include('Axioms/measure+2.ax').
fof(ne_ty_2Ereal__topology_2Enet,axiom,( ! [A0] : ( ne(A0) => ne(ty_2Ereal__topology_2Enet(A0)) ) )).
fof(mem_c_2Ereal__topology_2E_2D_2D_3E,axiom,( ! [A_27a] : ( ne(A_27a) => 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)))) ) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2EClosed,axiom,( mem(c_2Ereal__topology_2EClosed,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2EDist,axiom,( mem(c_2Ereal__topology_2EDist,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2EOpen,axiom,( mem(c_2Ereal__topology_2EOpen,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Eat,axiom,( mem(c_2Ereal__topology_2Eat,arr(ty_2Erealax_2Ereal,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal))) )).
fof(mem_c_2Ereal__topology_2Eat__infinity,axiom,( mem(c_2Ereal__topology_2Eat__infinity,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal)) )).
fof(mem_c_2Ereal__topology_2Eat__neginfinity,axiom,( mem(c_2Ereal__topology_2Eat__neginfinity,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal)) )).
fof(mem_c_2Ereal__topology_2Eat__posinfinity,axiom,( mem(c_2Ereal__topology_2Eat__posinfinity,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal)) )).
fof(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))) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Ebilinear,axiom,( mem(c_2Ereal__topology_2Ebilinear,arr(arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),bool)) )).
fof(mem_c_2Ereal__topology_2Ebounded__def,axiom,( mem(c_2Ereal__topology_2Ebounded__def,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Ecauchy,axiom,( mem(c_2Ereal__topology_2Ecauchy,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) )).
fof(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))) )).
fof(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))) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Eclosure,axiom,( mem(c_2Ereal__topology_2Eclosure,arr(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool))) )).
fof(mem_c_2Ereal__topology_2Ecollinear,axiom,( mem(c_2Ereal__topology_2Ecollinear,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Ecompact,axiom,( mem(c_2Ereal__topology_2Ecompact,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Ecomplete,axiom,( mem(c_2Ereal__topology_2Ecomplete,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Ecomponents,axiom,( mem(c_2Ereal__topology_2Ecomponents,arr(arr(ty_2Erealax_2Ereal,bool),arr(arr(ty_2Erealax_2Ereal,bool),bool))) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Econnected,axiom,( mem(c_2Ereal__topology_2Econnected,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(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)))) )).
fof(mem_c_2Ereal__topology_2Econtinuous,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Econtinuous(A_27a),arr(arr(A_27a,ty_2Erealax_2Ereal),arr(ty_2Ereal__topology_2Enet(A_27a),bool))) ) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Edependent,axiom,( mem(c_2Ereal__topology_2Edependent,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Ediameter,axiom,( mem(c_2Ereal__topology_2Ediameter,arr(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal)) )).
fof(mem_c_2Ereal__topology_2Edim,axiom,( mem(c_2Ereal__topology_2Edim,arr(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum)) )).
fof(mem_c_2Ereal__topology_2Eeuclidean,axiom,( mem(c_2Ereal__topology_2Eeuclidean,ty_2Etopology_2Etopology(ty_2Erealax_2Ereal)) )).
fof(mem_c_2Ereal__topology_2Eeventually,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Eeventually(A_27a),arr(arr(A_27a,bool),arr(ty_2Ereal__topology_2Enet(A_27a),bool))) ) )).
fof(mem_c_2Ereal__topology_2Efrom,axiom,( mem(c_2Ereal__topology_2Efrom,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) )).
fof(mem_c_2Ereal__topology_2Efrontier,axiom,( mem(c_2Ereal__topology_2Efrontier,arr(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool))) )).
fof(mem_c_2Ereal__topology_2Efsigma,axiom,( mem(c_2Ereal__topology_2Efsigma,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Egdelta,axiom,( mem(c_2Ereal__topology_2Egdelta,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(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)) )).
fof(mem_c_2Ereal__topology_2Ehomeomorphic,axiom,( mem(c_2Ereal__topology_2Ehomeomorphic,arr(arr(ty_2Erealax_2Ereal,bool),arr(arr(ty_2Erealax_2Ereal,bool),bool))) )).
fof(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))) )).
fof(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)))) )).
fof(mem_c_2Ereal__topology_2Eindependent,axiom,( mem(c_2Ereal__topology_2Eindependent,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Einterior,axiom,( mem(c_2Ereal__topology_2Einterior,arr(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool))) )).
fof(mem_c_2Ereal__topology_2Eis__interval,axiom,( mem(c_2Ereal__topology_2Eis__interval,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Eisnet,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => mem(c_2Ereal__topology_2Eisnet(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ) ) )).
fof(mem_c_2Ereal__topology_2Elim__def,axiom,( ! [A_27a] : ( ne(A_27a) => 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))) ) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Elinear,axiom,( mem(c_2Ereal__topology_2Elinear,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool)) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Emidpoint,axiom,( mem(c_2Ereal__topology_2Emidpoint,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) )).
fof(mem_c_2Ereal__topology_2Emk__net,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Emk__net(A_27a),arr(arr(A_27a,arr(A_27a,bool)),ty_2Ereal__topology_2Enet(A_27a))) ) )).
fof(mem_c_2Ereal__topology_2Enetlimit,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Enetlimit(A_27a),arr(ty_2Ereal__topology_2Enet(A_27a),A_27a)) ) )).
fof(mem_c_2Ereal__topology_2Enetord,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Enetord(A_27a),arr(ty_2Ereal__topology_2Enet(A_27a),arr(A_27a,arr(A_27a,bool)))) ) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Epairwise,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Epairwise(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Ereal__topology_2Epermutes,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Epermutes(A_27a),arr(arr(A_27a,A_27a),arr(arr(A_27a,bool),bool))) ) )).
fof(mem_c_2Ereal__topology_2Esequentially,axiom,( mem(c_2Ereal__topology_2Esequentially,ty_2Ereal__topology_2Enet(ty_2Enum_2Enum)) )).
fof(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)) )).
fof(mem_c_2Ereal__topology_2Espan,axiom,( mem(c_2Ereal__topology_2Espan,arr(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool))) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Esubspace,axiom,( mem(c_2Ereal__topology_2Esubspace,arr(arr(ty_2Erealax_2Ereal,bool),bool)) )).
fof(mem_c_2Ereal__topology_2Esubtopology,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Esubtopology(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(arr(A_27a,bool),ty_2Etopology_2Etopology(A_27a)))) ) )).
fof(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))) )).
fof(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)))) )).
fof(mem_c_2Ereal__topology_2Etrivial__limit,axiom,( ! [A_27a] : ( ne(A_27a) => mem(c_2Ereal__topology_2Etrivial__limit(A_27a),arr(ty_2Ereal__topology_2Enet(A_27a),bool)) ) )).
fof(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))) )).
fof(mem_c_2Ereal__topology_2Ewithin,axiom,( ! [A_27a] : ( ne(A_27a) => 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)))) ) )).
fof(conj_thm_2Ereal__topology_2EREAL__LE__SQUARE__ABS,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Ereal_2Eabs,V1y))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Epow,V0x),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,V1y),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__EQ__SQUARE__ABS,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal_2Eabs,V0x) = ap(c_2Ereal_2Eabs,V1y) ) <=> ( ap(ap(c_2Ereal_2Epow,V0x),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = ap(ap(c_2Ereal_2Epow,V1y),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__HALF,lemma,(( ( ! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2E_2F,V0e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))))) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0e)) ) ) ) & ( ( ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2E_2F,V1e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(ap(c_2Ereal_2E_2F,V1e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) = V1e ) ) ) & ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2E_2F,V2e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) = V2e ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__SUBSET__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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] : ( 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) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EEXISTS__FINITE__SUBSET__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( ? [V3t] : ( 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] : ( 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))) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EFORALL__FINITE__SUBSET__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ! [V1f] : ( mem(V1f,arr(A_27b,A_27a)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( ! [V3t] : ( 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] : ( 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))) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EFORALL__IN__GSPEC,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) =>  ! [A_27e]: ( ne(A_27e) =>  ! [A_27f]: ( ne(A_27f) =>  ! [A_27g]: ( ne(A_27g) => ! [V0Q] : ( mem(V0Q,arr(A_27b,bool)) => ( ( ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( ( ! [V3z] : ( 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] : ( mem(V5x,A_27a) => ( p(ap(V1P,V5x)) => p(ap(V0Q,ap(V2f,V5x))) ) ) ) ) ) ) & ( ( ! [V6P] : ( mem(V6P,arr(A_27c,arr(A_27d,bool))) => ! [V7f] : ( mem(V7f,arr(A_27c,arr(A_27d,A_27b))) => ( ( ! [V8z] : ( 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] : ( mem(V11x,A_27c) => ! [V12y] : ( mem(V12y,A_27d) => ( p(ap(ap(V6P,V11x),V12y)) => p(ap(V0Q,ap(ap(V7f,V11x),V12y))) ) ) ) ) ) ) ) & ! [V13P] : ( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool)))) => ! [V14f] : ( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b)))) => ( ( ! [V15z] : ( 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] : ( mem(V19w,A_27e) => ! [V20x] : ( mem(V20x,A_27f) => ! [V21y] : ( mem(V21y,A_27g) => ( p(ap(ap(ap(V13P,V19w),V20x),V21y)) => p(ap(V0Q,ap(ap(ap(V14f,V19w),V20x),V21y))) ) ) ) ) ) ) ) ) ) )))))))))).
fof(conj_thm_2Ereal__topology_2EEXISTS__IN__GSPEC,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) =>  ! [A_27e]: ( ne(A_27e) =>  ! [A_27f]: ( ne(A_27f) =>  ! [A_27g]: ( ne(A_27g) => ! [V0Q] : ( mem(V0Q,arr(A_27b,bool)) => ( ( ! [V1P] : ( mem(V1P,arr(A_27a,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,A_27b)) => ( ( ? [V3z] : ( 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] : ( mem(V5x,A_27a) & ( p(ap(V1P,V5x)) & p(ap(V0Q,ap(V2f,V5x))) ) ) ) ) ) ) & ( ( ! [V6P] : ( mem(V6P,arr(A_27c,arr(A_27d,bool))) => ! [V7f] : ( mem(V7f,arr(A_27c,arr(A_27d,A_27b))) => ( ( ? [V8z] : ( 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] : ( mem(V11x,A_27c) & ? [V12y] : ( mem(V12y,A_27d) & ( p(ap(ap(V6P,V11x),V12y)) & p(ap(V0Q,ap(ap(V7f,V11x),V12y))) ) ) ) ) ) ) ) & ! [V13P] : ( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool)))) => ! [V14f] : ( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b)))) => ( ( ? [V15z] : ( 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] : ( mem(V19w,A_27e) & ? [V20x] : ( mem(V20x,A_27f) & ? [V21y] : ( mem(V21y,A_27g) & ( p(ap(ap(ap(V13P,V19w),V20x),V21y)) & p(ap(V0Q,ap(ap(ap(V14f,V19w),V20x),V21y))) ) ) ) ) ) ) ) ) ) )))))))))).
fof(conj_thm_2Ereal__topology_2EEMPTY__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ( ( ap(c_2Epred__set_2EBIGUNION(A_27a),V0s) = c_2Epred__set_2EEMPTY(A_27a) ) <=> ! [V1t] : ( 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) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EREAL__LT__RNEG,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V0x),ap(c_2Erealax_2Ereal__neg,V1y))) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Enum_2Enum)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ? [V2a] : ( mem(V2a,ty_2Enum_2Enum) & ! [V3x] : ( 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)),V2a)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EREAL__BOUNDS__LT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1k] : ( mem(V1k,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Erealax_2Ereal__neg,V1k)),V0x)) & p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V1k)) ) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V0x)),V1k)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBIGUNION__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,bool))) => ! [V1s] : ( 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)) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EBIGINTER__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,bool))) => ! [V1s] : ( 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)) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EREAL__LE__LMUL1,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V2z)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)),ap(ap(c_2Erealax_2Ereal__mul,V0x),V2z))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELE__EXISTS,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) <=> ? [V2d] : ( mem(V2d,ty_2Enum_2Enum) & ( V1n = ap(ap(c_2Earithmetic_2E_2B,V0m),V2d) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDS__LINEAR,lemma,(! [V0A] : ( mem(V0A,ty_2Enum_2Enum) => ! [V1B] : ( mem(V1B,ty_2Enum_2Enum) => ! [V2C] : ( mem(V2C,ty_2Enum_2Enum) => ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Earithmetic_2E_2A,V0A),V3n)),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,V1B),V3n)),V2C))) ) ) <=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0A),V1B)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDS__LINEAR__0,lemma,(! [V0A] : ( mem(V0A,ty_2Enum_2Enum) => ! [V1B] : ( mem(V1B,ty_2Enum_2Enum) => ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Earithmetic_2E_2A,V0A),V2n)),V1B)) ) ) <=> ( V0A = c_2Enum_2E0 ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LE__BETWEEN,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V1b)) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V2x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EWLOG__LE,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => ( ( ( ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(V0P,V1m),V2n)) <=> p(ap(ap(V0P,V2n),V1m)) ) ) ) ) & ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n)) => p(ap(ap(V0P,V3m),V4n)) ) ) ) ) => ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => p(ap(ap(V0P,V5m),V6n)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__POWERSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( 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)))) ) )))).
fof(conj_thm_2Ereal__topology_2ELE__ADD,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),ap(ap(c_2Earithmetic_2E_2B,V0m),V1n))) ) ))).
fof(conj_thm_2Ereal__topology_2ELE__ADDR,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),ap(ap(c_2Earithmetic_2E_2B,V0m),V1n))) ) ))).
fof(conj_thm_2Ereal__topology_2EADD__SUB2,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V0m),V1n)),V0m) = V1n ) ) ))).
fof(conj_thm_2Ereal__topology_2EADD__SUBR2,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ap(ap(c_2Earithmetic_2E_2D,V0m),ap(ap(c_2Earithmetic_2E_2B,V0m),V1n)) = c_2Enum_2E0 ) ) ))).
fof(conj_thm_2Ereal__topology_2EADD__SUBR,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ap(ap(c_2Earithmetic_2E_2D,V1n),ap(ap(c_2Earithmetic_2E_2B,V0m),V1n)) = c_2Enum_2E0 ) ) ))).
fof(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE__EQ,lemma,(! [V0R] : ( mem(V0R,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => ( ( ( ! [V1x] : ( mem(V1x,ty_2Enum_2Enum) => p(ap(ap(V0R,V1x),V1x)) ) ) & ! [V2x] : ( mem(V2x,ty_2Enum_2Enum) => ! [V3y] : ( mem(V3y,ty_2Enum_2Enum) => ! [V4z] : ( mem(V4z,ty_2Enum_2Enum) => ( ( p(ap(ap(V0R,V2x),V3y)) & p(ap(ap(V0R,V3y),V4z)) ) => p(ap(ap(V0R,V2x),V4z)) ) ) ) ) ) => ( ( ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V5m),V6n)) => p(ap(ap(V0R,V5m),V6n)) ) ) ) ) <=> ! [V7n] : ( mem(V7n,ty_2Enum_2Enum) => p(ap(ap(V0R,V7n),ap(c_2Enum_2ESUC,V7n))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE,lemma,(! [V0R] : ( mem(V0R,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => ( ( ( ! [V1x] : ( mem(V1x,ty_2Enum_2Enum) => p(ap(ap(V0R,V1x),V1x)) ) ) & ( ( ! [V2x] : ( mem(V2x,ty_2Enum_2Enum) => ! [V3y] : ( mem(V3y,ty_2Enum_2Enum) => ! [V4z] : ( mem(V4z,ty_2Enum_2Enum) => ( ( p(ap(ap(V0R,V2x),V3y)) & p(ap(ap(V0R,V3y),V4z)) ) => p(ap(ap(V0R,V2x),V4z)) ) ) ) ) ) & ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => p(ap(ap(V0R,V5n),ap(c_2Enum_2ESUC,V5n))) ) ) ) => ! [V6m] : ( mem(V6m,ty_2Enum_2Enum) => ! [V7n] : ( mem(V7n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V6m),V7n)) => p(ap(ap(V0R,V6m),V7n)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELAMBDA__PAIR,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0P] : ( 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) ) )))))).
fof(conj_thm_2Ereal__topology_2ENOT__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,A_27a) => ! [V1b] : ( mem(V1b,A_27a) => ( ( ~~ ( V0a = V1b ) ) <=> ( ~~ ( V0a = V1b ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EABS__LE__0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) <=> ( V0x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__OF__NUM__GE,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Ereal__of__num,V0m)),ap(c_2Ereal_2Ereal__of__num,V1n))) <=> p(ap(ap(c_2Earithmetic_2E_3E_3D,V0m),V1n)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPOWERSET__CLAUSES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( ( 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] : ( mem(V1a,A_27a) => ! [V2t] : ( 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)))) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EREAL__LT__LCANCEL__IMP,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)),ap(ap(c_2Erealax_2Ereal__mul,V0x),V2z))) ) => p(ap(ap(c_2Erealax_2Ereal__lt,V1y),V2z)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESIMPLE__IMAGE__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1P] : ( 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))) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ESUBSET__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(arr(A_27a,bool),bool)) => ! [V1g] : ( 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))) ) ) )))).
fof(conj_thm_2Ereal__topology_2EREAL__LT__POW2,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V0n))) ))).
fof(conj_thm_2Ereal__topology_2EFUN__IN__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2x] : ( 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))) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ESUBSET__ANTISYM__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( 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 ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EDIFF__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0u] : ( mem(V0u,arr(A_27a,bool)) => ! [V1s] : ( 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))) ) ) )))).
fof(conj_thm_2Ereal__topology_2EBIGINTER__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( 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)))) ) )))).
fof(conj_thm_2Ereal__topology_2EBIGUNION__BIGINTER,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( 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)))) ) )))).
fof(conj_thm_2Ereal__topology_2EREAL__POW__1__LE,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V1x),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Epow,V1x),V0n)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__POW__LE__1,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1x)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,V1x),V0n))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LT__INV2,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) & p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V1y)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Erealax_2Einv,V1y)),ap(c_2Erealax_2Einv,V0x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__INV__1__LE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V0x),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Erealax_2Einv,V0x))) ) ))).
fof(conj_thm_2Ereal__topology_2ESUM__GP__BASIC,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( 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,c_2Earithmetic_2EZERO)))),V0x)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V1n)),f3396(V0x))) = ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,V0x),ap(c_2Enum_2ESUC,V1n))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUM__GP__MULTIPLIED,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)) => ( 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,c_2Earithmetic_2EZERO)))),V0x)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3610(V0x))) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,V0x),V1m)),ap(ap(c_2Ereal_2Epow,V0x),ap(c_2Enum_2ESUC,V2n))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUM__GP,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3610(V0x)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Eprim__rec_2E_3C,V2n),V1m)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1m))),ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,V0x),V1m)),ap(ap(c_2Ereal_2Epow,V0x),ap(c_2Enum_2ESUC,V2n)))),ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__SYM,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUM__ABS__TRIANGLE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2b] : ( mem(V2b,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))),V2b)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f))),V2b)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EIMAGE__SING,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1a] : ( 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)) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EREAL__WLOG__LE,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(V0P,V1x),V2y)) <=> p(ap(ap(V0P,V2y),V1x)) ) ) ) ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V4y)) => p(ap(ap(V0P,V3x),V4y)) ) ) ) ) => ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ! [V6y] : ( mem(V6y,ty_2Erealax_2Ereal) => p(ap(ap(V0P,V5x),V6y)) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Epairwise,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0r] : ( mem(V0r,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ereal__topology_2Epairwise(A_27a),V0r),V1s)) <=> ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( 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)) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EPAIRWISE__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0r] : ( 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 ) )))).
fof(conj_thm_2Ereal__topology_2EPAIRWISE__SING,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0r] : ( mem(V0r,arr(A_27a,arr(A_27a,bool))) => ! [V1x] : ( 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 ) ) )))).
fof(conj_thm_2Ereal__topology_2EPAIRWISE__MONO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0r] : ( mem(V0r,arr(A_27a,arr(A_27a,bool))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( 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)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EPAIRWISE__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0r] : ( mem(V0r,arr(A_27a,arr(A_27a,bool))) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2s] : ( 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] : ( 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)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EPAIRWISE__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27b,bool)) => ! [V1r] : ( mem(V1r,arr(A_27a,arr(A_27a,bool))) => ! [V2f] : ( 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)) ) ) ) ))))).
fof(ax_thm_2Ereal__topology_2Epermutes,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( mem(V0p,arr(A_27a,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ereal__topology_2Epermutes(A_27a),V0p),V1s)) <=> ( ( ! [V2x] : ( mem(V2x,A_27a) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) ) => ( ap(V0p,V2x) = V2x ) ) ) ) & ! [V3y] : ( mem(V3y,A_27a) => p(ap(c_2Ebool_2E_3F_21(A_27a),f3618(A_27a,V0p,V3y))) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EPERMUTES__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,arr(A_27a,A_27a)) => ! [V1s] : ( 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 ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EPERMUTES__INJECTIVE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,arr(A_27a,A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Ereal__topology_2Epermutes(A_27a),V0p),V1s)) => ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ( ( ap(V0p,V2x) = ap(V0p,V3y) ) <=> ( V2x = V3y ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EEXISTS__IN__INSERT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1a] : ( mem(V1a,A_27a) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( ? [V3x] : ( 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] : ( mem(V4x,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s)) & p(ap(V0P,V4x)) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EDEPENDENT__CHOICE__FIXED,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V1R] : ( mem(V1R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27a,bool)))) => ! [V2a] : ( mem(V2a,A_27a) => ( ( p(ap(ap(V0P,c_2Enum_2E0),V2a)) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(V0P,V3n),V4x)) => ? [V5y] : ( mem(V5y,A_27a) & ( p(ap(ap(V0P,ap(c_2Enum_2ESUC,V3n)),V5y)) & p(ap(ap(ap(V1R,V3n),V4x),V5y)) ) ) ) ) ) ) => ? [V6f] : ( mem(V6f,arr(ty_2Enum_2Enum,A_27a)) & ( ( ap(V6f,c_2Enum_2E0) = V2a ) & ( ( ! [V7n] : ( mem(V7n,ty_2Enum_2Enum) => p(ap(ap(V0P,V7n),ap(V6f,V7n))) ) ) & ! [V8n] : ( mem(V8n,ty_2Enum_2Enum) => p(ap(ap(ap(V1R,V8n),ap(V6f,V8n)),ap(V6f,ap(c_2Enum_2ESUC,V8n)))) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EDEPENDENT__CHOICE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,arr(A_27a,bool))) => ! [V1R] : ( mem(V1R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27a,bool)))) => ( ( ( ? [V2a] : ( mem(V2a,A_27a) & p(ap(ap(V0P,c_2Enum_2E0),V2a)) ) ) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(V0P,V3n),V4x)) => ? [V5y] : ( mem(V5y,A_27a) & ( p(ap(ap(V0P,ap(c_2Enum_2ESUC,V3n)),V5y)) & p(ap(ap(ap(V1R,V3n),V4x),V5y)) ) ) ) ) ) ) => ? [V6f] : ( mem(V6f,arr(ty_2Enum_2Enum,A_27a)) & ( ( ! [V7n] : ( mem(V7n,ty_2Enum_2Enum) => p(ap(ap(V0P,V7n),ap(V6f,V7n))) ) ) & ! [V8n] : ( mem(V8n,ty_2Enum_2Enum) => p(ap(ap(ap(V1R,V8n),ap(V6f,V8n)),ap(V6f,ap(c_2Enum_2ESUC,V8n)))) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EBIGUNION__MONO__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,bool))) => ! [V2g] : ( mem(V2g,arr(A_27a,arr(A_27b,bool))) => ( ( ! [V3x] : ( 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)))) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EBIGUNION__MONO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ! [V1t] : ( mem(V1t,arr(arr(A_27a,bool),bool)) => ( ( ! [V2x] : ( mem(V2x,arr(A_27a,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2x),V0s)) => ? [V3y] : ( 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))) ) ) )))).
