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,conjecture,(( ! [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))))))))).
