)))),ap(V1g,inj__ty_2Enum_2Enum(V4i))))))) => (? [V5l:tp__ty_2Erealax_2Ereal] : (p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),inj__ty_2Erealax_2Ereal(V5l)),V2s)) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V5l))),inj__ty_2Erealax_2Ereal(V3a))))))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__COMPARISON__UNIFORM,lemma,(( ! [A_27a:del]: ( ! [V0f:$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)))))))))))))))))))))))).
tff(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)))))))))))).
tff(conj_thm_2Ereal__topology_2EBOUNDED__PARTIAL__SUMS,lemma,(( ! [V0f:$i] : (mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ! [V1k:tp__ty_2Enum_2Enum] : (p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f4021(V1k,V0f)))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),ap(c_2Epair_2EUNCURRY(ty_2Enum_2Enum,ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f4026(V0f))))))))))).
tff(conj_thm_2Ereal__topology_2ESUMMABLE__BILINEAR__PARTIAL__PRE,lemma,(( ! [V0f:$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),f4027(V2h,V0f,V1g)),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))),f4028(V2h,V0f,V1g))))) => p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,inj__ty_2Enum_2Enum(V4k))),f4029(V2h,V0f,V1g)))))))))))))).
tff(conj_thm_2Ereal__topology_2ESERIES__DIRICHLET__BILINEAR,conjecture,(( ! [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),f4030(V4m,V0f)))) & (p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,inj__ty_2Enum_2Enum(V5p))),f4031(V1g))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f4032(V2h,V1g,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))),f4033(V2h,V1g,V0f)))))))))))))))).