fof(conj_thm_2Ereal__topology_2EREAL__WLOG__LT,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(ap(V0P,V1x),V1x)) ) ) & ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(V0P,V2x),V3y)) <=> p(ap(ap(V0P,V3y),V2x)) ) ) ) ) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V5y)) => p(ap(ap(V0P,V4x),V5y)) ) ) ) ) ) => ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ! [V7y] : ( mem(V7y,ty_2Erealax_2Ereal) => p(ap(ap(V0P,V6x),V7y)) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Edist,axiom, ! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)) = ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y)) ) ) )).
fof(conj_thm_2Ereal__topology_2EDIST__REFL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V0x)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__SYM,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1y),V0x)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,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),V0x),V2z))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1y),V2z))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ALT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,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),V1y),V2z))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2z))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__EQ__0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( V0x = V1y ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__POS__LT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ~~ ( V0x = V1y ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__NZ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ~~ ( V0x = V1y ) ) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__LE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,ty_2Erealax_2Ereal) => ! [V3e] : ( mem(V3e,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),V0x),V2z))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1y),V2z)))),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),V0x),V1y))),V3e)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__LT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,ty_2Erealax_2Ereal) => ! [V3e] : ( mem(V3e,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),V0x),V2z))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1y),V2z)))),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),V0x),V1y))),V3e)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__L,lemma,(! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ! [V1x1] : ( mem(V1x1,ty_2Erealax_2Ereal) => ! [V2x2] : ( mem(V2x2,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,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),V1x1),V3y))),ap(ap(c_2Ereal_2E_2F,V0e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,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),V2x2),V3y))),ap(ap(c_2Ereal_2E_2F,V0e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,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),V1x1),V2x2))),V0e)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__R,lemma,(! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ! [V1x1] : ( mem(V1x1,ty_2Erealax_2Ereal) => ! [V2x2] : ( mem(V2x2,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,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),V3y),V1x1))),ap(ap(c_2Ereal_2E_2F,V0e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,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),V3y),V2x2))),ap(ap(c_2Ereal_2E_2F,V0e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,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),V1x1),V2x2))),V0e)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ADD,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1x_27] : ( mem(V1x_27,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ! [V3y_27] : ( mem(V3y_27,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,V0x),V2y)),ap(ap(c_2Erealax_2Ereal__add,V1x_27),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),V0x),V1x_27))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2y),V3y_27))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__MUL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,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,V2c),V0x)),ap(ap(c_2Erealax_2Ereal__mul,V2c),V1y))) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V2c)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__ADD__HALF,lemma,(! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2x_27] : ( mem(V2x_27,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ! [V4y_27] : ( mem(V4y_27,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),V1x),V2x_27))),ap(ap(c_2Ereal_2E_2F,V0e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,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),V3y),V4y_27))),ap(ap(c_2Ereal_2E_2F,V0e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,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,V1x),V3y)),ap(ap(c_2Erealax_2Ereal__add,V2x_27),V4y_27)))),V0e)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__LE__0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,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),V0x),V1y))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) <=> ( V0x = V1y ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__POS__LE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)))) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__EQ,lemma,(! [V0w] : ( mem(V0w,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ! [V3z] : ( mem(V3z,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0w),V1x)) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2y),V3z)) ) <=> ( ap(ap(c_2Ereal_2Epow,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0w),V1x))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = ap(ap(c_2Ereal_2Epow,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2y),V3z))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) = ap(c_2Ereal_2Eabs,V0x) ) & ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) = ap(c_2Ereal_2Eabs,V0x) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__CHOOSE__DIST,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) => ? [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) & ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2y)) = V1e ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Elinear,axiom, ! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) <=> ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ap(V0f,ap(ap(c_2Erealax_2Ereal__add,V1x),V2y)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V1x)),ap(V0f,V2y)) ) ) ) ) & ! [V3c] : ( mem(V3c,ty_2Erealax_2Ereal) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( ap(V0f,ap(ap(c_2Erealax_2Ereal__mul,V3c),V4x)) = ap(ap(c_2Erealax_2Ereal__mul,V3c),ap(V0f,V4x)) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ELINEAR__SCALING,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Elinear,f3619(V0c))) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__CMUL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(c_2Ereal__topology_2Elinear,f3620(V0f,V1c))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__NEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(c_2Ereal__topology_2Elinear,f3621(V0f))) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__ADD,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__SUB,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__ID,lemma,(p(ap(c_2Ereal__topology_2Elinear,i(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ELINEAR__ZERO,lemma,(p(ap(c_2Ereal__topology_2Elinear,k(ty_2Erealax_2Ereal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))))).
fof(conj_thm_2Ereal__topology_2ELINEAR__NEGATION,lemma,(p(ap(c_2Ereal__topology_2Elinear,f3624)))).
fof(conj_thm_2Ereal__topology_2ELINEAR__COMPOSE__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) & ! [V2a] : ( 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))) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELINEAR__MUL__COMPONENT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1v] : ( mem(V1v,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(c_2Ereal__topology_2Elinear,f3627(V0f,V1v))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__0,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ( ap(V0f,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__CMUL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ( ap(V0f,ap(ap(c_2Erealax_2Ereal__mul,V1c),V2x)) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(V0f,V2x)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__NEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ( ap(V0f,ap(c_2Erealax_2Ereal__neg,V1x)) = ap(c_2Erealax_2Ereal__neg,ap(V0f,V1x)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__ADD,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ( ap(V0f,ap(ap(c_2Erealax_2Ereal__add,V1x),V2y)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V1x)),ap(V0f,V2y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__SUB,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ( ap(V0f,ap(ap(c_2Ereal_2Ereal__sub,V1x),V2y)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V1x)),ap(V0f,V2y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) ) => ( ap(V0f,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1g)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELINEAR__SUM__MUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2c] : ( mem(V2c,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3v] : ( mem(V3v,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) ) => ( ap(V0f,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3628(A_27a,V2c,V3v))) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3629(A_27a,V0f,V3v,V2c)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELINEAR__BOUNDED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ? [V1B] : ( mem(V1B,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V2x))),ap(ap(c_2Erealax_2Ereal__mul,V1B),ap(c_2Ereal_2Eabs,V2x)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__BOUNDED__POS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ? [V1B] : ( mem(V1B,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1B)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V2x))),ap(ap(c_2Erealax_2Ereal__mul,V1B),ap(c_2Ereal_2Eabs,V2x)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESYMMETRIC__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V2x)),V1s)) ) ) ) & p(ap(c_2Ereal__topology_2Elinear,V0f)) ) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(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,V3x)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Ebilinear,axiom, ! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0f)) <=> ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Elinear,f3630(V1x,V0f))) ) ) & ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Elinear,f3631(V0f,V3y))) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EBILINEAR__SWAP,lemma,(! [V0op] : ( 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)) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__LADD,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ! [V3z] : ( mem(V3z,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,ap(ap(c_2Erealax_2Ereal__add,V1x),V2y)),V3z) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(V0h,V1x),V3z)),ap(ap(V0h,V2y),V3z)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__RADD,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ! [V3z] : ( mem(V3z,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,V1x),ap(ap(c_2Erealax_2Ereal__add,V2y),V3z)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(V0h,V1x),V2y)),ap(ap(V0h,V1x),V3z)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__LMUL,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,ap(ap(c_2Erealax_2Ereal__mul,V1c),V2x)),V3y) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(ap(V0h,V2x),V3y)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__RMUL,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,V2x),ap(ap(c_2Erealax_2Ereal__mul,V1c),V3y)) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(ap(V0h,V2x),V3y)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__LNEG,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,ap(c_2Erealax_2Ereal__neg,V1x)),V2y) = ap(c_2Erealax_2Ereal__neg,ap(ap(V0h,V1x),V2y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__RNEG,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,V1x),ap(c_2Erealax_2Ereal__neg,V2y)) = ap(c_2Erealax_2Ereal__neg,ap(ap(V0h,V1x),V2y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__LZERO,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__RZERO,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,V1x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__LSUB,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ! [V3z] : ( mem(V3z,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,ap(ap(c_2Ereal_2Ereal__sub,V1x),V2y)),V3z) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(V0h,V1x),V3z)),ap(ap(V0h,V2y),V3z)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__RSUB,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ! [V3z] : ( mem(V3z,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ( ap(ap(V0h,V1x),ap(ap(c_2Ereal_2Ereal__sub,V2y),V3z)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(V0h,V1x),V2y)),ap(ap(V0h,V1x),V3z)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__SUM,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3g] : ( mem(V3g,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V4h] : ( 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)) ) ) => ( ap(ap(V4h,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f)),ap(ap(c_2Eiterate_2ESum(A_27b),V1t),V3g)) = 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))) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__BOUNDED,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ? [V1B] : ( mem(V1B,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(V0h,V2x),V3y))),ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Erealax_2Ereal__mul,V1B),ap(c_2Ereal_2Eabs,V2x))),ap(c_2Ereal_2Eabs,V3y)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__BOUNDED__POS,lemma,(! [V0h] : ( mem(V0h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( p(ap(c_2Ereal__topology_2Ebilinear,V0h)) => ? [V1B] : ( mem(V1B,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1B)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(V0h,V2x),V3y))),ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Erealax_2Ereal__mul,V1B),ap(c_2Ereal_2Eabs,V2x))),ap(c_2Ereal_2Eabs,V3y)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__SUC,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(c_2Ereal__topology_2Ebilinear,V2h)) => ( ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V3m),V4n)),f3636(V2h,V0f,V1g)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(V2h,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V4n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V1g,ap(ap(c_2Earithmetic_2E_2B,V4n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),ap(ap(V2h,ap(V0f,V3m)),ap(V1g,V3m)))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V3m),V4n)),f3637(V2h,V0f,V1g)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__PRE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(c_2Ereal__topology_2Ebilinear,V2h)) => ( ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V3m),V4n)),f3638(V2h,V0f,V1g)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(V2h,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V4n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V1g,V4n))),ap(ap(V2h,ap(V0f,V3m)),ap(V1g,ap(ap(c_2Earithmetic_2E_2D,V3m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V3m),V4n)),f3639(V2h,V0f,V1g)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Esubspace,axiom, ! [V0s] : ( 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,c_2Enum_2E0)),V0s)) & ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V1x),V2y)),V0s)) ) ) ) ) & ! [V3c] : ( mem(V3c,ty_2Erealax_2Ereal) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,V3c),V4x)),V0s)) ) ) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Espan,axiom, ! [V0s] : ( 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) ) )).
fof(ax_thm_2Ereal__topology_2Edependent,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Edependent,V0s)) <=> ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V0s),V1a)))) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Eindependent,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Eindependent,V0s)) <=> ( ~~ p(ap(c_2Ereal__topology_2Edependent,V0s)) ) ) )).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__UNIV,lemma,(p(ap(c_2Ereal__topology_2Esubspace,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__IMP__NONEMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Esubspace,V0s)) => ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__0,lemma,(! [V0s] : ( 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,c_2Enum_2E0)),V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__ADD,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Esubspace,V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),V2s)) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__MUL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Esubspace,V2s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2s)) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,V1c),V0x)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__NEG,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Esubspace,V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V1s)) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V0x)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__SUB,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Esubspace,V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),V2s)) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Ereal__topology_2Esubspace,V0s)) & ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2t)) & ! [V3x] : ( 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)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__LINEAR__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( 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)))) ) ) ))).
fof(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,c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__BIGINTER,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ! [V1s] : ( 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))) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__0__SUBSPACE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & p(ap(c_2Ereal__topology_2Esubspace,V1s)) ) => ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1s)) & ( ap(V0f,V2x) = ap(V0f,V3y) ) ) ) => ( V2x = V3y ) ) ) ) ) <=> ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) & ( ap(V0f,V4x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( V4x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__UNION__CHAIN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__SPAN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Espan,ap(c_2Ereal__topology_2Espan,V0s)) = ap(c_2Ereal__topology_2Espan,V0s) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__MONO,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__SPAN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2Esubspace,ap(c_2Ereal__topology_2Espan,V0s))) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__CLAUSES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V2s)) => p(ap(ap(c_2Ebool_2EIN(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,c_2Enum_2E0)),ap(c_2Ereal__topology_2Espan,V0s))) & ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ! [V5s] : ( mem(V5s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),ap(c_2Ereal__topology_2Espan,V5s))) & p(ap(ap(c_2Ebool_2EIN(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,V3x),V4y)),ap(c_2Ereal__topology_2Espan,V5s))) ) ) ) ) ) & ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ! [V7c] : ( mem(V7c,ty_2Erealax_2Ereal) => ! [V8s] : ( mem(V8s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(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,V7c),V6x)),ap(c_2Ereal__topology_2Espan,V8s))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__INDUCT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1h] : ( mem(V1h,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1h)) ) ) ) & p(ap(c_2Ereal__topology_2Esubspace,V1h)) ) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),ap(c_2Ereal__topology_2Espan,V0s))) => p(ap(V1h,V3x)) ) ) ) ) ))).
fof(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,c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__NONZERO,lemma,(! [V0s] : ( 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,c_2Enum_2E0)),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDEPENDENT__MONO,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__SUBSPACE,lemma,(! [V0b] : ( mem(V0b,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( 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 ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__INDUCT__ALT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1h] : ( mem(V1h,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(V1h,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) & ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) & p(ap(V1h,V4y)) ) => p(ap(V1h,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V2c),V3x)),V4y))) ) ) ) ) ) => ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),ap(c_2Ereal__topology_2Espan,V0s))) => p(ap(V1h,V5x)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__SUPERSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2Espan,V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__INC,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Espan,V0s))) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__UNION__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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)))) ) ))).
fof(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) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__0,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2Espan,V0s))) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__ADD,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Espan,V2s))) & p(ap(ap(c_2Ebool_2EIN(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,V0x),V1y)),ap(c_2Ereal__topology_2Espan,V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__MUL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(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,V1c),V0x)),ap(c_2Ereal__topology_2Espan,V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__MUL__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ~~ ( V1c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,V1c),V0x)),ap(c_2Ereal__topology_2Espan,V2s))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Espan,V2s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__NEG,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Espan,V1s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V0x)),ap(c_2Ereal__topology_2Espan,V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__NEG__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V0x)),ap(c_2Ereal__topology_2Espan,V1s))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Espan,V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__SUB,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Espan,V2s))) & p(ap(ap(c_2Ebool_2EIN(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,V0x),V1y)),ap(c_2Ereal__topology_2Espan,V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2t)) & ! [V3x] : ( 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))) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ESPAN__ADD__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(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,V1x),V2y)),ap(c_2Ereal__topology_2Espan,V0s))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),ap(c_2Ereal__topology_2Espan,V0s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__EQ__SELF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Espan,V0s) = V0s ) <=> p(ap(c_2Ereal__topology_2Esubspace,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__SUBSET__SUBSPACE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESURJECTIVE__IMAGE__EQ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( ( ! [V3y] : ( mem(V3y,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V2t)) => ? [V4x] : ( mem(V4x,A_27a) & ( ap(V0f,V4x) = V3y ) ) ) ) ) & ! [V5x] : ( 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 ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Esubspace,V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) ) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3641(V1a)),V0s) = V0s ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,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),V1a),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Ereal_2Eabs,V1y)) ) <=> ( ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V0x)),V1y) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V1y)),V0x) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__ABS__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal_2Eabs,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Ereal_2Eabs,V1y)) ) <=> ( ( ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V0x)),V1y) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V1y)),V0x) ) | ( ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V0x)),V1y) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Eabs,V1y))),V0x) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__EQ__LINV,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ( ap(c_2Erealax_2Ereal__neg,V0x) = V0x ) <=> ( V0x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__EQ__RINV,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ( V0x = ap(c_2Erealax_2Ereal__neg,V0x) ) <=> ( V0x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ))).
fof(conj_thm_2Ereal__topology_2EABS__TRIANGLE__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal_2Eabs,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)) = ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Ereal_2Eabs,V1y)) ) <=> ( ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V0x)),V1y) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V1y)),V0x) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__TRIANGLE__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2z)) = ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1y),V2z))) ) <=> ( ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y))),ap(ap(c_2Ereal_2Ereal__sub,V1y),V2z)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1y),V2z))),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y)) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Ecollinear,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecollinear,V0s)) <=> ? [V1u] : ( mem(V1u,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) ) => ? [V4c] : ( mem(V4c,ty_2Erealax_2Ereal) & ( ap(ap(c_2Ereal_2Ereal__sub,V2x),V3y) = ap(ap(c_2Erealax_2Ereal__mul,V4c),V1u) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Ecollinear,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__SING,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__2,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__SMALL,lemma,(! [V0s] : ( 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,c_2Earithmetic_2EZERO)))) ) => p(ap(c_2Ereal__topology_2Ecollinear,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__3,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),ap(ap(c_2Epred__set_2EINSERT(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,c_2Enum_2E0)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V2z),V1y)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__LEMMA,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) <=> ( ( V0x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) | ( ( V1y = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) | ? [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) & ( V1y = ap(ap(c_2Erealax_2Ereal__mul,V2c),V0x) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__LEMMA__ALT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) <=> ( ( V0x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) | ? [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) & ( V1y = ap(ap(c_2Erealax_2Ereal__mul,V2c),V0x) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQUAL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal_2Eabs,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Ereal_2Eabs,V1y)) ) <=> p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EMUL__CAUCHY__SCHWARZ__EQUAL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Erealax_2Ereal__mul,V0x),V0x)),ap(ap(c_2Erealax_2Ereal__mul,V1y),V1y)) ) <=> p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__3__EXPAND,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2c),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) <=> ( ( V0a = V2c ) | ? [V3u] : ( mem(V3u,ty_2Erealax_2Ereal) & ( V1b = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V3u),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,c_2Earithmetic_2EZERO)))),V3u)),V2c)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__TRIPLES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ~~ ( V1a = V2b ) ) => ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2b),V0s)))) <=> ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2b),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__4__3,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = V1b ) ) => ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2c),ap(ap(c_2Epred__set_2EINSERT(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),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),ap(ap(c_2Epred__set_2EINSERT(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),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3d),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__3__TRANS,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),ap(ap(c_2Epred__set_2EINSERT(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),V1b),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2c),ap(ap(c_2Epred__set_2EINSERT(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),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3d),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Ebetween,axiom, ! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Ebetween,V0x),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b))) <=> ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)) = ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V0x))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2b))) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EBETWEEN__REFL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Ebetween,V0a),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) & ( p(ap(ap(c_2Ereal__topology_2Ebetween,V1b),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) & p(ap(ap(c_2Ereal__topology_2Ebetween,V0a),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V0a))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__REFL__EQ,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Ebetween,V1x),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V0a))) <=> ( V1x = V0a ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__SYM,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Ebetween,V2x),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) <=> p(ap(ap(c_2Ereal__topology_2Ebetween,V2x),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V0a))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__ANTISYM,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Ebetween,V0a),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V2c))) & p(ap(ap(c_2Ereal__topology_2Ebetween,V1b),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2c))) ) => ( V0a = V1b ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__TRANS,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Ebetween,V0a),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V2c))) & p(ap(ap(c_2Ereal__topology_2Ebetween,V3d),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2c))) ) => p(ap(ap(c_2Ereal__topology_2Ebetween,V3d),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V2c))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__TRANS__2,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Ebetween,V0a),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V2c))) & p(ap(ap(c_2Ereal__topology_2Ebetween,V3d),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) ) => p(ap(ap(c_2Ereal__topology_2Ebetween,V0a),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__ABS,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Ebetween,V2x),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) <=> ( ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V2x),V0a))),ap(ap(c_2Ereal_2Ereal__sub,V1b),V2x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1b),V2x))),ap(ap(c_2Ereal_2Ereal__sub,V2x),V0a)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__IMP__COLLINEAR,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Ebetween,V2x),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) => p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__BETWEEN__CASES,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2c),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) <=> ( p(ap(ap(c_2Ereal__topology_2Ebetween,V0a),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V2c))) | ( p(ap(ap(c_2Ereal__topology_2Ebetween,V1b),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V0a))) | p(ap(ap(c_2Ereal__topology_2Ebetween,V2c),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__BETWEEN,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(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),V2x),V0a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),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),V2x),V1b))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) => p(ap(ap(c_2Ereal__topology_2Ebetween,V2x),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__1,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2Ecollinear,V0s)) ))).
fof(ax_thm_2Ereal__topology_2Emidpoint,axiom, ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(ap(c_2Erealax_2Ereal__add,V0a),V1b)) ) ) )).
fof(conj_thm_2Ereal__topology_2EMIDPOINT__REFL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V0x)) = V0x ) ))).
fof(conj_thm_2Ereal__topology_2EMIDPOINT__SYM,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V0a)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__MIDPOINT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ) & ( ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ) & ( ( 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),V0a),V1b))),V0a)) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ) & ( 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),V0a),V1b))),V1b)) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EMIDPOINT__EQ__ENDPOINT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ( ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = V0a ) <=> ( V0a = V1b ) ) & ( ( ( ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = V1b ) <=> ( V0a = V1b ) ) & ( ( ( V0a = ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) ) <=> ( V0a = V1b ) ) & ( ( V1b = ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) ) <=> ( V0a = V1b ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__MIDPOINT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,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),V0a),V1b))),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),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),V0a),V1b))),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V0a))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EMIDPOINT__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ( ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V1a)),ap(V0f,V2b))) = ap(V0f,ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__MIDPOINT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(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),V0a),V1b))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) ) ))).
fof(conj_thm_2Ereal__topology_2EMIDPOINT__COLLINEAR,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = V2c ) ) => ( ( V1b = ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2c)) ) <=> ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2c),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) & ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V2c)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__MONO,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__BREAKDOWN,lemma,(! [V0b] : ( mem(V0b,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0b),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2a),ap(c_2Ereal__topology_2Espan,V1s))) ) => ? [V3k] : ( mem(V3k,ty_2Erealax_2Ereal) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V2a),ap(ap(c_2Erealax_2Ereal__mul,V3k),V0b))),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V1s),V0b)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__SPAN__INSERT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),V2s)))) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2Espan,V2s))) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),V2s)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__INSERT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Eindependent,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),V1s))) <=> p(ap(ap(ap(c_2Ebool_2ECOND(bool),ap(ap(c_2Ebool_2EIN(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),V0a),ap(c_2Ereal__topology_2Espan,V1s)))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Eindependent,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__SING,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Eindependent,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) <=> ( ~~ ( V0x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ))).
fof(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))))).
fof(conj_thm_2Ereal__topology_2ESPANNING__SUBSET__INDEPENDENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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 ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__SPAN__DELETE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2Espan,V2s))) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V2s),V1b)))) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V2s),V1b))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__TRANS,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Espan,V2s))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),V2s)))) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),ap(c_2Ereal__topology_2Espan,V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEXCHANGE__LEMMA,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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] : ( 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))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECARD__STDBASIS,lemma,(( 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,c_2Earithmetic_2EZERO)))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__SPAN__BOUND,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__BOUND,lemma,(! [V0s] : ( 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,c_2Earithmetic_2EZERO)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET__EXTEND,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1v] : ( 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] : ( 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))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET,lemma,(! [V0v] : ( mem(V0v,arr(ty_2Erealax_2Ereal,bool)) => ? [V1b] : ( 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))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__BREAKDOWN__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Espan,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),V2s)))) <=> ? [V3k] : ( mem(V3k,ty_2Erealax_2Ereal) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V0x),ap(ap(c_2Erealax_2Ereal__mul,V3k),V1a))),ap(c_2Ereal__topology_2Espan,V2s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND__LEMMA,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1b] : ( 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] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),ap(c_2Ereal__topology_2Espan,V1b))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),ap(c_2Ereal__topology_2Espan,V1b))) ) => ( ap(V2g,ap(ap(c_2Erealax_2Ereal__add,V3x),V4y)) = ap(ap(c_2Erealax_2Ereal__add,ap(V2g,V3x)),ap(V2g,V4y)) ) ) ) ) ) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ! [V6c] : ( mem(V6c,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),ap(c_2Ereal__topology_2Espan,V1b))) => ( ap(V2g,ap(ap(c_2Erealax_2Ereal__mul,V6c),V5x)) = ap(ap(c_2Erealax_2Ereal__mul,V6c),ap(V2g,V5x)) ) ) ) ) ) & ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V1b)) => ( ap(V2g,V7x) = ap(V0f,V7x) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1b] : ( mem(V1b,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Eindependent,V1b)) => ? [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(c_2Ereal__topology_2Elinear,V2g)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1b)) => ( ap(V2g,V3x) = ap(V0f,V3x) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__KERNEL,lemma,(! [V0f] : ( 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)))) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__EQ__0__SPAN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1b] : ( mem(V1b,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1b)) => ( ap(V0f,V2x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),ap(c_2Ereal__topology_2Espan,V1b))) => ( ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__EQ__0,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1b] : ( mem(V1b,arr(ty_2Erealax_2Ereal,bool)) => ! [V2s] : ( 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] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1b)) => ( ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V2s)) => ( ap(V0f,V4x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2b] : ( mem(V2b,arr(ty_2Erealax_2Ereal,bool)) => ! [V3s] : ( 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] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V2b)) => ( ap(V0f,V4x) = ap(V1g,V4x) ) ) ) ) ) ) => ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V3s)) => ( ap(V0f,V5x) = ap(V1g,V5x) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__EQ__STDBASIS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( p(ap(c_2Ereal__topology_2Elinear,V1g)) & ! [V2i] : ( mem(V2i,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2i)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2i),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) => ( ap(V0f,V2i) = ap(V1g,V2i) ) ) ) ) ) => ( V0f = V1g ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__LEFT__INVERSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) => ( V1x = V2y ) ) ) ) ) => ? [V3g] : ( 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) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Edim,axiom, ! [V0v] : ( mem(V0v,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Edim,V0v) = ap(c_2Emin_2E_40(ty_2Enum_2Enum),f3647(V0v)) ) )).
fof(conj_thm_2Ereal__topology_2EBASIS__EXISTS,lemma,(! [V0v] : ( mem(V0v,arr(ty_2Erealax_2Ereal,bool)) => ? [V1b] : ( 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))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECARD__GE__DIM__INDEPENDENT,lemma,(! [V0v] : ( mem(V0v,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__EXPLICIT,lemma,(! [V0p] : ( 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)) ) ))).
fof(conj_thm_2Ereal__topology_2EDEPENDENT__EXPLICIT,lemma,(! [V0p] : ( mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Edependent,V0p)) <=> ? [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) & ? [V2u] : ( 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] : ( mem(V3v,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3v),V1s)) & ( ~~ ( ap(V2u,V3v) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) & ( ap(ap(c_2Eiterate_2ESum(ty_2Erealax_2Ereal),V1s),f3652(V2u)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE__GEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Eindependent,V1s)) & ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Espan,V1s))) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),ap(c_2Ereal__topology_2Espan,V1s))) & ( ap(V0f,V2x) = ap(V0f,V3y) ) ) ) => ( V2x = V3y ) ) ) ) ) ) => p(ap(c_2Ereal__topology_2Eindependent,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Eindependent,V1s)) & ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) ) => p(ap(c_2Ereal__topology_2Eindependent,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( 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)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__IMP__SURJECTIVE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) => ( V1x = V2y ) ) ) ) ) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( ap(V0f,V4x) = V3y ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW__POS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( 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] : ( mem(V2B,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2B)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,V2B),ap(c_2Ereal_2Eabs,V3x))),ap(c_2Ereal_2Eabs,ap(V0f,V3x)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( 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] : ( mem(V2B,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,V2B),ap(c_2Ereal_2Eabs,V3x))),ap(c_2Ereal_2Eabs,ap(V0f,V3x)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INJECTIVE__BOUNDED__BELOW__POS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) => ( V1x = V2y ) ) ) ) ) => ? [V3B] : ( mem(V3B,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3B)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,V4x)),V3B)),ap(c_2Ereal_2Eabs,ap(V0f,V4x)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINDEPENDENT__CARD__LE__DIM,lemma,(! [V0v] : ( mem(V0v,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( 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))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPAN__CARD__GE__DIM,lemma,(! [V0v] : ( mem(V0v,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBASIS__CARD__EQ__DIM,lemma,(! [V0v] : ( mem(V0v,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( 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)) & ( ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V1b) = ap(c_2Ereal__topology_2Edim,V0v) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBASIS__HAS__SIZE__DIM,lemma,(! [V0v] : ( mem(V0v,arr(ty_2Erealax_2Ereal,bool)) => ! [V1b] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIM__UNIQUE,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ! [V1v] : ( mem(V1v,arr(ty_2Erealax_2Ereal,bool)) => ! [V2b] : ( 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),V0n)) ) ) ) => ( ap(c_2Ereal__topology_2Edim,V1v) = V0n ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIM__LE__CARD,lemma,(! [V0s] : ( 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))) ) ))).
fof(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) ))).
fof(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,c_2Earithmetic_2EZERO)))))).
fof(conj_thm_2Ereal__topology_2EDIM__UNIV,lemma,(( ap(c_2Ereal__topology_2Edim,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ))).
fof(conj_thm_2Ereal__topology_2EDIM__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( 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))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIM__SUBSET__UNIV,lemma,(! [V0s] : ( 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,c_2Earithmetic_2EZERO)))) ))).
fof(ax_thm_2Ereal__topology_2Esubtopology,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1u] : ( 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))) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ESUBSET__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,A_27b)) => ! [V1s] : ( mem(V1s,arr(A_27b,bool)) => ! [V2t] : ( 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] : ( 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) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EINTER__BIGUNION,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( ( ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ! [V1t] : ( 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] : ( mem(V3s,arr(arr(A_27b,bool),bool)) => ! [V4t] : ( 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))) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EISTOPLOGY__SUBTOPOLOGY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1u] : ( 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)))) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2s] : ( 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] : ( 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) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ETOPSPACE__SUBTOPOLOGY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1u] : ( 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) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V1u)),V2s)) <=> ? [V3t] : ( mem(V3t,arr(A_27a,bool)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V3t)) & ( V2s = ap(ap(c_2Epred__set_2EINTER(A_27a),V3t),V1u) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),c_2Epred__set_2EEMPTY(A_27a))),V1s)) <=> ( V1s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__EMPTY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),c_2Epred__set_2EEMPTY(A_27a))),V1s)) <=> ( V1s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V1u)),V1u)) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1u),ap(c_2Etopology_2Etopspace(A_27a),V0top))) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__REFL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1u] : ( mem(V1u,arr(A_27a,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V1u)),V1u)) <=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1u),ap(c_2Etopology_2Etopspace(A_27a),V0top))) ) ) )))).
fof(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__SUPERSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Etopology_2Etopspace(A_27a),V0top)),V1s)) => ( ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V1s) = V0top ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__TOPSPACE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ( ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),ap(c_2Etopology_2Etopspace(A_27a),V0top)) = V0top ) )))).
fof(conj_thm_2Ereal__topology_2ESUBTOPOLOGY__UNIV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ( ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),c_2Epred__set_2EUNIV(A_27a)) = V0top ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__IMP__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V1s)),V2t)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__IMP__SUBSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V1s)),V2t)) => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( mem(V3u,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V2t)),V1s)) & p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V3u)),V1s)) ) => p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),ap(ap(c_2Epred__set_2EUNION(A_27a),V2t),V3u))),V1s)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__UNION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0top] : ( mem(V0top,ty_2Etopology_2Etopology(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ! [V3u] : ( mem(V3u,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V2t)),V1s)) & p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),V3u)),V1s)) ) => p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),ap(ap(c_2Ereal__topology_2Esubtopology(A_27a),V0top),ap(ap(c_2Epred__set_2EUNION(A_27a),V2t),V3u))),V1s)) ) ) ) ) )))).
fof(ax_thm_2Ereal__topology_2Eopen__def,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ? [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & ! [V3x_27] : ( mem(V3x_27,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),V3x_27),V1x))),V2e)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x_27),V0s)) ) ) ) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Eclosed__def,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) <=> p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) ) )).
fof(ax_thm_2Ereal__topology_2Eeuclidean,axiom, ( c_2Ereal__topology_2Eeuclidean = ap(c_2Etopology_2Etopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2EOpen) )).
fof(conj_thm_2Ereal__topology_2EOPEN__EMPTY,lemma,(p(ap(c_2Ereal__topology_2EOpen,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2EOPEN__UNIV,lemma,(p(ap(c_2Ereal__topology_2EOpen,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__BIGUNION,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => p(ap(c_2Ereal__topology_2EOpen,V1s)) ) ) ) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f))) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__EXISTS__IN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => ( ( ! [V2a] : ( mem(V2a,A_27a) => ( p(ap(V0P,V2a)) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3656(A_27a,V1Q,V2a)))) ) ) ) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3658(A_27a,V1Q,V0P)))) ) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__EXISTS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0Q] : ( mem(V0Q,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => ( ( ! [V1a] : ( mem(V1a,A_27a) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3659(A_27a,V0Q,V1a)))) ) ) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3661(A_27a,V0Q)))) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN,lemma,(( ap(c_2Etopology_2Etopspace(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ))).
fof(conj_thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN__SUBTOPOLOGY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Etopology_2Etopspace(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)) = V0s ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__REFL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V0s)) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__REFL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V0s)) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) <=> p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__SUB__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ? [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t),V0s)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__EMPTY,lemma,(p(ap(c_2Ereal__topology_2EClosed,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ECLOSED__UNIV,lemma,(p(ap(c_2Ereal__topology_2EClosed,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ECLOSED__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__BIGINTER,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => p(ap(c_2Ereal__topology_2EClosed,V1s)) ) ) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f))) ) ))).
fof(conj_thm_2Ereal__topology_2EBIGINTER__GSPEC,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) =>  ! [A_27e]: ( ne(A_27e) =>  ! [A_27f]: ( ne(A_27f) =>  ! [A_27g]: ( ne(A_27g) =>  ! [A_27h]: ( ne(A_27h) =>  ! [A_27i]: ( ne(A_27i) => ( ( ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,bool))) => ( ap(c_2Epred__set_2EBIGINTER(A_27b),ap(c_2Epred__set_2EGSPEC(arr(A_27b,bool),A_27a),f3662(A_27b,A_27a,V1f,V0P))) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3664(A_27a,A_27b,V1f,V0P)) ) ) ) ) & ( ( ! [V5P] : ( mem(V5P,arr(A_27c,arr(A_27d,bool))) => ! [V6f] : ( mem(V6f,arr(A_27c,arr(A_27d,arr(A_27e,bool)))) => ( ap(c_2Epred__set_2EBIGINTER(A_27e),ap(c_2Epred__set_2EGSPEC(arr(A_27e,bool),ty_2Epair_2Eprod(A_27c,A_27d)),ap(c_2Epair_2EUNCURRY(A_27c,A_27d,ty_2Epair_2Eprod(arr(A_27e,bool),bool)),f3666(A_27e,A_27c,A_27d,V6f,V5P)))) = ap(c_2Epred__set_2EGSPEC(A_27e,A_27e),f3669(A_27c,A_27d,A_27e,V5P,V6f)) ) ) ) ) & ! [V12P] : ( mem(V12P,arr(A_27f,arr(A_27g,arr(A_27h,bool)))) => ! [V13f] : ( mem(V13f,arr(A_27f,arr(A_27g,arr(A_27h,arr(A_27i,bool))))) => ( ap(c_2Epred__set_2EBIGINTER(A_27i),ap(c_2Epred__set_2EGSPEC(arr(A_27i,bool),ty_2Epair_2Eprod(A_27f,ty_2Epair_2Eprod(A_27g,A_27h))),ap(c_2Epair_2EUNCURRY(A_27f,ty_2Epair_2Eprod(A_27g,A_27h),ty_2Epair_2Eprod(arr(A_27i,bool),bool)),f3672(A_27f,A_27h,A_27i,A_27g,V12P,V13f)))) = ap(c_2Epred__set_2EGSPEC(A_27i,A_27i),f3676(A_27f,A_27h,A_27g,A_27i,V13f,V12P)) ) ) ) ) )))))))))))).
fof(conj_thm_2Ereal__topology_2ECLOSED__FORALL__IN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => ( ( ! [V2a] : ( mem(V2a,A_27a) => ( p(ap(V0P,V2a)) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3656(A_27a,V1Q,V2a)))) ) ) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3678(A_27a,V1Q,V0P)))) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECLOSED__FORALL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0Q] : ( mem(V0Q,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => ( ( ! [V1a] : ( mem(V1a,A_27a) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3659(A_27a,V0Q,V1a)))) ) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3679(A_27a,V0Q)))) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__DIFF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__DIFF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__BIGINTER,lemma,(! [V0s] : ( mem(V0s,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0s)) & ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1t),V0s)) => p(ap(c_2Ereal__topology_2EOpen,V1t)) ) ) ) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__BIGUNION,lemma,(! [V0s] : ( mem(V0s,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0s)) & ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1t),V0s)) => p(ap(c_2Ereal__topology_2EClosed,V1t)) ) ) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0s))) ) ))).
fof(ax_thm_2Ereal__topology_2Eball,axiom, ! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3680(V1e,V0x)) ) ) )).
fof(ax_thm_2Ereal__topology_2Ecball,axiom, ! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3681(V1e,V0x)) ) ) )).
fof(ax_thm_2Ereal__topology_2Esphere,axiom, ! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3682(V1e,V0x)) ) ) )).
fof(conj_thm_2Ereal__topology_2EIN__BALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e)))) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y))),V2e)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y))),V2e)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__SPHERE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e)))) <=> ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)) = V2e ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__BALL__0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)))) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V0x)),V1e)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__CBALL__0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V0x)),V1e)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__SPHERE__0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)))) <=> ( ap(c_2Ereal_2Eabs,V0x) = V1e ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__TRIVIAL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__TRIVIAL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ))).
fof(conj_thm_2Ereal__topology_2ECENTRE__IN__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__SUBSET__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE__SUBSET__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSET__BALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1d] : ( mem(V1d,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V1d),V2e)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1d))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSET__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1d] : ( mem(V1d,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V1d),V2e)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1d))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__MAX__UNION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),ap(ap(c_2Ereal_2Emax,V1r),V2s))) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__MIN__INTER,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),ap(ap(c_2Ereal_2Emin,V1r),V2s))) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__MAX__UNION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),ap(ap(c_2Ereal_2Emax,V1r),V2s))) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__MIN__INTER,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1d] : ( mem(V1d,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(ap(c_2Ereal_2Emin,V1d),V2e))) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1d))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V0a),V1x)),V2r)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3683(V0a)),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V0a),V1x)),V2r)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3683(V0a)),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V0a),V1x)),V2r)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3683(V0a)),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ( ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( ap(V0f,V4x) = V3y ) ) ) ) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal_2Eabs,ap(V0f,V5x)) = ap(c_2Ereal_2Eabs,V5x) ) ) ) ) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V1x)),V2r)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ( ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( ap(V0f,V4x) = V3y ) ) ) ) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal_2Eabs,ap(V0f,V5x)) = ap(c_2Ereal_2Eabs,V5x) ) ) ) ) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V1x)),V2r)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ( ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( ap(V0f,V4x) = V3y ) ) ) ) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal_2Eabs,ap(V0f,V5x)) = ap(c_2Ereal_2Eabs,V5x) ) ) ) ) => ( ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V1x)),V2r)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__SCALING,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0c)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,V0c),V1x)),ap(ap(c_2Erealax_2Ereal__mul,V0c),V2r))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3684(V0c)),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__SCALING,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0c)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,V0c),V1x)),ap(ap(c_2Erealax_2Ereal__mul,V0c),V2r))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3684(V0c)),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__DIFF__BALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))) = ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__UNION__SPHERE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))) = ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE__UNION__BALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))) = ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__DIFF__SPHERE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))) = ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__BALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECENTRE__IN__BALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__BALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ? [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))),V0s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__BALL__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) <=> ? [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))),V0s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__EQ__EMPTY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V1e),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__EMPTY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V1e),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ? [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))),V0s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) <=> ? [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))),V0s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE__EQ__EMPTY,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> p(ap(ap(c_2Erealax_2Ereal__lt,V1r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE__EMPTY,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V1r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) => ( ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ENEGATIONS__BALL,lemma,(! [V0r] : ( mem(V0r,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0r))) = ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0r)) ) ))).
fof(conj_thm_2Ereal__topology_2ENEGATIONS__CBALL,lemma,(! [V0r] : ( mem(V0r,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0r))) = ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0r)) ) ))).
fof(conj_thm_2Ereal__topology_2ENEGATIONS__SPHERE,lemma,(! [V0r] : ( mem(V0r,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0r))) = ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0r)) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1u] : ( mem(V1u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1u)),V0s)) <=> ? [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2t)) & ( V0s = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1u),V2t) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__INTER__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__INTER,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V1s)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0u),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__OPEN__IN__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & ( p(ap(c_2Ereal__topology_2EOpen,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2EOpen,V0s)) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1u] : ( mem(V1u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1u)),V0s)) <=> ? [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EClosed,V2t)) & ( V0s = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1u),V2t) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__SUBSET__EQ,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V1s)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) <=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V0u)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__INTER,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V1s)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0u),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2EClosed,V0s)) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__SUBSET__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)) ) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__SUBSET__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)) ) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2Eopen__in,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V0u)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => ? [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) & ! [V4x_27] : ( mem(V4x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x_27),V0u)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x_27),V2x))),V3e)) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x_27),V1s)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__CONTAINS__BALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => ? [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3e))),V1t)),V0s)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__CONTAINS__CBALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => ? [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3e))),V1t)),V0s)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V1t)) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__TRANS__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V2u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) ) ) ) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => p(ap(c_2Ereal__topology_2EOpen,V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V1t)) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__TRANS__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V2u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) ) ) ) <=> p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__INTER__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1u] : ( mem(V1u,arr(ty_2Erealax_2Ereal,bool)) => ! [V2v] : ( mem(V2v,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1u)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1u),V0s))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2v),V1u)) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2v)),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2v),V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__OPEN__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) <=> ( p(ap(c_2Ereal__topology_2EOpen,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__CLOSED__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) <=> ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Eclosed__segment,axiom, ! [V0l] : ( mem(V0l,ty_2Elist_2Elist(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ap(c_2Ereal__topology_2Eclosed__segment,V0l) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3686(V0l)) ) )).
fof(ax_thm_2Ereal__topology_2Eopen__segment,axiom, ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) )).
fof(conj_thm_2Ereal__topology_2EOPEN__SEGMENT__ALT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = V1b ) ) => ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3687(V0a,V1b)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2Esegment,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3688(V0a,V1b)) ) & ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEGMENT__REFL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V0a)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) & ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V1a)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> ? [V3u] : ( mem(V3u,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3u)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V3u),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) & ( V2x = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V3u)),V0a)),ap(ap(c_2Erealax_2Ereal__mul,V3u),V1b)) ) ) ) ) ) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) <=> ( ( ~~ ( V0a = V1b ) ) & ? [V4u] : ( mem(V4u,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4u)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,V4u),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) & ( V2x = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V4u)),V0a)),ap(ap(c_2Erealax_2Ereal__mul,V4u),V1b)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEGMENT__SYM,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V0a)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3b),V2a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EENDS__IN__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EENDS__NOT__IN__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEGMENT__CLOSED__OPEN,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEGMENT__OPEN__SUBSET__CLOSED,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ))).
fof(conj_thm_2Ereal__topology_2EMIDPOINT__IN__SEGMENT,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) <=> ( ~~ ( V2a = V3b ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBETWEEN__IN__SEGMENT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Ebetween,V0x),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__CONVEX__BOUND__LE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3u] : ( mem(V3u,ty_2Erealax_2Ereal) => ! [V4v] : ( mem(V4v,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V2a)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V2a)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3u)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4v)) & ( ap(ap(c_2Erealax_2Ereal__add,V3u),V4v) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V3u),V0x)),ap(ap(c_2Erealax_2Ereal__mul,V4v),V1y))),V2a)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__SEGMENT__COMPONENT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3i] : ( mem(V3i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Emin,V0a),V1b)),V2x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2x),ap(ap(c_2Ereal_2Emax,V0a),V1b))) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ESEGMENT__TRANSLATION,lemma,(( ( ! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V0c),V1a)),ap(ap(c_2Erealax_2Ereal__add,V0c),V2b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3689(V0c)),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) ) ) ) ) ) & ! [V4c] : ( mem(V4c,ty_2Erealax_2Ereal) => ! [V5a] : ( mem(V5a,ty_2Erealax_2Ereal) => ! [V6b] : ( mem(V6b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V4c),V5a)),ap(ap(c_2Erealax_2Ereal__add,V4c),V6b))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3690(V4c)),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5a),V6b))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__SEGMENT__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => ( ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V1a)),ap(V0f,V2b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__SEGMENT__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V3x) = ap(V0f,V4y) ) => ( V3x = V4y ) ) ) ) ) => ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V1a)),ap(V0f,V2b))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__OPEN__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) <=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) & ( ( ~~ ( V2x = V0a ) ) & ( ~~ ( V2x = V1b ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__OPEN__SEGMENT__ALT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) <=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) & ( ( ~~ ( V2x = V0a ) ) & ( ( ~~ ( V2x = V1b ) ) & ( ~~ ( V0a = V1b ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__CLOSED__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(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),V2x),V0a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),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),V2x),V1b))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__OPEN__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ecollinear,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))))) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V0a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V1b))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__IN__OPEN__CLOSED__SEGMENT,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V0a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),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),V2x),V1b))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ) ) ) ) & ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3a),V4b)))) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x),V3a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3a),V4b)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x),V4b))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3a),V4b)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__IN__CLOSED__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V0a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),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),V2x),V1b))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__IN__OPEN__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V0a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V1b))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Econnected,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ~~ ? [V1e1] : ( mem(V1e1,arr(ty_2Erealax_2Ereal,bool)) & ? [V2e2] : ( mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V1e1)) & ( p(ap(c_2Ereal__topology_2EOpen,V2e2)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2))) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2)),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2e2),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ~~ ? [V1e1] : ( mem(V1e1,arr(ty_2Erealax_2Ereal,bool)) & ? [V2e2] : ( mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EClosed,V1e1)) & ( p(ap(c_2Ereal__topology_2EClosed,V2e2)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2))) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2)),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2e2),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__IN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ~~ ? [V1e1] : ( mem(V1e1,arr(ty_2Erealax_2Ereal,bool)) & ? [V2e2] : ( mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1e1)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V2e2)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2))) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ( ~~ ( V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__IN__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ~~ ? [V1e1] : ( mem(V1e1,arr(ty_2Erealax_2Ereal,bool)) & ? [V2e2] : ( mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1e1)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V2e2)) & ( ( ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2) = V0s ) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ( ~~ ( V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__IN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ~~ ? [V1e1] : ( mem(V1e1,arr(ty_2Erealax_2Ereal,bool)) & ? [V2e2] : ( mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1e1)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V2e2)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2))) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ( ~~ ( V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__IN__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ~~ ? [V1e1] : ( mem(V1e1,arr(ty_2Erealax_2Ereal,bool)) & ? [V2e2] : ( mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1e1)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V2e2)) & ( ( ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2) = V0s ) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ( ~~ ( V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEXISTS__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(arr(A_27a,bool),bool)) => ( ( ? [V1s] : ( mem(V1s,arr(A_27a,bool)) & p(ap(V0P,ap(ap(c_2Epred__set_2EDIFF(A_27a),c_2Epred__set_2EUNIV(A_27a)),V1s))) ) ) <=> ? [V2s] : ( mem(V2s,arr(A_27a,bool)) & p(ap(V0P,V2s)) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CLOPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) ) => ( ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( V1t = V0s ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__SET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ~~ ? [V1e1] : ( mem(V1e1,arr(ty_2Erealax_2Ereal,bool)) & ? [V2e2] : ( mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EClosed,V1e1)) & ( p(ap(c_2Ereal__topology_2EClosed,V2e2)) & ( ( ~~ ( V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2) = V0s ) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__SET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ( ~~ ? [V1e1] : ( mem(V1e1,arr(ty_2Erealax_2Ereal,bool)) & ? [V2e2] : ( mem(V2e2,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V1e1)) & ( p(ap(c_2Ereal__topology_2EOpen,V2e2)) & ( ( ~~ ( V1e1 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V2e2 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1e1),V2e2) = V0s ) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1e1),V2e2) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTABLE__POINTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2b),V0s)) ) => ? [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2Econnected,V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V3t)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2b),V3t)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Econnected,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__REAL__LEMMA,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ! [V3e1] : ( mem(V3e1,arr(ty_2Erealax_2Ereal,bool)) => ! [V4e2] : ( mem(V4e2,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V1a)),V3e1)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V2b)),V4e2)) & ( ( ! [V5e] : ( mem(V5e,ty_2Erealax_2Ereal) => ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V6x)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V6x),V2b)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5e)) ) ) => ? [V7d] : ( mem(V7d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V7d)) & ! [V8y] : ( mem(V8y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V8y),V6x))),V7d)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V8y)),ap(V0f,V6x)))),V5e)) ) ) ) ) ) ) ) ) & ( ( ! [V9y] : ( mem(V9y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9y),V3e1)) => ? [V10e] : ( mem(V10e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V10e)) & ! [V11y_27] : ( mem(V11y_27,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),V11y_27),V9y))),V10e)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V11y_27),V3e1)) ) ) ) ) ) ) ) & ( ( ! [V12y] : ( mem(V12y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V12y),V4e2)) => ? [V13e] : ( mem(V13e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V13e)) & ! [V14y_27] : ( mem(V14y_27,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),V14y_27),V12y))),V13e)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V14y_27),V4e2)) ) ) ) ) ) ) ) & ( ~~ ? [V15x] : ( mem(V15x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V15x)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V15x),V2b)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V15x)),V3e1)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V15x)),V4e2)) ) ) ) ) ) ) ) ) ) ) ) => ? [V16x] : ( mem(V16x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V16x)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V16x),V2b)) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V16x)),V3e1)) ) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V16x)),V4e2)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__SEGMENT,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__UNIV,lemma,(p(ap(c_2Ereal__topology_2Econnected,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ECLOPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2EOpen,V0s)) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( V0s = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__BIGUNION,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0P)) => p(ap(c_2Ereal__topology_2Econnected,V1s)) ) ) ) & ( ~~ ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0P) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0P))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( p(ap(c_2Ereal__topology_2Econnected,V1t)) & ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__DIFF__OPEN__FROM__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)) & ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ( p(ap(c_2Ereal__topology_2Econnected,V2u)) & p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V1t),V0s))) ) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V2u),V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__DISJOINT__BIGUNION__OPEN__UNIQUE,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1f_27] : ( mem(V1f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Epairwise(arr(ty_2Erealax_2Ereal,bool)),c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal)),V0f)) & ( p(ap(ap(c_2Ereal__topology_2Epairwise(arr(ty_2Erealax_2Ereal,bool)),c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal)),V1f_27)) & ( ( ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2s),V0f)) => ( p(ap(c_2Ereal__topology_2EOpen,V2s)) & ( p(ap(c_2Ereal__topology_2Econnected,V2s)) & ( ~~ ( V2s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) & ( ( ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3s),V1f_27)) => ( p(ap(c_2Ereal__topology_2EOpen,V3s)) & ( p(ap(c_2Ereal__topology_2Econnected,V3s)) & ( ~~ ( V3s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) & ( ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1f_27) ) ) ) ) ) => ( V0f = V1f_27 ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__FROM__CLOSED__UNION__AND__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ( p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) & p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & p(ap(c_2Ereal__topology_2Econnected,V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__FROM__OPEN__UNION__AND__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & ( p(ap(c_2Ereal__topology_2EOpen,V1t)) & ( p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) & p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & p(ap(c_2Ereal__topology_2Econnected,V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__INDUCTION,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ! [V1Q] : ( mem(V1Q,arr(ty_2Erealax_2Ereal,bool)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V2s)) & ( ( ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V3t)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4a),V3t)) ) => ? [V5z] : ( mem(V5z,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5z),V3t)) & p(ap(V0P,V5z)) ) ) ) ) ) ) & ! [V6a] : ( mem(V6a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6a),V2s)) => ? [V7t] : ( mem(V7t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V7t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6a),V7t)) & ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ! [V9y] : ( mem(V9y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),V7t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9y),V7t)) & ( p(ap(V0P,V8x)) & ( p(ap(V0P,V9y)) & p(ap(V1Q,V8x)) ) ) ) ) => p(ap(V1Q,V9y)) ) ) ) ) ) ) ) ) ) ) => ! [V10a] : ( mem(V10a,ty_2Erealax_2Ereal) => ! [V11b] : ( mem(V11b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10a),V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V11b),V2s)) & ( p(ap(V0P,V10a)) & ( p(ap(V0P,V11b)) & p(ap(V1Q,V10a)) ) ) ) ) => p(ap(V1Q,V11b)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION__GEN,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ! [V1R] : ( mem(V1R,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V2s)) & ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(V1R,V3x),V4y)) => p(ap(ap(V1R,V4y),V3x)) ) ) ) ) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ! [V6y] : ( mem(V6y,ty_2Erealax_2Ereal) => ! [V7z] : ( mem(V7z,ty_2Erealax_2Ereal) => ( ( p(ap(ap(V1R,V5x),V6y)) & p(ap(ap(V1R,V6y),V7z)) ) => p(ap(ap(V1R,V5x),V7z)) ) ) ) ) ) & ( ( ! [V8t] : ( mem(V8t,arr(ty_2Erealax_2Ereal,bool)) => ! [V9a] : ( mem(V9a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V8t)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9a),V8t)) ) => ? [V10z] : ( mem(V10z,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10z),V8t)) & p(ap(V0P,V10z)) ) ) ) ) ) ) & ! [V11a] : ( mem(V11a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V11a),V2s)) => ? [V12t] : ( mem(V12t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V12t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V11a),V12t)) & ! [V13x] : ( mem(V13x,ty_2Erealax_2Ereal) => ! [V14y] : ( mem(V14y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V13x),V12t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V14y),V12t)) & ( p(ap(V0P,V13x)) & p(ap(V0P,V14y)) ) ) ) => p(ap(ap(V1R,V13x),V14y)) ) ) ) ) ) ) ) ) ) ) ) ) => ! [V15a] : ( mem(V15a,ty_2Erealax_2Ereal) => ! [V16b] : ( mem(V16b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V15a),V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V16b),V2s)) & ( p(ap(V0P,V15a)) & p(ap(V0P,V16b)) ) ) ) => p(ap(ap(V1R,V15a),V16b)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__INDUCTION__SIMPLE,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2a),V1s)) => ? [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2a),V3t)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V3t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V3t)) & p(ap(V0P,V4x)) ) ) => p(ap(V0P,V5y)) ) ) ) ) ) ) ) ) ) => ! [V6a] : ( mem(V6a,ty_2Erealax_2Ereal) => ! [V7b] : ( mem(V7b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6a),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7b),V1s)) & p(ap(V0P,V6a)) ) ) => p(ap(V0P,V7b)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION,lemma,(! [V0R] : ( mem(V0R,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(V0R,V2x),V3y)) => p(ap(ap(V0R,V3y),V2x)) ) ) ) ) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ! [V6z] : ( mem(V6z,ty_2Erealax_2Ereal) => ( ( p(ap(ap(V0R,V4x),V5y)) & p(ap(ap(V0R,V5y),V6z)) ) => p(ap(ap(V0R,V4x),V6z)) ) ) ) ) ) & ! [V7a] : ( mem(V7a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7a),V1s)) => ? [V8t] : ( mem(V8t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V8t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7a),V8t)) & ! [V9x] : ( mem(V9x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9x),V8t)) => p(ap(ap(V0R,V7a),V9x)) ) ) ) ) ) ) ) ) ) ) => ! [V10a] : ( mem(V10a,ty_2Erealax_2Ereal) => ! [V11b] : ( mem(V11b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10a),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V11b),V1s)) ) => p(ap(ap(V0R,V10a),V11b)) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Elimit__point__of,axiom, ! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2t)) & p(ap(c_2Ereal__topology_2EOpen,V2t)) ) => ? [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) & ( ( ~~ ( V3y = V0x ) ) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V2t)) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ELIMPT__SUBSET,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V2t)) ) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__APPROACHABLE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3x_27] : ( mem(V3x_27,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x_27),V1s)) & ( ( ~~ ( V3x_27 = V0x ) ) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x_27),V0x))),V2e)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EAPPROACHABLE__LT__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( ? [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V3x)),V2d)) => p(ap(V0P,V3x)) ) ) ) ) ) <=> ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ! [V5x] : ( mem(V5x,A_27a) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V5x)),V4d)) => p(ap(V0P,V5x)) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIMPT__APPROACHABLE__LE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3x_27] : ( mem(V3x_27,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x_27),V1s)) & ( ( ~~ ( V3x_27 = V0x ) ) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x_27),V0x))),V2e)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__CHOOSE__SIZE,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0c)) => ? [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) & ( ap(c_2Ereal_2Eabs,V1x) = V0c ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__UNIV,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__LIMPT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__EMPTY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) ) ))).
fof(conj_thm_2Ereal__topology_2ENO__LIMIT__POINT__IMP__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ~~ ? [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) ) ) => p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__POSITIVE__ORTHANT,lemma,(p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3691))))).
fof(conj_thm_2Ereal__topology_2EFINITE__SET__AVOID,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s)) => ? [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & ( ~~ ( V3x = V0a ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2d),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V3x)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMIT__POINT__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => ( ~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1a),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__SING,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMIT__POINT__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) <=> ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V0s)) | p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V1t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__INSERT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2y),V0s))) <=> p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__OF__LIMPTS,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3692(V1s)))) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__LIMPTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3693(V0s)))) ))).
fof(conj_thm_2Ereal__topology_2EDISCRETE__IMP__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3y),V2x))),V1e)) ) ) => ( V3y = V2x ) ) ) ) ) => p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__OF__UNIV,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__OF__OPEN__IN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) & ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1t)) ) ) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__OF__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__SING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) <=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) & ( ~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1a),V0s)) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Einterior,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,V0s) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3695(V0s)) ) )).
fof(conj_thm_2Ereal__topology_2EINTERIOR__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Einterior,V0s) = V0s ) <=> p(ap(c_2Ereal__topology_2EOpen,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( ap(c_2Ereal__topology_2Einterior,V0s) = V0s ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__EMPTY,lemma,(( ap(c_2Ereal__topology_2Einterior,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__UNIV,lemma,(( ap(c_2Ereal__topology_2Einterior,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Ereal__topology_2Einterior,V0s))) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Einterior,V0s)) = ap(c_2Ereal__topology_2Einterior,V0s) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,V0s)),V0s)) ))).
fof(conj_thm_2Ereal__topology_2ESUBSET__INTERIOR__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Einterior,V0s))) <=> p(ap(c_2Ereal__topology_2EOpen,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSET__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,V0s)),ap(c_2Ereal__topology_2Einterior,V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__MAXIMAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),ap(c_2Ereal__topology_2Einterior,V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__MAXIMAL__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Einterior,V1t))) <=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__UNIQUE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & ( p(ap(c_2Ereal__topology_2EOpen,V1t)) & ! [V2t_27] : ( mem(V2t_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t_27),V0s)) & p(ap(c_2Ereal__topology_2EOpen,V2t_27)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t_27),V1t)) ) ) ) ) => ( ap(c_2Ereal__topology_2Einterior,V0s) = V1t ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__INTERIOR,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Einterior,V1s))) <=> ? [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e))),V1s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__SUBSET__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Einterior,V1t))) <=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,V0s)),ap(c_2Ereal__topology_2Einterior,V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__FINITE__BIGINTER,lemma,(! [V0s] : ( mem(V0s,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0s)) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0s)) = ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),c_2Ereal__topology_2Einterior),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__BIGINTER__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f))),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),c_2Ereal__topology_2Einterior),V0f)))) ))).
fof(conj_thm_2Ereal__topology_2EUNION__INTERIOR__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,V0s)),ap(c_2Ereal__topology_2Einterior,V1t))),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)))) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Einterior,V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) => ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY__ALT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Einterior,V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V1t)) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ~~ ( ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V1t),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__LIMIT__POINT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2Einterior,V0s))) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__UNION__EMPTY__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ap(c_2Ereal__topology_2Einterior,V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)) = ap(c_2Ereal__topology_2Einterior,V0s) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__UNION__EQ__EMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) | p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => ( ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( ( ap(c_2Ereal__topology_2Einterior,V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ap(c_2Ereal__topology_2Einterior,V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__UNIONS__OPEN__SUBSETS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),f3696(V0s))) = ap(c_2Ereal__topology_2Einterior,V0s) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__ARCH__INV,lemma,(! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0e)) <=> ? [V1n] : ( mem(V1n,ty_2Enum_2Enum) & ( ( ~~ ( V1n = c_2Enum_2E0 ) ) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,V1n)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,V1n))),V0e)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__POW__LBOUND,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V1n)),V0x))),ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x)),V1n))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__ARCH__POW,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x)) => ? [V2n] : ( mem(V2n,ty_2Enum_2Enum) & p(ap(ap(c_2Erealax_2Ereal__lt,V1y),ap(ap(c_2Ereal_2Epow,V0x),V2n))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__ARCH__POW2,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ? [V1n] : ( mem(V1n,ty_2Enum_2Enum) & p(ap(ap(c_2Erealax_2Ereal__lt,V0x),ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V1n))) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__ARCH__POW__INV,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1y)) & p(ap(ap(c_2Erealax_2Ereal__lt,V0x),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) => ? [V2n] : ( mem(V2n,ty_2Enum_2Enum) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Ereal_2Epow,V0x),V2n)),V1y)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFORALL__POS__MONO,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V1d] : ( mem(V1d,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1d),V2e)) & p(ap(V0P,V1d)) ) => p(ap(V0P,V2e)) ) ) ) ) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( ~~ ( V3n = c_2Enum_2E0 ) ) => p(ap(V0P,ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,V3n)))) ) ) ) => ! [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) => p(ap(V0P,V4e)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFORALL__SUC,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,bool)) => ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ( ~~ ( V1n = c_2Enum_2E0 ) ) => p(ap(V0P,V1n)) ) ) ) <=> ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(V0P,ap(c_2Enum_2ESUC,V2n))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELT__NZ,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n)) <=> ( ~~ ( V0n = c_2Enum_2E0 ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__ARCH__RDIV__EQ__0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1c)) & ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2m)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V2m)),V0x)),V1c)) ) ) ) ) => ( V0x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Eclosure,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,V0s) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3693(V0s))) ) )).
fof(conj_thm_2Ereal__topology_2ECLOSURE__APPROACHABLE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Eclosure,V1s))) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3y),V0x))),V2e)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__NONEMPTY__OPEN__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2Eclosure,V0s))) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V2t)) & p(ap(c_2Ereal__topology_2EOpen,V2t)) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V2t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,V0s) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,V0s) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2Eclosure,V0s))) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__HULL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,V0s) = ap(ap(c_2Etopology_2Ehull(ty_2Erealax_2Ereal),c_2Ereal__topology_2EClosed),V0s) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Eclosure,V0s) = V0s ) <=> p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => ( ap(c_2Ereal__topology_2Eclosure,V0s) = V0s ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Eclosure,V0s)) = ap(c_2Ereal__topology_2Eclosure,V0s) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V0s))) ))).
fof(conj_thm_2Ereal__topology_2ESUBSET__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__INTER__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__BIGINTER__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f))),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),c_2Ereal__topology_2Eclosure),V0f)))) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__MINIMAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__MINIMAL__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V1t)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),V1t)) <=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__UNIQUE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ! [V2t_27] : ( mem(V2t_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V2t_27)) & p(ap(c_2Ereal__topology_2EClosed,V2t_27)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2t_27)) ) ) ) ) => ( ap(c_2Ereal__topology_2Eclosure,V0s) = V1t ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__EMPTY,lemma,(( ap(c_2Ereal__topology_2Eclosure,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__UNIV,lemma,(( ap(c_2Ereal__topology_2Eclosure,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__BIGUNION,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0f)) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f)) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),f3697(V0f))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__EQ__EMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Eclosure,V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__SUBSET__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),V0s)) <=> p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__EQ__EMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__IN__INTER__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V1t))) = ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V1t))) = ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V1t))),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__SUPERSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V1t))) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)) = ap(c_2Ereal__topology_2Eclosure,V0s) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__COMPLEMENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Einterior,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__COMPLEMENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Eclosure,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__INTERMEDIATE__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),ap(c_2Ereal__topology_2Eclosure,V0s))) ) ) => p(ap(c_2Ereal__topology_2Econnected,V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Ereal__topology_2Eclosure,V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__UNION__STRONG,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( p(ap(c_2Ereal__topology_2Econnected,V1t)) & ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__DIFF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__OF__CLOSURE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),ap(c_2Ereal__topology_2Eclosure,V1s))) <=> p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__LIMPT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1t)) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) <=> ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V1t)) = V1t ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE__IDEMP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s)))) = ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR__IDEMP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V0s)))) = ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ENOWHERE__DENSE__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ENOWHERE__DENSE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V1t)) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2u)) & ( ( ~~ ( V2u = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2u),V1t)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2u),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__CLOSURE__INTER__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s))),ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V1t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__INTERIOR__UNION__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V0s))),ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V1t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREGULAR__OPEN__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s)) = V0s ) & ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V1t)) = V1t ) ) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREGULAR__CLOSED__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V0s)) = V0s ) & ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V1t)) = V1t ) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREGULAR__CLOSED__BIGUNION,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0f)) & ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1t),V0f)) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V1t)) = V1t ) ) ) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f))) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIFF__CLOSURE__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t))),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t)))) ) ))).
fof(conj_thm_2Ereal__topology_2EDENSE__OPEN__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)) ) | ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V2u)) ) ) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2u),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))) <=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2u),ap(c_2Ereal__topology_2Eclosure,V0s))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2u),ap(c_2Ereal__topology_2Eclosure,V1t))) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Efrontier,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Efrontier,V0s) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Einterior,V0s)) ) )).
fof(conj_thm_2Ereal__topology_2EFRONTIER__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2Efrontier,V0s))) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__CLOSURES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Efrontier,V0s) = ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__STRADDLE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2Efrontier,V1s))) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ( ( ? [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V3x))),V2e)) ) ) ) & ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) ) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V4x))),V2e)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__EMPTY,lemma,(( ap(c_2Ereal__topology_2Efrontier,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__UNIV,lemma,(( ap(c_2Ereal__topology_2Efrontier,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),V0s)) <=> p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__COMPLEMENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s)) = ap(c_2Ereal__topology_2Efrontier,V0s) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__DISJOINT__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> p(ap(c_2Ereal__topology_2EOpen,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),ap(c_2Ereal__topology_2Efrontier,V1t)))) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__UNION__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),ap(c_2Ereal__topology_2Efrontier,V1t)))) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__INTERIORS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Efrontier,V0s) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Einterior,V0s))),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Efrontier,V0s))),ap(c_2Ereal__topology_2Efrontier,V0s))) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Efrontier,V0s)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eclosure,V0s))),ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Einterior,V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__FRONTIER__EMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) | p(ap(c_2Ereal__topology_2EClosed,V0s)) ) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Efrontier,V0s)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) | p(ap(c_2Ereal__topology_2EClosed,V0s)) ) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Efrontier,V0s)) = ap(c_2Ereal__topology_2Efrontier,V0s) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__FRONTIER__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Efrontier,V0s))) = ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Efrontier,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EUNION__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),ap(c_2Ereal__topology_2Efrontier,V1t)) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))),ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t)))),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),ap(c_2Ereal__topology_2Efrontier,V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__INTER__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Efrontier,V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__EQ__EMPTY__AS__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ap(c_2Ereal__topology_2Einterior,V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) <=> ? [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V1t)) & ( V0s = ap(c_2Ereal__topology_2Efrontier,V1t) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) => ( ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),ap(c_2Ereal__topology_2Efrontier,V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__UNION__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,V0s) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Efrontier,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__INTERIOR__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Einterior,V0s))),ap(c_2Ereal__topology_2Efrontier,V0s))) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__CLOSURE__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Eclosure,V0s))),ap(c_2Ereal__topology_2Efrontier,V0s))) ))).
fof(conj_thm_2Ereal__topology_2ESET__DIFF__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Efrontier,V0s)) = ap(c_2Ereal__topology_2Einterior,V0s) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Efrontier,V1t))),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t))))) ) ))).
fof(ax_thm_2Ereal__topology_2Eisnet,axiom,( ! [A_27a] : ( ne(A_27a) => ! [A_27b] : ( ne(A_27b) => ! [V0g] : ( mem(V0g,arr(A_27a,arr(A_27b,bool))) => ( p(ap(c_2Ereal__topology_2Eisnet(A_27a,A_27b),V0g)) <=> ! [V1x] : ( mem(V1x,A_27b) => ! [V2y] : ( mem(V2y,A_27b) => ( ( ! [V3z] : ( mem(V3z,A_27a) => ( p(ap(ap(V0g,V3z),V1x)) => p(ap(ap(V0g,V3z),V2y)) ) ) ) | ! [V4z] : ( mem(V4z,A_27a) => ( p(ap(ap(V0g,V4z),V2y)) => p(ap(ap(V0g,V4z),V1x)) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Enet__TY__DEF,axiom,( ! [A_27a] : ( ne(A_27a) => ? [V0rep] : ( mem(V0rep,arr(ty_2Ereal__topology_2Enet(A_27a),arr(A_27a,arr(A_27a,bool)))) & p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(arr(A_27a,arr(A_27a,bool)),ty_2Ereal__topology_2Enet(A_27a)),c_2Ereal__topology_2Eisnet(A_27a,A_27a)),V0rep)) ) ) )).
fof(conj_thm_2Ereal__topology_2Enet__tybij,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ( ( ! [V0a] : ( mem(V0a,ty_2Ereal__topology_2Enet(A_27a)) => ( ap(c_2Ereal__topology_2Emk__net(A_27a),ap(c_2Ereal__topology_2Enetord(A_27a),V0a)) = V0a ) ) ) & ! [V1r] : ( mem(V1r,arr(A_27b,arr(A_27b,bool))) => ( ( ! [V2x] : ( mem(V2x,A_27b) => ! [V3y] : ( mem(V3y,A_27b) => ( ( ! [V4z] : ( mem(V4z,A_27b) => ( p(ap(ap(V1r,V4z),V2x)) => p(ap(ap(V1r,V4z),V3y)) ) ) ) | ! [V5z] : ( mem(V5z,A_27b) => ( p(ap(ap(V1r,V5z),V3y)) => p(ap(ap(V1r,V5z),V2x)) ) ) ) ) ) ) <=> ( ap(c_2Ereal__topology_2Enetord(A_27b),ap(c_2Ereal__topology_2Emk__net(A_27b),V1r)) = V1r ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ENET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0n] : ( mem(V0n,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ( ! [V3z] : ( mem(V3z,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V3z),V1x)) => p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V3z),V2y)) ) ) ) | ! [V4z] : ( mem(V4z,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V4z),V2y)) => p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V4z),V1x)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EOLDNET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0n] : ( mem(V0n,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1x] : ( mem(V1x,A_27a) => ! [V2y] : ( mem(V2y,A_27a) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V1x),V1x)) & p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V2y),V2y)) ) => ? [V3z] : ( mem(V3z,A_27a) & ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V3z),V3z)) & ! [V4w] : ( mem(V4w,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V4w),V3z)) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V4w),V1x)) & p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V4w),V2y)) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ENET__DILEMMA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1Q] : ( mem(V1Q,arr(A_27a,bool)) => ! [V2net] : ( mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ( ( ( ? [V3a] : ( mem(V3a,A_27a) & ( ( ? [V4x] : ( mem(V4x,A_27a) & p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V4x),V3a)) ) ) & ! [V5x] : ( mem(V5x,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V5x),V3a)) => p(ap(V0P,V5x)) ) ) ) ) ) & ? [V6b] : ( mem(V6b,A_27a) & ( ( ? [V7x] : ( mem(V7x,A_27a) & p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V7x),V6b)) ) ) & ! [V8x] : ( mem(V8x,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V8x),V6b)) => p(ap(V1Q,V8x)) ) ) ) ) ) => ? [V9c] : ( mem(V9c,A_27a) & ( ( ? [V10x] : ( mem(V10x,A_27a) & p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V10x),V9c)) ) ) & ! [V11x] : ( mem(V11x,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V11x),V9c)) => ( p(ap(V0P,V11x)) & p(ap(V1Q,V11x)) ) ) ) ) ) ) ) ) )))).
fof(ax_thm_2Ereal__topology_2Eat,axiom, ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eat,V0a) = ap(c_2Ereal__topology_2Emk__net(ty_2Erealax_2Ereal),f3699(V0a)) ) )).
fof(ax_thm_2Ereal__topology_2Eat__infinity,axiom, ( c_2Ereal__topology_2Eat__infinity = ap(c_2Ereal__topology_2Emk__net(ty_2Erealax_2Ereal),f3701) )).
fof(ax_thm_2Ereal__topology_2Eat__posinfinity,axiom, ( c_2Ereal__topology_2Eat__posinfinity = ap(c_2Ereal__topology_2Emk__net(ty_2Erealax_2Ereal),f3703) )).
fof(ax_thm_2Ereal__topology_2Eat__neginfinity,axiom, ( c_2Ereal__topology_2Eat__neginfinity = ap(c_2Ereal__topology_2Emk__net(ty_2Erealax_2Ereal),f3705) )).
fof(ax_thm_2Ereal__topology_2Esequentially,axiom, ( c_2Ereal__topology_2Esequentially = ap(c_2Ereal__topology_2Emk__net(ty_2Enum_2Enum),f3707) )).
fof(ax_thm_2Ereal__topology_2Ewithin,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( ap(ap(c_2Ereal__topology_2Ewithin(A_27a),V0net),V1s) = ap(c_2Ereal__topology_2Emk__net(A_27a),f3709(A_27a,V0net,V1s)) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Ein__direction,axiom, ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1v] : ( mem(V1v,ty_2Erealax_2Ereal) => ( ap(ap(c_2Ereal__topology_2Ein__direction,V0a),V1v) = ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3711(V1v,V0a))) ) ) )).
fof(conj_thm_2Ereal__topology_2EAT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),V1x),V2y)) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V0a)))) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V0a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2y),V0a)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EAT__INFINITY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__infinity),V0x),V1y)) <=> p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Ereal_2Eabs,V1y))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EAT__POSINFINITY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__posinfinity),V0x),V1y)) <=> p(ap(ap(c_2Ereal_2Ereal__ge,V0x),V1y)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EAT__NEGINFINITY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__neginfinity),V0x),V1y)) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEQUENTIALLY,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(ty_2Enum_2Enum),c_2Ereal__topology_2Esequentially),V0m),V1n)) <=> p(ap(ap(c_2Earithmetic_2E_3E_3D,V0m),V1n)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EWITHIN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0n] : ( mem(V0n,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2x] : ( mem(V2x,A_27a) => ! [V3y] : ( mem(V3y,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),ap(ap(c_2Ereal__topology_2Ewithin(A_27a),V0n),V1s)),V2x),V3y)) <=> ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0n),V2x),V3y)) & p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EIN__DIRECTION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1v] : ( mem(V1v,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ein__direction,V0a),V1v)),V2x),V3y)) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V0a)))) & ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V0a))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3y),V0a)))) & ? [V4c] : ( mem(V4c,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4c)) & ( ap(ap(c_2Ereal_2Ereal__sub,V2x),V0a) = ap(ap(c_2Erealax_2Ereal__mul,V4c),V1v) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EWITHIN__UNIV,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0x)),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) = ap(c_2Ereal__topology_2Eat,V0x) ) ))).
fof(conj_thm_2Ereal__topology_2EWITHIN__WITHIN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ap(ap(c_2Ereal__topology_2Ewithin(A_27a),ap(ap(c_2Ereal__topology_2Ewithin(A_27a),V0net),V1s)),V2t) = ap(ap(c_2Ereal__topology_2Ewithin(A_27a),V0net),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)) ) ) ) )))).
fof(ax_thm_2Ereal__topology_2Etrivial__limit,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) <=> ( ( ! [V1a] : ( mem(V1a,A_27a) => ! [V2b] : ( mem(V2b,A_27a) => ( V1a = V2b ) ) ) ) | ? [V3a] : ( mem(V3a,A_27a) & ? [V4b] : ( mem(V4b,A_27a) & ( ( ~~ ( V3a = V4b ) ) & ! [V5x] : ( mem(V5x,A_27a) => ( ( ~~ p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0net),V5x),V3a)) ) & ( ~~ p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V0net),V5x),V4b)) ) ) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__WITHIN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1a)),V0s))) <=> ( ~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1a),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a))) ) ))).
fof(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__INFINITY,lemma,(( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__infinity)) ))).
fof(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__POSINFINITY,lemma,(( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__posinfinity)) ))).
fof(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__NEGINFINITY,lemma,(( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eat__neginfinity)) ))).
fof(conj_thm_2Ereal__topology_2ETRIVIAL__LIMIT__SEQUENTIALLY,lemma,(( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Enum_2Enum),c_2Ereal__topology_2Esequentially)) ))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN__CLOSED__TRIVIAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V1s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),V1s)) ) ) => p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ENONTRIVIAL__LIMIT__WITHIN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) => p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),ap(ap(c_2Ereal__topology_2Ewithin(A_27a),V0net),V1s))) ) ) )))).
fof(ax_thm_2Ereal__topology_2Eeventually,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0p] : ( mem(V0p,arr(A_27a,bool)) => ! [V1net] : ( mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V0p),V1net)) <=> ( p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V1net)) | ? [V2y] : ( mem(V2y,A_27a) & ( ( ? [V3x] : ( mem(V3x,A_27a) & p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V1net),V3x),V2y)) ) ) & ! [V4x] : ( mem(V4x,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V1net),V4x),V2y)) => p(ap(V0p,V4x)) ) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__HAPPENS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1p] : ( mem(V1p,arr(A_27a,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V1p),V0net)) => ( p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) | ? [V2x] : ( mem(V2x,A_27a) & p(ap(V1p,V2x)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2p] : ( mem(V2p,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V2p),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1a)),V0s))) <=> ? [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V1a)))) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V1a))),V3d)) ) ) => p(ap(V2p,V4x)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2p] : ( mem(V2p,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V2p),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1a)),V0s))) <=> ? [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V1a)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V1a))),V3d)) ) ) => p(ap(V2p,V4x)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__AT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1p] : ( mem(V1p,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V1p),ap(c_2Ereal__topology_2Eat,V0a))) <=> ? [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V0a)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V0a))),V2d)) ) => p(ap(V1p,V3x)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__SEQUENTIALLY,lemma,(! [V0p] : ( mem(V0p,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Enum_2Enum),V0p),c_2Ereal__topology_2Esequentially)) <=> ? [V1N] : ( mem(V1N,ty_2Enum_2Enum) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1N),V2n)) => p(ap(V0p,V2n)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY,lemma,(! [V0p] : ( mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),c_2Ereal__topology_2Eat__infinity)) <=> ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,V2x)),V1b)) => p(ap(V0p,V2x)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__POSINFINITY,lemma,(! [V0p] : ( mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),c_2Ereal__topology_2Eat__posinfinity)) <=> ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__ge,V2x),V1b)) => p(ap(V0p,V2x)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__NEGINFINITY,lemma,(! [V0p] : ( mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),c_2Ereal__topology_2Eat__neginfinity)) <=> ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) => p(ap(V0p,V2x)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY__POS,lemma,(! [V0p] : ( mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),c_2Ereal__topology_2Eat__infinity)) <=> ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1b)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,V2x)),V1b)) => p(ap(V0p,V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EALWAYS__EVENTUALLY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,arr(A_27a,bool)) => ! [V1net] : ( mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => ( ( ! [V2x] : ( mem(V2x,A_27a) => p(ap(V0p,V2x)) ) ) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V0p),V1net)) ) ) )))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__AND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1p] : ( mem(V1p,arr(A_27a,bool)) => ! [V2q] : ( mem(V2q,arr(A_27a,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3712(A_27a,V1p,V2q)),V0net)) <=> ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V1p),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V2q),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__MONO,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1p] : ( mem(V1p,arr(A_27a,bool)) => ! [V2q] : ( mem(V2q,arr(A_27a,bool)) => ( ( ( ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(V1p,V3x)) => p(ap(V2q,V3x)) ) ) ) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V1p),V0net)) ) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V2q),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__MP,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1p] : ( mem(V1p,arr(A_27a,bool)) => ! [V2q] : ( mem(V2q,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3713(A_27a,V1p,V2q)),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V1p),V0net)) ) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V2q),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__FALSE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),k(A_27a,c_2Ebool_2EF)),V0net)) <=> p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) )))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__TRUE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),k(A_27a,c_2Ebool_2ET)),V0net)) <=> $true ) )))).
fof(conj_thm_2Ereal__topology_2ENOT__EVENTUALLY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1p] : ( mem(V1p,arr(A_27a,bool)) => ( ( ( ! [V2x] : ( mem(V2x,A_27a) => ( ~~ p(ap(V1p,V2x)) ) ) ) & ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) ) => ( ~~ p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),V1p),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__FORALL,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1p] : ( mem(V1p,arr(A_27b,arr(A_27a,bool))) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s)) & ( ~~ ( V2s = c_2Epred__set_2EEMPTY(A_27b) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3715(A_27a,A_27b,V2s,V1p)),V0net)) <=> ! [V5a] : ( mem(V5a,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5a),V2s)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),ap(V1p,V5a)),V0net)) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EFORALL__EVENTUALLY,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1p] : ( mem(V1p,arr(A_27b,arr(A_27a,bool))) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s)) & ( ~~ ( V2s = c_2Epred__set_2EEMPTY(A_27b) ) ) ) => ( ( ! [V3a] : ( mem(V3a,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3a),V2s)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),ap(V1p,V3a)),V0net)) ) ) ) <=> p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3717(A_27a,A_27b,V2s,V1p)),V0net)) ) ) ) ) ))))).
fof(ax_thm_2Ereal__topology_2Etendsto,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2net] : ( mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V0f),V1l),V2net)) <=> ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3718(A_27a,V1l,V0f,V3e)),V2net)) ) ) ) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Elim__def,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ap(ap(c_2Ereal__topology_2Elim__def(A_27a),V0net),V1f) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3719(A_27a,V1f,V0net)) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ELIM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2net] : ( mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V0f),V1l),V2net)) <=> ( p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V2net)) | ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4y] : ( mem(V4y,A_27a) & ( ( ? [V5x] : ( mem(V5x,A_27a) & p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V5x),V4y)) ) ) & ! [V6x] : ( mem(V6x,A_27a) => ( p(ap(ap(ap(c_2Ereal__topology_2Enetord(A_27a),V2net),V6x),V4y)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V6x)),V1l))),V3e)) ) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN__LE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V3s))) <=> ! [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) => ? [V5d] : ( mem(V5d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5d)) & ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V3s)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6x),V2a)))) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6x),V2a))),V5d)) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V6x)),V1l))),V4e)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V3s))) <=> ! [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) => ? [V5d] : ( mem(V5d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5d)) & ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V3s)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6x),V2a)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6x),V2a))),V5d)) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V6x)),V1l))),V4e)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__AT__LE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(c_2Ereal__topology_2Eat,V2a))) <=> ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x),V2a)))) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x),V2a))),V4d)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V5x)),V1l))),V3e)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__AT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(c_2Ereal__topology_2Eat,V2a))) <=> ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x),V2a)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x),V2a))),V4d)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V5x)),V1l))),V3e)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__AT__INFINITY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),c_2Ereal__topology_2Eat__infinity)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,V4x)),V3b)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V4x)),V1l))),V2e)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__AT__INFINITY__POS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),c_2Ereal__topology_2Eat__infinity)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3b)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,V4x)),V3b)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V4x)),V1l))),V2e)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__AT__POSINFINITY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),c_2Ereal__topology_2Eat__posinfinity)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__ge,V4x),V3b)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V4x)),V1l))),V2e)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__AT__NEGINFINITY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),c_2Ereal__topology_2Eat__neginfinity)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V3b)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V4x)),V1l))),V2e)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__SEQUENTIALLY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),V1l),c_2Ereal__topology_2Esequentially)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3N] : ( mem(V3N,ty_2Enum_2Enum) & ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3N),V4n)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0s,V4n)),V1l))),V2e)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__EVENTUALLY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3720(A_27a,V1f,V2l)),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__POSINFINITY__SEQUENTIALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),c_2Ereal__topology_2Eat__posinfinity)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3721(V0f)),V1l),c_2Ereal__topology_2Esequentially)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__INFINITY__POSINFINITY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),c_2Ereal__topology_2Eat__infinity)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),c_2Ereal__topology_2Eat__posinfinity)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN__EMPTY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN__SUBSET,lemma,(! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4s] : ( mem(V4s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),V2l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3a)),V4s))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0t),V4s)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),V2l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3a)),V0t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__UNION,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V2l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V3s))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V2l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V4t))) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V2l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__UNION__UNIV,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V2l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V3s))) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V2l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V4t))) & ( ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V2l),ap(c_2Ereal__topology_2Eat,V1x))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__COMPOSE__WITHIN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ! [V5z] : ( mem(V5z,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V4y),V0net)) & ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3722(A_27a,V5z,V4y,V2g,V3s,V1f)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),V5z),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V4y)),V3s))) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2g),V1f)),V5z),V0net)) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__COMPOSE__AT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ! [V4z] : ( mem(V4z,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3y),V0net)) & ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3723(A_27a,V1f,V2g,V3y,V4z)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),V4z),ap(c_2Ereal__topology_2Eat,V3y))) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2g),V1f)),V4z),V0net)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__AT__WITHIN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(c_2Ereal__topology_2Eat,V2a))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V3s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2a),V3s)) & p(ap(c_2Ereal__topology_2EOpen,V3s)) ) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V3s))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(c_2Ereal__topology_2Eat,V2a))) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Efrom,axiom, ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ap(c_2Ereal__topology_2Efrom,V0n) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3724(V0n)) ) )).
fof(conj_thm_2Ereal__topology_2EFROM__0,lemma,(( ap(c_2Ereal__topology_2Efrom,c_2Enum_2E0) = c_2Epred__set_2EUNIV(ty_2Enum_2Enum) ))).
fof(conj_thm_2Ereal__topology_2EIN__FROM,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0m),ap(c_2Ereal__topology_2Efrom,V1n))) <=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V0m)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG__GEN,lemma,(! [V0k] : ( mem(V0k,ty_2Enum_2Enum) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),ap(c_2Ereal__topology_2Efrom,V0k)),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Eprim__rec_2E_3C,V1m),V0k)),ap(ap(c_2Eiterate_2E_2E_2E,V0k),V2n)),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG__MAX,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ! [V2p] : ( mem(V2p,ty_2Enum_2Enum) => ( ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),ap(c_2Ereal__topology_2Efrom,V2p)),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)) = ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2EMAX,V2p),V0m)),V1n) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFROM__INTER__NUMSEG,lemma,(! [V0k] : ( mem(V0k,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),ap(c_2Ereal__topology_2Efrom,V0k)),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V1n)) = ap(ap(c_2Eiterate_2E_2E_2E,V0k),V1n) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFINITE__FROM,lemma,(! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Ereal__topology_2Efrom,V0n))) ) ))).
fof(conj_thm_2Ereal__topology_2EWLOG__LT,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => ( ( ( ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => p(ap(ap(V0P,V1m),V1m)) ) ) & ( ( ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(V0P,V2m),V3n)) <=> p(ap(ap(V0P,V3n),V2m)) ) ) ) ) & ! [V4m] : ( mem(V4m,ty_2Enum_2Enum) => ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V4m),V5n)) => p(ap(ap(V0P,V4m),V5n)) ) ) ) ) ) => ! [V6m] : ( mem(V6m,ty_2Enum_2Enum) => ! [V7y] : ( mem(V7y,ty_2Enum_2Enum) => p(ap(ap(V0P,V6m),V7y)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELT__EXISTS,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) <=> ? [V2d] : ( mem(V2d,ty_2Enum_2Enum) & ( V1n = ap(ap(c_2Earithmetic_2E_2B,V0m),ap(c_2Enum_2ESUC,V2d)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT__EQ,lemma,(! [V0R] : ( mem(V0R,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => ( ( ! [V1x] : ( mem(V1x,ty_2Enum_2Enum) => ! [V2y] : ( mem(V2y,ty_2Enum_2Enum) => ! [V3z] : ( mem(V3z,ty_2Enum_2Enum) => ( ( p(ap(ap(V0R,V1x),V2y)) & p(ap(ap(V0R,V2y),V3z)) ) => p(ap(ap(V0R,V1x),V3z)) ) ) ) ) ) => ( ( ! [V4m] : ( mem(V4m,ty_2Enum_2Enum) => ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V4m),V5n)) => p(ap(ap(V0R,V4m),V5n)) ) ) ) ) <=> ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => p(ap(ap(V0R,V6n),ap(c_2Enum_2ESUC,V6n))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT,lemma,(! [V0R] : ( mem(V0R,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) => ( ( ( ! [V1x] : ( mem(V1x,ty_2Enum_2Enum) => ! [V2y] : ( mem(V2y,ty_2Enum_2Enum) => ! [V3z] : ( mem(V3z,ty_2Enum_2Enum) => ( ( p(ap(ap(V0R,V1x),V2y)) & p(ap(ap(V0R,V2y),V3z)) ) => p(ap(ap(V0R,V1x),V3z)) ) ) ) ) ) & ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(V0R,V4n),ap(c_2Enum_2ESUC,V4n))) ) ) => ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V5m),V6n)) => p(ap(ap(V0R,V5m),V6n)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__SEQUENTIAL__INJ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) <=> ? [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2f,V3n)),ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V1s),V0x))) ) ) & ( ( ! [V4m] : ( mem(V4m,ty_2Enum_2Enum) => ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( ( ap(V2f,V4m) = ap(V2f,V5n) ) <=> ( V4m = V5n ) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V2f),V0x),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__SEQUENTIAL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) <=> ? [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2f,V3n)),ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V1s),V0x))) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V2f),V0x),c_2Ereal__topology_2Esequentially)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFINITE__SUPERSET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(A_27a,bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) & p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN__BALL__CBALL,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V2t)) & p(ap(c_2Ereal__topology_2EOpen,V2t)) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V2t))) ) ) ) ) ) ) ) & ( ( ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V4x),V3s)) <=> ! [V5e] : ( mem(V5e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5e)) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3s),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V5e))))) ) ) ) ) ) ) ) & ! [V6s] : ( mem(V6s,arr(ty_2Erealax_2Ereal,bool)) => ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V7x),V6s)) <=> ! [V8e] : ( mem(V8e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V8e)) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V6s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V7x),V8e))))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V2t)) & p(ap(c_2Ereal__topology_2EOpen,V2t)) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V2t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__BALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__INFINITE__CBALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFINITE__OPEN__IN,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V0u)) ) ) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s)) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Econdensation__point__of,axiom, ! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,V0x),V1s)) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2t)) & p(ap(c_2Ereal__topology_2EOpen,V2t)) ) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1s),V2t))) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__OF__SUBSET,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,V0x),V1s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V2t)) ) => p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,V0x),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__IMP__LIMPT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,V0x),V1s)) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL__CBALL,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,V1x),V0s)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))))) ) ) ) ) ) ) ) & ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,V4x),V3s)) <=> ! [V5e] : ( mem(V5e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5e)) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V5e))))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,V1x),V0s)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__CBALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econdensation__point__of,V1x),V0s)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2e))))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__LINEAR,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1h] : ( mem(V1h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2f),V3l),V0net)) & p(ap(c_2Ereal__topology_2Elinear,V1h)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3725(A_27a,V2f,V1h)),ap(V1h,V3l)),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__CONST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),k(A_27a,V1a)),V1a),V0net)) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__CMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3c] : ( mem(V3c,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3726(A_27a,V1f,V3c)),ap(ap(c_2Erealax_2Ereal__mul,V3c),V2l)),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__CMUL__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3c] : ( mem(V3c,ty_2Erealax_2Ereal) => ( ( ~~ ( V3c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3726(A_27a,V1f,V3c)),ap(ap(c_2Erealax_2Ereal__mul,V3c),V2l)),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NEG,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3727(A_27a,V1f)),ap(c_2Erealax_2Ereal__neg,V2l)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NEG__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3727(A_27a,V1f)),ap(c_2Erealax_2Ereal__neg,V2l)),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__ADD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4m] : ( mem(V4m,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V4m),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3728(A_27a,V1f,V2g)),ap(ap(c_2Erealax_2Ereal__add,V3l),V4m)),V0net)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__ABS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3729(A_27a,V1f)),ap(c_2Ereal_2Eabs,V2l)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__SUB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4m] : ( mem(V4m,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V4m),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3730(A_27a,V1f,V2g)),ap(ap(c_2Ereal_2Ereal__sub,V3l),V4m)),V0net)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__MAX,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4m] : ( mem(V4m,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V4m),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3731(A_27a,V1f,V2g)),ap(ap(c_2Ereal_2Emax,V3l),V4m)),V0net)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__MIN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4m] : ( mem(V4m,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V4m),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3732(A_27a,V1f,V2g)),ap(ap(c_2Ereal_2Emin,V3l),V4m)),V0net)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3733(A_27a,V1f,V2l)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL__ABS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3734(A_27a,V1f)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( ~~ ( V2c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3172(A_27a,V1f,V2c)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3172(A_27a,V1f,V2c)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL__ADD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3735(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL__SUB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3736(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL__COMPARISON,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3737(A_27a,V1f,V2g)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3738(A_27a,V2g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__COMPONENT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2i] : ( mem(V2i,A_27b) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3739(A_27a,V1f)),V3l),V0net)) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__BOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3740(A_27a,V1f,V2g)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL__CMUL__BOUNDED,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3B] : ( mem(V3B,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3741(A_27a,V3B,V1f,V2g)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3742(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__SUM,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27b)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal))) => ! [V2l] : ( mem(V2l,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s)) & ! [V4i] : ( mem(V4i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4i),V3s)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27b),ap(V1f,V4i)),ap(V2l,V4i)),V0net)) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27b),f3744(A_27b,A_27a,V1f,V3s)),ap(ap(c_2Eiterate_2ESum(A_27a),V3s),V2l)),V0net)) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ELIM__NULL__SUM,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal))) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s)) & ! [V3a] : ( mem(V3a,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3a),V2s)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3745(A_27a,A_27b,V1f,V3a)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3746(A_27a,A_27b,V1f,V2s)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ELIM__IN__CLOSED__SET,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V2s)) & ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3747(A_27a,V1f,V2s)),V0net)) & ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3l),V2s)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__ABS__UBOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3748(A_27a,V1f,V3b)),V0net)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2l)),V3b)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__ABS__LBOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3749(A_27a,V1f,V3b)),V0net)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3b),ap(c_2Ereal_2Eabs,V2l))) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__UNIQUE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3l_27] : ( mem(V3l_27,ty_2Erealax_2Ereal) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l_27),V0net)) ) ) => ( V2l = V3l_27 ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ETENDSTO__LIM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) ) => ( ap(ap(c_2Ereal__topology_2Elim__def(A_27a),V0net),V1f) = V2l ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__CONST__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),k(A_27a,V1c)),V2d),V0net)) <=> ( p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) | ( V1c = V2d ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EUNIFORM__LIM__ADD,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1P] : ( mem(V1P,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27b,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V3g] : ( mem(V3g,arr(A_27b,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V4l] : ( mem(V4l,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V5m] : ( mem(V5m,arr(A_27b,ty_2Erealax_2Ereal)) => ( ( ( ! [V6e] : ( mem(V6e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V6e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3751(A_27a,A_27b,V1P,V4l,V2f,V6e)),V0net)) ) ) ) & ! [V9e] : ( mem(V9e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V9e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3753(A_27a,A_27b,V1P,V5m,V3g,V9e)),V0net)) ) ) ) => ! [V12e] : ( mem(V12e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V12e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3755(A_27a,A_27b,V1P,V5m,V4l,V3g,V2f,V12e)),V0net)) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EUNIFORM__LIM__SUB,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1P] : ( mem(V1P,arr(A_27b,bool)) => ! [V2f] : ( mem(V2f,arr(A_27b,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V3g] : ( mem(V3g,arr(A_27b,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V4l] : ( mem(V4l,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V5m] : ( mem(V5m,arr(A_27b,ty_2Erealax_2Ereal)) => ( ( ( ! [V6e] : ( mem(V6e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V6e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3751(A_27a,A_27b,V1P,V4l,V2f,V6e)),V0net)) ) ) ) & ! [V9e] : ( mem(V9e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V9e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3753(A_27a,A_27b,V1P,V5m,V3g,V9e)),V0net)) ) ) ) => ! [V12e] : ( mem(V12e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V12e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3757(A_27a,A_27b,V1P,V5m,V4l,V3g,V2f,V12e)),V0net)) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EUNIFORM__LIM__BILINEAR,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1P] : ( mem(V1P,arr(A_27b,bool)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V3f] : ( mem(V3f,arr(A_27b,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V4g] : ( mem(V4g,arr(A_27b,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V5l] : ( mem(V5l,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V6m] : ( mem(V6m,arr(A_27b,ty_2Erealax_2Ereal)) => ! [V7b1] : ( mem(V7b1,ty_2Erealax_2Ereal) => ! [V8b2] : ( mem(V8b2,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ebilinear,V2h)) & ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),k(A_27a,ap(c_2Ebool_2E_21(A_27b),f3758(A_27b,V7b1,V5l,V1P)))),V0net)) & ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),k(A_27a,ap(c_2Ebool_2E_21(A_27b),f3759(A_27b,V8b2,V6m,V1P)))),V0net)) & ( ( ! [V13e] : ( mem(V13e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V13e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3761(A_27a,A_27b,V1P,V5l,V3f,V13e)),V0net)) ) ) ) & ! [V16e] : ( mem(V16e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V16e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3763(A_27a,A_27b,V1P,V6m,V4g,V16e)),V0net)) ) ) ) ) ) ) => ! [V19e] : ( mem(V19e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V19e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3765(A_27a,A_27b,V1P,V6m,V5l,V2h,V4g,V3f,V19e)),V0net)) ) ) ) ) ) ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ELIM__BILINEAR,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1h] : ( mem(V1h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V2f] : ( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3g] : ( mem(V3g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V4l] : ( mem(V4l,ty_2Erealax_2Ereal) => ! [V5m] : ( mem(V5m,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2f),V4l),V0net)) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V3g),V5m),V0net)) & p(ap(c_2Ereal__topology_2Ebilinear,V1h)) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3766(A_27a,V1h,V2f,V3g)),ap(ap(V1h,V4l),V5m)),V0net)) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN__ID,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),V0a),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),V1s))) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__AT__ID,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),V0a),ap(c_2Ereal__topology_2Eat,V0a))) ))).
fof(conj_thm_2Ereal__topology_2ELIM__AT__ZERO,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(c_2Ereal__topology_2Eat,V2a))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),f3767(V2a,V0f)),V1l),ap(c_2Ereal__topology_2Eat,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Enetlimit,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ( ap(c_2Ereal__topology_2Enetlimit(A_27a),V0net) = ap(c_2Emin_2E_40(A_27a),f3769(A_27a,V0net)) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ENETLIMIT__WITHIN,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),V1s))) ) => ( ap(c_2Ereal__topology_2Enetlimit(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),V1s)) = V0a ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ENETLIMIT__AT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Enetlimit(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)) = V0a ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3770(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V3l),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__EVENTUALLY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3771(A_27a,V1f,V2g)),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V3l),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN,lemma,(! [V0l] : ( mem(V0l,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4s] : ( mem(V4s,arr(ty_2Erealax_2Ereal,bool)) => ! [V5d] : ( mem(V5d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5d)) & ( ( ! [V6x_27] : ( mem(V6x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x_27),V4s)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6x_27),V3x)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6x_27),V3x))),V5d)) ) ) => ( ap(V1f,V6x_27) = ap(V2g,V6x_27) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),V0l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3x)),V4s))) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),V0l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3x)),V4s))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AT,lemma,(! [V0l] : ( mem(V0l,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ( ( ! [V5x_27] : ( mem(V5x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x_27),V3x)))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x_27),V3x))),V4d)) ) => ( ap(V1f,V5x_27) = ap(V2g,V5x_27) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),V0l),ap(c_2Ereal__topology_2Eat,V3x))) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),V0l),ap(c_2Ereal__topology_2Eat,V3x))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__EQ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3770(A_27a,V1f,V2g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0net)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V3l),V0net)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET,lemma,(! [V0l] : ( mem(V0l,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),f3772(V3s,V4t)),ap(c_2Ereal__topology_2Eat,V2a))) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),V0l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V3s))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),V0l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V4t))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET__IMP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),f3773(V4t,V3s)),ap(c_2Ereal__topology_2Eat,V2a))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V3s))) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V4t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__WITHIN,lemma,(! [V0l] : ( mem(V0l,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ! [V5s] : ( mem(V5s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V3a = V4b ) ) & ( ( ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V5s)) & ( ( ~~ ( V6x = V3a ) ) & ( ~~ ( V6x = V4b ) ) ) ) => ( ap(V1f,V6x) = ap(V2g,V6x) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),V0l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3a)),V5s))) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),V0l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3a)),V5s))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__AT,lemma,(! [V0l] : ( mem(V0l,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V3a = V4b ) ) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V5x = V3a ) ) & ( ~~ ( V5x = V4b ) ) ) => ( ap(V1f,V5x) = ap(V2g,V5x) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),V0l),ap(c_2Ereal__topology_2Eat,V3a))) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),V0l),ap(c_2Ereal__topology_2Eat,V3a))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4l] : ( mem(V4l,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EOpen,V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3a),V2s)) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V2s)) & ( ~~ ( V5x = V3a ) ) ) => ( ap(V0f,V5x) = ap(V1g,V5x) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V4l),ap(c_2Ereal__topology_2Eat,V3a))) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1g),V4l),ap(c_2Ereal__topology_2Eat,V3a))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN__IN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5l] : ( mem(V5l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4a),V2s)) & ( ( ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V2s)) & ( ~~ ( V6x = V4a ) ) ) => ( ap(V0f,V6x) = ap(V1g,V6x) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V5l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V4a)),V3t))) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1g),V5l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V4a)),V3t))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__CASES__FINITE__SEQUENTIALLY,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3774(V0P)))) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3775(V1f,V0P,V2g)),V3l),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V2g),V3l),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__CASES__COFINITE__SEQUENTIALLY,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Enum_2Enum,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3776(V0P)))) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3775(V1f,V0P,V2g)),V3l),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1f),V3l),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__CASES__SEQUENTIALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3777(V0f,V3m,V1g)),V2l),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),V2l),c_2Ereal__topology_2Esequentially)) ) & ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3778(V0f,V3m,V1g)),V2l),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),V2l),c_2Ereal__topology_2Esequentially)) ) & ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3779(V0f,V3m,V1g)),V2l),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1g),V2l),c_2Ereal__topology_2Esequentially)) ) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3780(V0f,V3m,V1g)),V2l),c_2Ereal__topology_2Esequentially)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1g),V2l),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__CONG__WITHIN,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4s] : ( mem(V4s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( ~~ ( V5x = V0a ) ) => ( ap(V1f,V5x) = ap(V2g,V5x) ) ) ) ) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),f3781(V1f)),V3l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),V4s))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),V3l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),V4s))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__CONG__AT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( ( ~~ ( V4x = V0a ) ) => ( ap(V1f,V4x) = ap(V2g,V4x) ) ) ) ) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),f3782(V1f)),V3l),ap(c_2Ereal__topology_2Eat,V0a))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2g),V3l),ap(c_2Ereal__topology_2Eat,V0a))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__SEQUENTIAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1l),ap(c_2Ereal__topology_2Eclosure,V0s))) <=> ? [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V3n)),V0s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V2x),V1l),c_2Ereal__topology_2Esequentially)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__CONTAINS__SEQUENTIAL__LIMIT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V1x,V3n)),V0s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1x),V2l),c_2Ereal__topology_2Esequentially)) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2l),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__SEQUENTIAL__LIMITS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) <=> ! [V1x] : ( mem(V1x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V1x,V3n)),V0s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1x),V2l),c_2Ereal__topology_2Esequentially)) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2l),V0s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__APPROACHABLE,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V1s)) => ( ( ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3y),V0x))),V2e)) ) ) ) ) ) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__CLOSURE__DELETE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V0s),V1x)))) <=> p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDENSE__IMP__PERFECT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Eclosure,V0s) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDENSE__LIMIT__POINTS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,A_27a) => ( ( ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3783(V0s)) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) <=> ( ap(c_2Ereal__topology_2Eclosure,V0s) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ESEQ__OFFSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2k] : ( mem(V2k,ty_2Enum_2Enum) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),V1l),c_2Ereal__topology_2Esequentially)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3784(V2k,V0f)),V1l),c_2Ereal__topology_2Esequentially)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEQ__OFFSET__NEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2k] : ( mem(V2k,ty_2Enum_2Enum) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),V1l),c_2Ereal__topology_2Esequentially)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3785(V2k,V0f)),V1l),c_2Ereal__topology_2Esequentially)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEQ__OFFSET__REV,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2k] : ( mem(V2k,ty_2Enum_2Enum) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3784(V2k,V0f)),V1l),c_2Ereal__topology_2Esequentially)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),V1l),c_2Ereal__topology_2Esequentially)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEQ__HARMONIC__OFFSET,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3786(V0a)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)) ))).
fof(conj_thm_2Ereal__topology_2ESEQ__HARMONIC,lemma,(p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3787),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)))).
fof(conj_thm_2Ereal__topology_2ECLOSED__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__INTERIOR__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Einterior,V1s))) <=> ? [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e))),V1s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__BALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1y),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e)))) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V2e)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__BALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e))) = ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__BALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))) = ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e))) = ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__BALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1e))) = ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1e)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__CBALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1e))) = ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1e)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__EQ__EMPTY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> p(ap(ap(c_2Erealax_2Ereal__lt,V1e),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__EMPTY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V1e),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__EQ__SING,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) <=> ( V1e = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__SING,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ( V1e = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE__SING,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ( V1e = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) => ( ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE__EQ__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) <=> ( ( V2x = V0a ) & ( V1r = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEVENTUALLY__WITHIN__INTERIOR,lemma,(! [V0p] : ( mem(V0p,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Einterior,V1s))) => ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V1s))) <=> p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),V0p),ap(c_2Ereal__topology_2Eat,V2x))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN__INTERIOR,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),ap(c_2Ereal__topology_2Einterior,V2s))) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3x)),V2s))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(c_2Ereal__topology_2Eat,V3x))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ENETLIMIT__WITHIN__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2Einterior,V0s))) => ( ap(c_2Ereal__topology_2Enetlimit(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V0s)) = V1x ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( ( ~~ ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ( V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) => p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ( V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Ebounded__def,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2x)),V1a)) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EBOUNDED__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Ebounded__def,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) ) => p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Einterior,V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Eclosure,V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__CLOSURE__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Eclosure,V0s))) <=> p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__BALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__IMP__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) <=> ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__BIGUNION,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0f)) & ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) ) ) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0f))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__POS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1b)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2x)),V1b)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__POS__LT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1b)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V2x)),V1b)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) | p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__DIFF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__INSERT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),V1s))) <=> p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__BIGINTER,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ? [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) ) ) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f))) ) ))).
fof(conj_thm_2Ereal__topology_2ENOT__BOUNDED__UNIV,lemma,(( ~~ p(ap(c_2Ereal__topology_2Ebounded__def,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ))).
fof(conj_thm_2Ereal__topology_2ECOBOUNDED__IMP__UNBOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) => ( ~~ p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) & p(ap(c_2Ereal__topology_2Elinear,V0f)) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SCALING,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3788(V0c)),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__NEGATIONS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__TRANSLATION__EQ,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))) <=> p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__DIFFS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3791(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,A_27a),f3792(A_27a,V0f,V2t)))) & p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,A_27a),f3793(A_27a,V1g,V2t)))) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,A_27a),f3794(A_27a,V0f,V1g,V2t)))) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGES,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal))) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s)) & ! [V3a] : ( mem(V3a,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3a),V2s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,A_27a),f3795(A_27b,A_27a,V0f,V3a,V1t)))) ) ) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,A_27a),f3796(A_27b,A_27a,V0f,V2s,V1t)))) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__BALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ? [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2r)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CBALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ? [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2r)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNBOUNDED__INTER__COBOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) & p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V1t))) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOBOUNDED__INTER__UNBOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) & ( ~~ p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__BOUNDED__EQ__TRIVIAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Esubspace,V0s)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> ( V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__COMPONENTWISE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__HAS__SUP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1x),ap(c_2Ereal_2Esup,V0s))) ) ) ) & ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V2b)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUP__INSERT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) => ( ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),V1s)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),V1s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0x),ap(ap(c_2Ereal_2Emax,V0x),ap(c_2Ereal_2Esup,V1s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__HAS__INF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V1x)) ) ) ) & ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2b),V3x)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2b),ap(c_2Eiterate_2Einf,V0s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINF__INSERT,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) => ( ap(c_2Eiterate_2Einf,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),V1s)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),V1s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0x),ap(ap(c_2Ereal_2Emin,V0x),ap(c_2Eiterate_2Einf,V1s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSET__BALLS,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1a_27] : ( mem(V1a_27,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ! [V3r_27] : ( mem(V3r_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a_27),V3r_27)))) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1a_27))),V2r)),V3r_27)) | p(ap(ap(c_2Ereal_2Ereal__lte,V2r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) & ( ( ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5a_27] : ( mem(V5a_27,ty_2Erealax_2Ereal) => ! [V6r] : ( mem(V6r,ty_2Erealax_2Ereal) => ! [V7r_27] : ( mem(V7r_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V6r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5a_27),V7r_27)))) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V5a_27))),V6r)),V7r_27)) | p(ap(ap(c_2Ereal_2Ereal__lte,V6r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) & ( ( ! [V8a] : ( mem(V8a,ty_2Erealax_2Ereal) => ! [V9a_27] : ( mem(V9a_27,ty_2Erealax_2Ereal) => ! [V10r] : ( mem(V10r,ty_2Erealax_2Ereal) => ! [V11r_27] : ( mem(V11r_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V10r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V9a_27),V11r_27)))) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V9a_27))),V10r)),V11r_27)) | p(ap(ap(c_2Erealax_2Ereal__lt,V10r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) & ! [V12a] : ( mem(V12a,ty_2Erealax_2Ereal) => ! [V13a_27] : ( mem(V13a_27,ty_2Erealax_2Ereal) => ! [V14r] : ( mem(V14r,ty_2Erealax_2Ereal) => ! [V15r_27] : ( mem(V15r_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V14r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V13a_27),V15r_27)))) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V13a_27))),V14r)),V15r_27)) | p(ap(ap(c_2Erealax_2Ereal__lt,V14r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTER__BALLS__EQ__EMPTY,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V3s))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V2r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V3s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) | p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,V2r),V3s)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ) ) ) ) ) ) & ( ( ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5b] : ( mem(V5b,ty_2Erealax_2Ereal) => ! [V6r] : ( mem(V6r,ty_2Erealax_2Ereal) => ! [V7s] : ( mem(V7s,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V6r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5b),V7s))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V6r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) | ( p(ap(ap(c_2Erealax_2Ereal__lt,V7s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) | p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,V6r),V7s)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V5b)))) ) ) ) ) ) ) ) ) & ( ( ! [V8a] : ( mem(V8a,ty_2Erealax_2Ereal) => ! [V9b] : ( mem(V9b,ty_2Erealax_2Ereal) => ! [V10r] : ( mem(V10r,ty_2Erealax_2Ereal) => ! [V11s] : ( mem(V11s,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V10r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V9b),V11s))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V10r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V11s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) | p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,V10r),V11s)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V9b)))) ) ) ) ) ) ) ) ) & ! [V12a] : ( mem(V12a,ty_2Erealax_2Ereal) => ! [V13b] : ( mem(V13b,ty_2Erealax_2Ereal) => ! [V14r] : ( mem(V14r,ty_2Erealax_2Ereal) => ! [V15s] : ( mem(V15s,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V14r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V13b),V15s))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V14r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) | ( p(ap(ap(c_2Erealax_2Ereal__lt,V15s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) | p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__add,V14r),V15s)),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V13b)))) ) ) ) ) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Ecompact,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) <=> ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V1f,V2n)),V0s)) ) ) => ? [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) & ? [V4r] : ( mem(V4r,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3l),V0s)) & ( ( ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V5m),V6n)) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(V4r,V5m)),ap(V4r,V6n))) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Enum_2Enum),V1f),V4r)),V3l),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EMONOTONE__BIGGER,lemma,(! [V0r] : ( mem(V0r,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ( ( ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V1m),V2n)) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(V0r,V1m)),ap(V0r,V2n))) ) ) ) ) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Earithmetic_2E_3C_3D,V3n),ap(V0r,V3n))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__SUBSEQUENCE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1r] : ( mem(V1r,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V3m),V4n)) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(V1r,V3m)),ap(V1r,V4n))) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),V2l),c_2Ereal__topology_2Esequentially)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Enum_2Enum),V0s),V1r)),V2l),c_2Ereal__topology_2Esequentially)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EMONOTONE__SUBSEQUENCE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ? [V1r] : ( mem(V1r,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) & ( ( ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V2m),V3n)) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(V1r,V2m)),ap(V1r,V3n))) ) ) ) ) & ( ( ! [V4m] : ( mem(V4m,ty_2Enum_2Enum) => ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V4m),V5n)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,ap(V1r,V4m))),ap(V0s,ap(V1r,V5n)))) ) ) ) ) | ! [V6m] : ( mem(V6m,ty_2Enum_2Enum) => ! [V7n] : ( mem(V7n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V6m),V7n)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,ap(V1r,V7n))),ap(V0s,ap(V1r,V6m)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONVERGENT__BOUNDED__INCREASING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ( ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V3n)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,V2m)),ap(V0s,V3n))) ) ) ) ) & ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0s,V4n))),V1b)) ) ) => ? [V5l] : ( mem(V5l,ty_2Erealax_2Ereal) & ! [V6e] : ( mem(V6e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V6e)) => ? [V7N] : ( mem(V7N,ty_2Enum_2Enum) & ! [V8n] : ( mem(V8n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V7N),V8n)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0s,V8n)),V5l))),V6e)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONVERGENT__BOUNDED__MONOTONE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0s,V2n))),V1b)) ) ) & ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,V3m)),ap(V0s,V4n))) ) ) ) ) | ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V5m),V6n)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,V6n)),ap(V0s,V5m))) ) ) ) ) ) => ? [V7l] : ( mem(V7l,ty_2Erealax_2Ereal) & ! [V8e] : ( mem(V8e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V8e)) => ? [V9N] : ( mem(V9N,ty_2Enum_2Enum) & ! [V10n] : ( mem(V10n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V9N),V10n)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0s,V10n)),V7l))),V8e)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__REAL__LEMMA,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0s,V2n))),V1b)) ) ) => ? [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) & ? [V4r] : ( mem(V4r,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) & ( ( ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V5m),V6n)) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(V4r,V5m)),ap(V4r,V6n))) ) ) ) ) & ! [V7e] : ( mem(V7e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V7e)) => ? [V8N] : ( mem(V8N,ty_2Enum_2Enum) & ! [V9n] : ( mem(V9n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V8N),V9n)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0s,ap(V4r,V9n))),V3l))),V7e)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0x,V2n)),V1s)) ) ) => ? [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) & ? [V4r] : ( mem(V4r,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) & ( ( ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V5m),V6n)) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(V4r,V5m)),ap(V4r,V6n))) ) ) ) ) & ! [V7e] : ( mem(V7e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V7e)) => ? [V8N] : ( mem(V8N,ty_2Enum_2Enum) & ! [V9n] : ( mem(V9n,ty_2Enum_2Enum) => ! [V10i] : ( mem(V10i,A_27a) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V8N),V9n)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0x,ap(V4r,V9n))),V3l))),V7e)) ) ) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__IMP__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V0s)) ) => p(ap(c_2Ereal__topology_2Ecompact,V0s)) ) ))).
fof(ax_thm_2Ereal__topology_2Ecauchy,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Ecauchy,V0s)) <=> ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) => ? [V2N] : ( mem(V2N,ty_2Enum_2Enum) & ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V3m),V2N)) & p(ap(ap(c_2Earithmetic_2E_3E_3D,V4n),V2N)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0s,V3m)),ap(V0s,V4n)))),V1e)) ) ) ) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Ecomplete,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecomplete,V0s)) <=> ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V1f,V2n)),V0s)) ) ) & p(ap(c_2Ereal__topology_2Ecauchy,V1f)) ) => ? [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3l),V0s)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1f),V3l),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ECAUCHY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Ecauchy,V0s)) <=> ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) => ? [V2N] : ( mem(V2N,ty_2Enum_2Enum) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V3n),V2N)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0s,V3n)),ap(V0s,V2N)))),V1e)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONVERGENT__IMP__CAUCHY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),V1l),c_2Ereal__topology_2Esequentially)) => p(ap(c_2Ereal__topology_2Ecauchy,V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EGREATER__EQ__REFL,lemma,(! [V0m] : ( mem(V0m,ty_2Enum_2Enum) => p(ap(ap(c_2Earithmetic_2E_3E_3D,V0m),V0m)) ))).
fof(conj_thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET__REAL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(A_27a,bool)) => ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3x)),V2a)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECAUCHY__IMP__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Ecauchy,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3798(V0s)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__IMP__COMPLETE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecomplete,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPLETE__UNIV,lemma,(p(ap(c_2Ereal__topology_2Ecomplete,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ECOMPLETE__EQ__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecomplete,V0s)) <=> p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECONVERGENT__EQ__CAUCHY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( ? [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),V1l),c_2Ereal__topology_2Esequentially)) ) ) <=> p(ap(c_2Ereal__topology_2Ecauchy,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECONVERGENT__IMP__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),V1l),c_2Ereal__topology_2Esequentially)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Erealax_2Ereal),V0s),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__IMP__TOTALLY__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) => ? [V2k] : ( mem(V2k,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V2k)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2k),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool)),f3799(V1e)),V2k)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHEINE__BOREL__LEMMA,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => ! [V1t] : ( mem(V1t,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1t))) & ! [V2b] : ( mem(V2b,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2b),V1t)) => p(ap(c_2Ereal__topology_2EOpen,V2b)) ) ) ) => ? [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) => ? [V5b] : ( mem(V5b,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V5b),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V3e))),V5b)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__IMP__HEINE__BOREL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => ! [V1f] : ( mem(V1f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2t),V1f)) => p(ap(c_2Ereal__topology_2EOpen,V2t)) ) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1f))) ) => ? [V3f_27] : ( mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f)) & ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V3f_27))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHEINE__BOREL__IMP__BOLZANO__WEIERSTRASS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V1f] : ( mem(V1f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2t),V1f)) => p(ap(c_2Ereal__topology_2EOpen,V2t)) ) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1f))) ) => ? [V3f_27] : ( mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f)) & ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V3f_27))) ) ) ) ) ) ) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V4t)) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V0s)) ) => ? [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V5x),V4t)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V1t)) ) ) ) ) => p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EINF__FINITE__LEMMA,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),V0s)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINF__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Eiterate_2Einf,V0s)),V0s)) & ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V1x)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LE__INF__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Eiterate_2Einf,V0s))) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__INF__LE__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V1a)) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1a)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LT__INF__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),ap(c_2Eiterate_2Einf,V0s))) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__INF__LT__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Eiterate_2Einf,V0s)),V1a)) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1a)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEQUENCE__INFINITE__LEMMA,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ~~ ( ap(V0f,V2n) = V1l ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),V1l),c_2Ereal__topology_2Esequentially)) ) => ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3801(V0f)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELE__1,lemma,(( ( ! [V0n] : ( mem(V0n,ty_2Enum_2Enum) => ( ( ~~ ( V0n = c_2Enum_2E0 ) ) => p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n)) ) ) ) & ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( ( ~~ ( V1n = c_2Enum_2E0 ) ) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V1n)) ) ) ) & ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2n)) => ( ~~ ( V2n = c_2Enum_2E0 ) ) ) ) ) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V3n)) => p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V3n)) ) ) ) & ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V4n)) => p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V4n)) ) ) ) & ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V5n)) => ( ~~ ( V5n = c_2Enum_2E0 ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__OF__SEQUENCE__SUBSEQUENCE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1l),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Erealax_2Ereal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) => ? [V2r] : ( mem(V2r,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) & ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V3m),V4n)) => p(ap(ap(c_2Eprim__rec_2E_3C,ap(V2r,V3m)),ap(V2r,V4n))) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Enum_2Enum),V0f),V2r)),V1l),c_2Ereal__topology_2Esequentially)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEQUENCE__UNIQUE__LIMPT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2l_27] : ( mem(V2l_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),V1l),c_2Ereal__topology_2Esequentially)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2l_27),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3801(V0f)))) ) => ( V2l_27 = V1l ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V1t)) ) ) ) ) ) => p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) <=> ! [V1f] : ( mem(V1f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2t),V1f)) => p(ap(c_2Ereal__topology_2EOpen,V2t)) ) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1f))) ) => ? [V3f_27] : ( mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f)) & ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V3f_27))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__EQ__BOLZANO__WEIERSTRASS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) <=> ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V1t)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__EQ__BOUNDED__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) <=> ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__IMP__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__IMP__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__SEQUENCE__WITH__LIMIT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),V1l),c_2Ereal__topology_2Esequentially)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1l),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Erealax_2Ereal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) ) => p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPACT__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1t)) <=> ( p(ap(c_2Ereal__topology_2Ecompact,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL__SUBTOPOLOGY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) <=> ! [V1f] : ( mem(V1f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2t),V1f)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V2t)) ) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1f))) ) => ? [V3f_27] : ( mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f)) & ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V3f_27))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Eclosure,V0s))) <=> p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__CONTRAPOS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => ( ~~ p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V1t)) ) ) ) ) ) => p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDISCRETE__BOUNDED__IMP__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) & ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3y),V2x))),V1e)) ) ) => ( V3y = V2x ) ) ) ) ) & p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ) => p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) ) ) => ? [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V1x),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__EQ__BOLZANO__WEIERSTRASS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) <=> ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) ) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V1t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Ecompact,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__UNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__INTER__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INTER__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__BIGINTER,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => p(ap(c_2Ereal__topology_2Ecompact,V1s)) ) ) ) & ( ~~ ( V0f = c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool)) ) ) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f))) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__IMP__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => p(ap(c_2Ereal__topology_2EClosed,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__IMP__CLOSED__IN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__IMP__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => p(ap(c_2Ereal__topology_2Ecompact,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__INSERT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V1s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__SING,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0u)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__SING,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INSERT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V1s)) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__CBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__FRONTIER__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Efrontier,V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Efrontier,V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Efrontier,V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__SUBSET__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s)),V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__DELETE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V0s),V1x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__DELETE,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V1s),V2a))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__BIGINTER__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) <=> ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e))),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__BIGUNION,lemma,(! [V0s] : ( mem(V0s,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V0s)) & ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1t),V0s)) => p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) ) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__DIFF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__SPHERE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SPHERE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__SPHERE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)))) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Efrontier,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__IMP__FIP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( ( ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2t),V1f)) => p(ap(c_2Ereal__topology_2EClosed,V2t)) ) ) ) & ! [V3f_27] : ( mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f)) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V3f_27)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V1f)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IMP__FIP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ( ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2t),V1f)) => p(ap(c_2Ereal__topology_2EClosed,V2t)) ) ) ) & ( ( ? [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3t),V1f)) & p(ap(c_2Ereal__topology_2Ebounded__def,V3t)) ) ) ) & ! [V4f_27] : ( mem(V4f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V4f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V4f_27),V1f)) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V4f_27)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V1f)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IMP__FIP__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ( ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2t),V1f)) => p(ap(c_2Ereal__topology_2Ecompact,V2t)) ) ) ) & ! [V3f_27] : ( mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V1f)) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V3f_27)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) => ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V1f)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__FIP,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1t),V0f)) => p(ap(c_2Ereal__topology_2EClosed,V1t)) ) ) ) & ( ( ? [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2t),V0f)) & p(ap(c_2Ereal__topology_2Ebounded__def,V2t)) ) ) ) & ! [V3f_27] : ( mem(V3f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V3f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V3f_27),V0f)) ) => ( ~~ ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V3f_27) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) => ( ~~ ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__FIP,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1t),V0f)) => p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) ) ) & ! [V2f_27] : ( mem(V2f_27,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),V2f_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V2f_27),V0f)) ) => ( ~~ ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V2f_27) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ( ~~ ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__NEST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => p(ap(c_2Ereal__topology_2EClosed,ap(V0s,V1n))) ) ) & ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ~~ ( ap(V0s,V2n) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) & ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0s,V4n)),ap(V0s,V3m))) ) ) ) ) & p(ap(c_2Ereal__topology_2Ebounded__def,ap(V0s,c_2Enum_2E0))) ) ) ) => ? [V5a] : ( mem(V5a,ty_2Erealax_2Ereal) & ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5a),ap(V0s,V6n))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDECREASING__CLOSED__NEST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => p(ap(c_2Ereal__topology_2EClosed,ap(V0s,V1n))) ) ) & ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ~~ ( ap(V0s,V2n) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) & ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0s,V4n)),ap(V0s,V3m))) ) ) ) ) & ! [V5e] : ( mem(V5e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5e)) => ? [V6n] : ( mem(V6n,ty_2Enum_2Enum) & ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ! [V8y] : ( mem(V8y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),ap(V0s,V6n))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8y),ap(V0s,V6n))) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V7x),V8y))),V5e)) ) ) ) ) ) ) ) ) ) => ? [V9a] : ( mem(V9a,ty_2Erealax_2Ereal) & ! [V10n] : ( mem(V10n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9a),ap(V0s,V10n))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDECREASING__CLOSED__NEST__SING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => p(ap(c_2Ereal__topology_2EClosed,ap(V0s,V1n))) ) ) & ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ~~ ( ap(V0s,V2n) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) & ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0s,V4n)),ap(V0s,V3m))) ) ) ) ) & ! [V5e] : ( mem(V5e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5e)) => ? [V6n] : ( mem(V6n,ty_2Enum_2Enum) & ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ! [V8y] : ( mem(V8y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),ap(V0s,V6n))) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8y),ap(V0s,V6n))) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V7x),V8y))),V5e)) ) ) ) ) ) ) ) ) ) => ? [V9a] : ( mem(V9a,ty_2Erealax_2Ereal) & ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),f3803(V0s))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V9a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__CHAIN,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1b] : ( mem(V1b,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2s),V0f)) => ( p(ap(c_2Ereal__topology_2EClosed,V2s)) & ( ~~ ( V2s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) & ( ( ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3s),V0f)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V4t),V0f)) ) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3s),V4t)) | p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V3s)) ) ) ) ) ) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1b),V0f)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1b)) ) ) ) => ( ~~ ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__CHAIN,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => ( p(ap(c_2Ereal__topology_2Ecompact,V1s)) & ( ~~ ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) & ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2s),V0f)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3t),V0f)) ) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2s),V3t)) | p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),V2s)) ) ) ) ) ) => ( ~~ ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__NEST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(c_2Ereal__topology_2Ecompact,ap(V0s,V1n))) & ( ~~ ( ap(V0s,V1n) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) & ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V3n)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0s,V3n)),ap(V0s,V2m))) ) ) ) ) => ( ~~ ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum),f3804(V0s))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Erealax_2Ereal))) => ( ( ? [V2l] : ( mem(V2l,arr(A_27a,ty_2Erealax_2Ereal)) & ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4N] : ( mem(V4N,ty_2Enum_2Enum) & ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ! [V6x] : ( mem(V6x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V4N),V5n)) & p(ap(V0P,V6x)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(V1s,V5n),V6x)),ap(V2l,V6x)))),V3e)) ) ) ) ) ) ) ) ) <=> ! [V7e] : ( mem(V7e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V7e)) => ? [V8N] : ( mem(V8N,ty_2Enum_2Enum) & ! [V9m] : ( mem(V9m,ty_2Enum_2Enum) => ! [V10n] : ( mem(V10n,ty_2Enum_2Enum) => ! [V11x] : ( mem(V11x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V8N),V9m)) & ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V8N),V10n)) & p(ap(V0P,V11x)) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(V1s,V9m),V11x)),ap(ap(V1s,V10n),V11x)))),V7e)) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY__ALT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Erealax_2Ereal))) => ( ( ? [V2l] : ( mem(V2l,arr(A_27a,ty_2Erealax_2Ereal)) & ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4N] : ( mem(V4N,ty_2Enum_2Enum) & ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ! [V6x] : ( mem(V6x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V4N),V5n)) & p(ap(V0P,V6x)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(V1s,V5n),V6x)),ap(V2l,V6x)))),V3e)) ) ) ) ) ) ) ) ) <=> ! [V7e] : ( mem(V7e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V7e)) => ? [V8N] : ( mem(V8N,ty_2Enum_2Enum) & ! [V9m] : ( mem(V9m,ty_2Enum_2Enum) => ! [V10n] : ( mem(V10n,ty_2Enum_2Enum) => ! [V11x] : ( mem(V11x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V8N),V9m)) & ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V8N),V10n)) & ( p(ap(ap(c_2Eprim__rec_2E_3C,V9m),V10n)) & p(ap(V0P,V11x)) ) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(V1s,V9m),V11x)),ap(ap(V1s,V10n),V11x)))),V7e)) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CAUCHY__IMP__UNIFORMLY__CONVERGENT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V2l] : ( mem(V2l,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( ( ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4N] : ( mem(V4N,ty_2Enum_2Enum) & ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ! [V7x] : ( mem(V7x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V4N),V5m)) & ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V4N),V6n)) & p(ap(V0P,V7x)) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(V1s,V5m),V7x)),ap(ap(V1s,V6n),V7x)))),V3e)) ) ) ) ) ) ) ) ) & ! [V8x] : ( mem(V8x,A_27a) => ( p(ap(V0P,V8x)) => ! [V9e] : ( mem(V9e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V9e)) => ? [V10N] : ( mem(V10N,ty_2Enum_2Enum) & ! [V11n] : ( mem(V11n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V10N),V11n)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(V1s,V11n),V8x)),ap(V2l,V8x)))),V9e)) ) ) ) ) ) ) ) ) => ! [V12e] : ( mem(V12e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V12e)) => ? [V13N] : ( mem(V13N,ty_2Enum_2Enum) & ! [V14n] : ( mem(V14n,ty_2Enum_2Enum) => ! [V15x] : ( mem(V15x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V13N),V14n)) & p(ap(V0P,V15x)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(V1s,V14n),V15x)),ap(V2l,V15x)))),V12e)) ) ) ) ) ) ) ) ) ) )))).
fof(ax_thm_2Ereal__topology_2Econtinuous,axiom,( ! [A_27a] : ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1net] : ( mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V1net)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V0f),ap(V0f,ap(c_2Ereal__topology_2Enetlimit(A_27a),V1net))),V1net)) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__TRIVIAL__LIMIT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1net] : ( mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V1net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V1net)) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V0s))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V1f),ap(V1f,V2x)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1x))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),ap(V0f,V1x)),ap(c_2Ereal__topology_2Eat,V1x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1x))) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__CLOSED__NONTRIVIAL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V2s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V2s)) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1a)),V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) & ( ( ! [V5x_27] : ( mem(V5x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x_27),V2s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x_27),V3x))),V4d)) ) => ( ap(V0f,V5x_27) = ap(V1g,V5x_27) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3x)),V2s))) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3x)),V2s))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__AT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & ( ( ! [V4x_27] : ( mem(V4x_27,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),V4x_27),V2x))),V3d)) => ( ap(V0f,V4x_27) = ap(V1g,V4x_27) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V2x))) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(c_2Ereal__topology_2Eat,V2x))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EOpen,V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3a),V2s)) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V2s)) => ( ap(V0f,V4x) = ap(V1g,V4x) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V3a))) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(c_2Ereal__topology_2Eat,V3a))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN__IN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4a),V2s)) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V2s)) => ( ap(V0f,V5x) = ap(V1g,V5x) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V4a)),V3t))) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V4a)),V3t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__SET__IMP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Erealax_2Ereal),f3805(V3t,V2s)),ap(c_2Ereal__topology_2Eat,V1a))) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1a)),V2s))) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1a)),V3t))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2Econtinuous__within,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V2s))) <=> ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ! [V5x_27] : ( mem(V5x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x_27),V2s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x_27),V1x))),V4d)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V5x_27)),ap(V0f,V1x)))),V3e)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2Econtinuous__at,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1x))) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & ! [V4x_27] : ( mem(V4x_27,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),V4x_27),V1x))),V3d)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V4x_27)),ap(V0f,V1x)))),V2e)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__BALL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V1s))) <=> ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V4d))),V1s))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V2x)),V3e)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__BALL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1x))) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V3d)))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V1x)),V2e)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPARISON,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3a)),V2s))) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V3a)),ap(V0f,V4x)))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V1g,V3a)),ap(V1g,V4x))))) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3a)),V2s))) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Econtinuous__on,axiom, ! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ! [V5x_27] : ( mem(V5x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x_27),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x_27),V2x))),V4d)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V5x_27)),ap(V0f,V2x)))),V3e)) ) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Euniformly__continuous__on,axiom, ! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5x_27] : ( mem(V5x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x_27),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x_27),V4x))),V3d)) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V5x_27)),ap(V0f,V4x)))),V2e)) ) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CONTINUOUS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__IMP__CONTINUOUS__ON,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V2x))) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__WITHIN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V1s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),ap(V0f,V2x)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V1s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__AT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V1s)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V2x))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3x)),V1s))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t),V1s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V3x)),V2t))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t),V1s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2t),V1s)) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INTERIOR,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Einterior,V1s))) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V2x))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) => ( ap(V0f,V3x) = ap(V1g,V3x) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) => ( ap(V0f,V3x) = ap(V1g,V3x) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1g),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SING,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__EMPTY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__NO__LIMPT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ~~ ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V2x),V1s)) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__FINITE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTRACTION__IMP__CONTINUOUS__ON,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V1f,V2x)),ap(V1f,V3y)))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3y)))) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EISOMETRY__ON__IMP__CONTINUOUS__ON,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) ) => ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V1f,V2x)),ap(V1f,V3y))) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3y)) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFORALL__POS__MONO__1,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V1d] : ( mem(V1d,ty_2Erealax_2Ereal) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1d),V2e)) & p(ap(V0P,V1d)) ) => p(ap(V0P,V2e)) ) ) ) ) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(V0P,ap(c_2Erealax_2Einv,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,V3n)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))))) ) ) => ! [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) => p(ap(V0P,V4e)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SEQUENTIALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V1s))) <=> ! [V3x] : ( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V3x,V4n)),V1s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V3x),V2a),c_2Ereal__topology_2Esequentially)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3x)),ap(V0f,V2a)),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__SEQUENTIALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1a))) <=> ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V2x),V1a),c_2Ereal__topology_2Esequentially)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x)),ap(V0f,V1a)),c_2Ereal__topology_2Esequentially)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SEQUENTIALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3a),V1s)) & ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V4n)),V1s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V2x),V3a),c_2Ereal__topology_2Esequentially)) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x)),ap(V0f,V3a)),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SEQUENTIALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) <=> ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3y] : ( mem(V3y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V4n)),V1s)) ) ) & ( ( ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V3y,V5n)),V1s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3806(V2x,V3y)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3807(V2x,V3y,V0f)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__CONTINUOUS__FUNCTION,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1net] : ( mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V3l))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V3l),V1net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3808(A_27a,V2g,V0f)),ap(V0f,V3l)),V1net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CONST,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),k(A_27a,V1c)),V0net)) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2net] : ( mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V2net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3356(A_27a,V0f,V1c)),V2net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__NEG,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1net] : ( mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V1net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3358(A_27a,V0f)),V1net)) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ADD,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2net] : ( mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V2net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1g),V2net)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3352(A_27a,V0f,V1g)),V2net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__SUB,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2net] : ( mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V2net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1g),V2net)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3359(A_27a,V0f,V1g)),V2net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ABS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1net] : ( mem(V1net,ty_2Ereal__topology_2Enet(A_27a)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V1net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3360(A_27a,V0f)),V1net)) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__MAX,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2net] : ( mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V2net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1g),V2net)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3809(A_27a,V0f,V1g)),V2net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__MIN,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2net] : ( mem(V2net,ty_2Ereal__topology_2Enet(A_27a)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V0f),V2net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1g),V2net)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3810(A_27a,V0f,V1g)),V2net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__SUM,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27b,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V2s] : ( mem(V2s,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s)) & ! [V3a] : ( mem(V3a,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3a),V2s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),ap(V1f,V3a)),V0net)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3812(A_27a,A_27b,V1f,V2s)),V0net)) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CONST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,k(ty_2Erealax_2Ereal,V1c)),V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CMUL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3813(V0f,V1c)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__NEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f2906(V0f)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ADD,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f2904(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUB,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f2908(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ABS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3814(V0f)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MAX,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3815(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MIN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3816(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ! [V3a] : ( mem(V3a,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3a),V2s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(V1f,V3a)),V0t)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3818(A_27a,V1f,V2s)),V0t)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CONST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,k(ty_2Erealax_2Ereal,V1c)),V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__UNIFORMLY__CONTINUOUS__ON,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1g),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__UNIFORMLY__CONTINUOUS__ON__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V3s)) & ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1g),V3s)) & ( p(ap(c_2Ereal__topology_2Ebilinear,V2h)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3s))) & p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3s))) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f3819(V2h,V0f,V1g)),V3s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__MUL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) & ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1g),V2s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))) & p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V2s))) ) ) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f2905(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CMUL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f3813(V0f,V1c)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__VMUL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2v] : ( mem(V2v,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1c),V0s)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f3820(V1c,V2v)),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__NEG,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f2906(V0f)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ADD,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1g),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f2904(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUB,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V1g),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f2908(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V2s] : ( mem(V2s,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s)) & ! [V3a] : ( mem(V3a,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3a),V2s)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,ap(V1f,V3a)),V0t)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f3818(A_27a,V1f,V2s)),V0t)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__ID,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V0a)),V1s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__ID,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Eat,V0a))) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ID,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,i(ty_2Erealax_2Ereal)),V0s)) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ID,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,i(ty_2Erealax_2Ereal)),V0s)) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V3s))) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,ap(V0f,V2x))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3s)))) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V3s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V2x))) & p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1g),ap(c_2Ereal__topology_2Eat,ap(V0f,V2x)))) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),ap(c_2Ereal__topology_2Eat,V2x))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V2u))) <=> ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V3t)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V1x)),V3t)) ) => ? [V4s] : ( mem(V4s,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V4s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V4s)) & ! [V5x_27] : ( mem(V5x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x_27),V4s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x_27),V2u)) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V5x_27)),V3t)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1x))) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V2t)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V1x)),V2t)) ) => ? [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V3s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V3s)) & ! [V4x_27] : ( mem(V4x_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x_27),V3s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V4x_27)),V2t)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__GEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V2t)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__GEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__OPEN__IN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V2t)) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED__GEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V2t)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__GEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__CLOSED__IN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V2t)) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(c_2Ereal__topology_2EOpen,V2t)) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(c_2Ereal__topology_2EClosed,V2t)) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( p(ap(c_2Ereal__topology_2EOpen,V1s)) & p(ap(c_2Ereal__topology_2EOpen,V2t)) ) ) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1s)) & p(ap(c_2Ereal__topology_2EClosed,V2t)) ) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE__UNIV,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V2x))) ) ) & p(ap(c_2Ereal__topology_2EOpen,V1s)) ) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3823(V1s,V0f)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__UNIV,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V2x))) ) ) & p(ap(c_2Ereal__topology_2EClosed,V1s)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3823(V1s,V0f)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V2t)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V2t)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3822(V1s,V0f,V2t)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__LIM__0,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2Eat,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__AT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1a))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__WITHIN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V1s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__COMPOSE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) & p(ap(c_2Ereal__topology_2Elinear,V2g)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3824(A_27a,V1f,V2g)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(c_2Ereal__topology_2Elinear,V1g)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f2910(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__COMPONENT__COMPOSE,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2i] : ( mem(V2i,A_27b) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3825(A_27a,V1f)),V0net)) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENT__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3826(V0f)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__COMPOSE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3h] : ( mem(V3h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V2g),V0net)) & p(ap(c_2Ereal__topology_2Ebilinear,V3h)) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3827(A_27a,V3h,V1f,V2g)),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__ON__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V3s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3s)) & p(ap(c_2Ereal__topology_2Ebilinear,V2h)) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3819(V2h,V0f,V1g)),V3s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBILINEAR__DOT,lemma,(p(ap(c_2Ereal__topology_2Ebilinear,f3829)))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__DOT2,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V2g),V0net)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3830(A_27a,V1f,V2g)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DOT2,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f2905(V0f,V1g)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(c_2Ereal__topology_2Ecompact,V1s)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__TRANSLATION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3641(V1a)),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__TRANSLATION__EQ,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))) <=> p(ap(c_2Ereal__topology_2Ecompact,V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V1s)) & p(ap(c_2Ereal__topology_2Elinear,V0f)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CONTINUOUS__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(c_2Ereal__topology_2Econnected,V1s)) ) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V1s)) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__TRANSLATION__EQ,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))) <=> p(ap(c_2Ereal__topology_2Econnected,V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & p(ap(c_2Ereal__topology_2Elinear,V0f)) ) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUASICOMPACT__OPEN__CLOSED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( ( ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) ) ) <=> ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5u),V2t)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3831(V1s,V0f,V5u)))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V5u)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) & ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__CLOSED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) & ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__IMP__QUOTIENT__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V2t)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2t))) ) ) ) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3832(V1s,V0f,V3t)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V3t)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__IMP__QUOTIENT__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V2t)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2t))) ) ) ) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3832(V1s,V0f,V3t)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),V3t)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__RIGHT__INVERSE__IMP__QUOTIENT__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V2s)) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V3t)) => ( ap(V0f,ap(V1g,V4y)) = V4y ) ) ) ) ) ) ) => ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5u),V3t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3833(V2s,V0f,V5u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V5u)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__LEFT__INVERSE__IMP__QUOTIENT__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) => ( ap(V1g,ap(V0f,V3x)) = V3x ) ) ) ) ) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4u),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3834(V2s,V0f,V4u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))),V4u)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__CLOSED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( ( ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) ) ) <=> ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5u),V2t)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3831(V1s,V0f,V5u)))) <=> p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V5u)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE__QUOTIENT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ( ( ! [V5v] : ( mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V3t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3835(V2s,V0f,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V5v)) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V2s)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFUNCTION__FACTORS__LEFT__GEN,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,A_27b)) => ! [V2g] : ( mem(V2g,arr(A_27a,A_27c)) => ( ( ! [V3x] : ( mem(V3x,A_27a) => ! [V4y] : ( mem(V4y,A_27a) => ( ( p(ap(V0P,V3x)) & ( p(ap(V0P,V4y)) & ( ap(V2g,V3x) = ap(V2g,V4y) ) ) ) => ( ap(V1f,V3x) = ap(V1f,V4y) ) ) ) ) ) <=> ? [V5h] : ( mem(V5h,arr(A_27c,A_27b)) & ! [V6x] : ( mem(V6x,A_27a) => ( p(ap(V0P,V6x)) => ( ap(V1f,V6x) = ap(V5h,ap(V2g,V6x)) ) ) ) ) ) ) ) )))))).
fof(conj_thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1h] : ( mem(V1h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s) = V3t ) & ( ( ! [V5v] : ( mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V3t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3835(V2s,V0f,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V5v)) ) ) ) ) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1h),V2s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1h),V2s) = V4u ) & ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ! [V8y] : ( mem(V8y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8y),V2s)) & ( ap(V0f,V7x) = ap(V0f,V8y) ) ) ) => ( ap(V1h,V7x) = ap(V1h,V8y) ) ) ) ) ) ) ) ) => ? [V9g] : ( mem(V9g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V9g),V3t)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V9g),V3t) = V4u ) & ! [V10x] : ( mem(V10x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10x),V2s)) => ( ap(V1h,V10x) = ap(V9g,ap(V0f,V10x)) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( ( ! [V5v] : ( mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V3t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3835(V2s,V0f,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V5v)) ) ) ) ) & ! [V7v] : ( mem(V7v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V7v),V4u)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3836(V3t,V1g,V7v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),V7v)) ) ) ) ) ) => ! [V9v] : ( mem(V9v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V9v),V4u)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3837(V2s,V0f,V1g,V9v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),V9v)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__COMPOSITION,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ! [V5v] : ( mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V4u)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3838(V2s,V0f,V1g,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),V5v)) ) ) ) ) ) ) ) => ! [V7v] : ( mem(V7v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V7v),V4u)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3836(V3t,V1g,V7v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),V7v)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2t)),V3u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V2t)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V3u ) & ! [V4v] : ( mem(V4v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4v),V3u)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3839(V1s,V0f,V4v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3u)),V4v)) ) ) ) ) ) ) ) => ! [V6v] : ( mem(V6v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6v),V3u)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3840(V2t,V0f,V6v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3u)),V6v)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__RESTRICT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3c] : ( mem(V3c,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) & ( ( ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3841(V1s,V0f,V4u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V4u)) ) ) ) ) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3c)) | p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3c)) ) ) ) => ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6u),V3c)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3842(V1s,V0f,V3c)))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3844(V6u,V0f,V3c,V1s)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3c)),V6u)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) ) ) & ( ( ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V2t)) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3845(V1s,V0f,V5y)))) ) ) ) & p(ap(c_2Ereal__topology_2Econnected,V2t)) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE__GEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3c] : ( mem(V3c,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3841(V1s,V0f,V4u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V4u)) ) ) ) ) & ( ( ! [V6y] : ( mem(V6y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6y),V2t)) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3846(V1s,V0f,V6y)))) ) ) ) & ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3c)) | p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3c)) ) & p(ap(c_2Ereal__topology_2Econnected,V3c)) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3847(V1s,V0f,V3c)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__CLOSURES,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V1s)) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ) <=> ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosure,V2s)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__INTERIORS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V1s)) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ) <=> ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Einterior,V2s))),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__RESTRICT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4u))) ) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t_27),V2t)) ) => ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3848(V1s,V0f,V3t_27)))),V5u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t_27)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__RESTRICT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4u))) ) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t_27),V2t)) ) => ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3848(V1s,V0f,V3t_27)))),V5u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t_27)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__MAP__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) & ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) ) ) => ( ( ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5k))) ) ) ) <=> ! [V6k] : ( mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3849(V1s,V0f,V6k)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EQUOTIENT__MAP__CLOSED__MAP__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) & ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) ) ) => ( ( ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5k))) ) ) ) <=> ! [V6k] : ( mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3849(V1s,V0f,V6k)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__IMP__OPEN__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3u))) ) ) ) & ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3850(V1s,V0f,V4u)))) ) ) ) ) => ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__IMP__CLOSED__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3u))) ) ) ) & ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3850(V1s,V0f,V4u)))) ) ) ) ) => ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__SURJECTIVE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s) = V3t ) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V5k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V5k))) ) ) ) ) ) => ! [V6k] : ( mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V6k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V6k))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__SURJECTIVE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s) = V3t ) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V5k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V5k))) ) ) ) ) ) => ! [V6k] : ( mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V6k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V6k))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__INJECTIVE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ! [V6y] : ( mem(V6y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V3t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6y),V3t)) & ( ap(V1g,V5x) = ap(V1g,V6y) ) ) ) => ( V5x = V6y ) ) ) ) ) & ! [V7k] : ( mem(V7k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V7k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V7k))) ) ) ) ) ) ) => ! [V8k] : ( mem(V8k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V8k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V8k))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__INJECTIVE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ! [V6y] : ( mem(V6y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V3t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6y),V3t)) & ( ap(V1g,V5x) = ap(V1g,V6y) ) ) ) => ( V5x = V6y ) ) ) ) ) & ! [V7k] : ( mem(V7k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V7k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V7k))) ) ) ) ) ) ) => ! [V8k] : ( mem(V8k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V8k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V8k))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ! [V4w] : ( mem(V4w,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5k))) ) ) ) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4w),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3851(V1s,V0f,V4w))),V3u)) ) ) ) => ? [V7v] : ( mem(V7v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V7v)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4w),V7v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3852(V1s,V0f,V7v))),V3u)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( ( ! [V3k] : ( mem(V3k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3k)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3k))) ) ) ) <=> ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ! [V5w] : ( mem(V5w,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5w),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3853(V1s,V0f,V5w))),V4u)) ) ) => ? [V7v] : ( mem(V7v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V7v)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5w),V7v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3852(V1s,V0f,V7v))),V4u)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ! [V4w] : ( mem(V4w,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5k))) ) ) ) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4w),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3851(V1s,V0f,V4w))),V3u)) ) ) ) => ? [V7v] : ( mem(V7v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V7v)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4w),V7v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3852(V1s,V0f,V7v))),V3u)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( ( ! [V3k] : ( mem(V3k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3k))) ) ) ) <=> ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ! [V5w] : ( mem(V5w,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5w),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3853(V1s,V0f,V5w))),V4u)) ) ) => ? [V7v] : ( mem(V7v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V7v)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5w),V7v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3852(V1s,V0f,V7v))),V4u)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBIGUNION__GSPEC,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) =>  ! [A_27c]: ( ne(A_27c) =>  ! [A_27d]: ( ne(A_27d) =>  ! [A_27e]: ( ne(A_27e) =>  ! [A_27f]: ( ne(A_27f) =>  ! [A_27g]: ( ne(A_27g) =>  ! [A_27h]: ( ne(A_27h) =>  ! [A_27i]: ( ne(A_27i) => ( ( ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,bool))) => ( ap(c_2Epred__set_2EBIGUNION(A_27b),ap(c_2Epred__set_2EGSPEC(arr(A_27b,bool),A_27a),f3662(A_27b,A_27a,V1f,V0P))) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3855(A_27a,A_27b,V1f,V0P)) ) ) ) ) & ( ( ! [V5P] : ( mem(V5P,arr(A_27c,arr(A_27d,bool))) => ! [V6f] : ( mem(V6f,arr(A_27c,arr(A_27d,arr(A_27e,bool)))) => ( ap(c_2Epred__set_2EBIGUNION(A_27e),ap(c_2Epred__set_2EGSPEC(arr(A_27e,bool),ty_2Epair_2Eprod(A_27c,A_27d)),ap(c_2Epair_2EUNCURRY(A_27c,A_27d,ty_2Epair_2Eprod(arr(A_27e,bool),bool)),f3666(A_27e,A_27c,A_27d,V6f,V5P)))) = ap(c_2Epred__set_2EGSPEC(A_27e,A_27e),f3858(A_27c,A_27d,A_27e,V5P,V6f)) ) ) ) ) & ! [V12P] : ( mem(V12P,arr(A_27f,arr(A_27g,arr(A_27h,bool)))) => ! [V13f] : ( mem(V13f,arr(A_27f,arr(A_27g,arr(A_27h,arr(A_27i,bool))))) => ( ap(c_2Epred__set_2EBIGUNION(A_27i),ap(c_2Epred__set_2EGSPEC(arr(A_27i,bool),ty_2Epair_2Eprod(A_27f,ty_2Epair_2Eprod(A_27g,A_27h))),ap(c_2Epair_2EUNCURRY(A_27f,ty_2Epair_2Eprod(A_27g,A_27h),ty_2Epair_2Eprod(arr(A_27i,bool),bool)),f3672(A_27f,A_27h,A_27i,A_27g,V12P,V13f)))) = ap(c_2Epred__set_2EGSPEC(A_27i,A_27i),f3862(A_27f,A_27h,A_27g,A_27i,V13f,V12P)) ) ) ) ) )))))))))))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__POINT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( ( ! [V3k] : ( mem(V3k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3k))) ) ) ) <=> ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3845(V1s,V0f,V5y))),V4u)) ) ) => ? [V7v] : ( mem(V7v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V7v)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V7v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3852(V1s,V0f,V7v))),V4u)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__OPEN__MONOTONE__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V3c] : ( mem(V3c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3c)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3c))) ) ) ) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V2t)) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3863(V1s,V0f,V4y)))) ) ) ) ) ) => ! [V6c] : ( mem(V6c,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V6c)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6c),V2t)) ) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3864(V1s,V0f,V6c)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CLOSED__MONOTONE__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V3c] : ( mem(V3c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3c)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3c))) ) ) ) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V2t)) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3863(V1s,V0f,V4y)))) ) ) ) ) ) => ! [V6c] : ( mem(V6c,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V6c)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6c),V2t)) ) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3864(V1s,V0f,V6c)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPROPER__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( ( ! [V3k] : ( mem(V3k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3k),V2t)) & p(ap(c_2Ereal__topology_2Ecompact,V3k)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3865(V1s,V0f,V3k)))) ) ) ) <=> ( ( ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5k)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5k))) ) ) ) & ! [V6a] : ( mem(V6a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6a),V2t)) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3866(V1s,V0f,V6a)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1s)) & ( ap(V0f,V2x) = ap(V0f,V3y) ) ) ) => ( V2x = V3y ) ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V4t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V1s)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPACT,lemma,(! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3k] : ( mem(V3k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V2s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V2s)),V0t)) & ( p(ap(c_2Ereal__topology_2Ecompact,V2s)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0t)),V3k)) ) ) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3867(V2s,V1f,V3k)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPROPER__MAP__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( ( ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5k),V3t)) & p(ap(c_2Ereal__topology_2Ecompact,V5k)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3868(V2s,V0f,V5k)))) ) ) ) & ! [V7k] : ( mem(V7k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V7k),V4u)) & p(ap(c_2Ereal__topology_2Ecompact,V7k)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3869(V3t,V1g,V7k)))) ) ) ) ) => ! [V9k] : ( mem(V9k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V9k),V4u)) & p(ap(c_2Ereal__topology_2Ecompact,V9k)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3870(V2s,V0f,V1g,V9k)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__LEFT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s) = V3t ) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5k),V4u)) & p(ap(c_2Ereal__topology_2Ecompact,V5k)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3871(V2s,V0f,V1g,V5k)))) ) ) ) ) ) ) => ! [V7k] : ( mem(V7k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V7k),V4u)) & p(ap(c_2Ereal__topology_2Ecompact,V7k)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3869(V3t,V1g,V7k)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__RIGHT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ! [V5k] : ( mem(V5k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5k),V4u)) & p(ap(c_2Ereal__topology_2Ecompact,V5k)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3871(V2s,V0f,V1g,V5k)))) ) ) ) ) ) ) => ! [V7k] : ( mem(V7k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V7k),V3t)) & p(ap(c_2Ereal__topology_2Ecompact,V7k)) ) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3872(V2s,V0f,V7k)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPASTING__LEMMA,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4k] : ( mem(V4k,arr(A_27a,bool)) => ( ( ( ! [V5i] : ( mem(V5i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5i),V4k)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3s)),ap(V2t,V5i))) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(V0f,V5i)),ap(V2t,V5i))) ) ) ) ) & ( ( ! [V6i] : ( mem(V6i,A_27a) => ! [V7j] : ( mem(V7j,A_27a) => ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6i),V4k)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7j),V4k)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3s),ap(V2t,V6i))),ap(V2t,V7j)))) ) ) => ( ap(ap(V0f,V6i),V8x) = ap(ap(V0f,V7j),V8x) ) ) ) ) ) ) & ! [V9x] : ( mem(V9x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9x),V3s)) => ? [V10j] : ( mem(V10j,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V10j),V4k)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9x),ap(V2t,V10j))) & ( ap(V1g,V9x) = ap(ap(V0f,V10j),V9x) ) ) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3s)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1t] : ( mem(V1t,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3k] : ( mem(V3k,arr(A_27a,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),A_27a),f3873(A_27a,V1t,V3k))))) & ( ( ! [V5i] : ( mem(V5i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5i),V3k)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(V1t,V5i))) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(V0f,V5i)),ap(V1t,V5i))) ) ) ) ) & ! [V6i] : ( mem(V6i,A_27a) => ! [V7j] : ( mem(V7j,A_27a) => ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6i),V3k)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7j),V3k)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2s),ap(V1t,V6i))),ap(V1t,V7j)))) ) ) => ( ap(ap(V0f,V6i),V8x) = ap(ap(V0f,V7j),V8x) ) ) ) ) ) ) ) => ? [V9g] : ( mem(V9g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V9g),V2s)) & ! [V10x] : ( mem(V10x,ty_2Erealax_2Ereal) => ! [V11i] : ( mem(V11i,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V11i),V3k)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10x),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2s),ap(V1t,V11i)))) ) => ( ap(V9g,V10x) = ap(ap(V0f,V11i),V10x) ) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL__OPEN,lemma,(! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V2s),V0t))),V2s)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V2s),V0t))),V0t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0t)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V2s),V0t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V1s)) & ( p(ap(c_2Ereal__topology_2EOpen,V2t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2t)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1s),V2t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL__OPEN,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))),V3s)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))),V4t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V3s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V4t)) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V3s)) & ( ~~ p(ap(V0P,V5x)) ) ) | ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V4t)) & p(ap(V0P,V5x)) ) ) => ( ap(V1f,V5x) = ap(V2g,V5x) ) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3874(V1f,V0P,V2g)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__OPEN,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V3s)) & ( p(ap(c_2Ereal__topology_2EOpen,V4t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V3s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V4t)) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V3s)) & ( ~~ p(ap(V0P,V5x)) ) ) | ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V4t)) & p(ap(V0P,V5x)) ) ) => ( ap(V1f,V5x) = ap(V2g,V5x) ) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3874(V1f,V0P,V2g)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPASTING__LEMMA__CLOSED,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4k] : ( mem(V4k,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V4k)) & ( ( ! [V5i] : ( mem(V5i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5i),V4k)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3s)),ap(V2t,V5i))) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(V0f,V5i)),ap(V2t,V5i))) ) ) ) ) & ( ( ! [V6i] : ( mem(V6i,A_27a) => ! [V7j] : ( mem(V7j,A_27a) => ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6i),V4k)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7j),V4k)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3s),ap(V2t,V6i))),ap(V2t,V7j)))) ) ) => ( ap(ap(V0f,V6i),V8x) = ap(ap(V0f,V7j),V8x) ) ) ) ) ) ) & ! [V9x] : ( mem(V9x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9x),V3s)) => ? [V10j] : ( mem(V10j,A_27a) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V10j),V4k)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9x),ap(V2t,V10j))) & ( ap(V1g,V9x) = ap(ap(V0f,V10j),V9x) ) ) ) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3s)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS__CLOSED,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1t] : ( mem(V1t,arr(A_27a,arr(ty_2Erealax_2Ereal,bool))) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3k] : ( mem(V3k,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3k)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),A_27a),f3873(A_27a,V1t,V3k))))) & ( ( ! [V5i] : ( mem(V5i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5i),V3k)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(V1t,V5i))) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(V0f,V5i)),ap(V1t,V5i))) ) ) ) ) & ! [V6i] : ( mem(V6i,A_27a) => ! [V7j] : ( mem(V7j,A_27a) => ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6i),V3k)) & ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7j),V3k)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2s),ap(V1t,V6i))),ap(V1t,V7j)))) ) ) => ( ap(ap(V0f,V6i),V8x) = ap(ap(V0f,V7j),V8x) ) ) ) ) ) ) ) ) => ? [V9g] : ( mem(V9g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V9g),V2s)) & ! [V10x] : ( mem(V10x,ty_2Erealax_2Ereal) => ! [V11i] : ( mem(V11i,A_27a) => ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V11i),V3k)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10x),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2s),ap(V1t,V11i)))) ) => ( ap(V9g,V10x) = ap(ap(V0f,V11i),V10x) ) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__LIFT__DOT,lemma,(! [V0l] : ( mem(V0l,ty_2Erealax_2Ereal) => ! [V1net] : ( mem(V1net,ty_2Ereal__topology_2Enet(ty_2Erealax_2Ereal)) => ! [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V2f),V0l),V1net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),f3875(V2f,V3a)),ap(ap(c_2Erealax_2Ereal__mul,V3a),V0l)),V1net)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__LIFT__DOT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),f3876(V0a)),ap(c_2Ereal__topology_2Eat,V1x))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__LIFT__DOT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3876(V0a)),V1s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__LEFT,lemma,(! [V0b] : ( mem(V0b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3877(V0b)))) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__RIGHT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3878(V0a)))) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__LE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3879(V1b,V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__GE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3880(V1b,V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__HYPERPLANE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3881(V1b,V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__HYPERPLANE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3881(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__STANDARD__HYPERPLANE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3883(V0a)))) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__LE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3884(V0a)))) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__GE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3885(V0a)))) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__LT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3886(V1b,V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__LT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3887(V0a)))) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__GT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3888(V1b,V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__GT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3889(V0a)))) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__POSITIVE__MULTIPLES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3891(V0s))))) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__LEFT,lemma,(! [V0b] : ( mem(V0b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3892(V0b)))) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__RIGHT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3263(V0a)))) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__POSITIVE__ORTHANT,lemma,(p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3893))))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__LE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3879(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3894(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__GE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3880(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3895(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__LE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3884(V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3896(V0a)) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__GE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3885(V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3897(V0a)) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__LT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3886(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3898(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__GT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3888(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3899(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__LT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3887(V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3900(V0a)) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__GT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3889(V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3901(V0a)) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__HYPERPLANE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3881(V1b,V0a))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) & ( V1b = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3879(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) & ( V1b = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3880(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) & ( V1b = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3886(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) & ( V1b = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3888(V1b,V0a))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__STANDARD__HYPERPLANE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3883(V0a))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ))).
fof(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3884(V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3885(V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3887(V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ~~ p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3889(V0a)))) ) ))).
fof(conj_thm_2Ereal__topology_2EFORALL__IN__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V2t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V3x)),V2t)) ) ) ) ) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),ap(c_2Ereal__topology_2Eclosure,V1s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V4x)),V2t)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFORALL__IN__CLOSURE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V2t)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) ) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),ap(c_2Ereal__topology_2Eclosure,V1s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V3x)),V2t)) ) ) ) <=> ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V4x)),V2t)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__LE__ON__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3x)),V2a)) ) ) ) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),ap(c_2Ereal__topology_2Eclosure,V1s))) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4x)),V2a)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__GE__ON__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2a),ap(V0f,V3x))) ) ) ) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),ap(c_2Ereal__topology_2Eclosure,V1s))) => p(ap(ap(c_2Ereal_2Ereal__lte,V2a),ap(V0f,V4x))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CONSTANT__ON__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) => ( ap(V0f,V3x) = V2a ) ) ) ) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),ap(c_2Ereal__topology_2Eclosure,V1s))) => ( ap(V0f,V4x) = V2a ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AGREE__ON__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(c_2Ereal__topology_2Eclosure,V0s))) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2h),ap(c_2Ereal__topology_2Eclosure,V0s))) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => ( ap(V1g,V3x) = ap(V2h,V3x) ) ) ) ) ) => ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),ap(c_2Ereal__topology_2Eclosure,V0s))) => ( ap(V1g,V4x) = ap(V2h,V4x) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__CONSTANT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3902(V1s,V0f,V2a)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__CONSTANT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V2s)) & p(ap(c_2Ereal__topology_2EClosed,V2s)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3903(V2s,V1f,V0a)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosure,V1s))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) ) => ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5y),V2x))),V4d)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V5y)),ap(V0f,V2x)))),V3e)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__SEQUENTIALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) <=> ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3a),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V4n)),V1s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V2x),V3a),c_2Ereal__topology_2Esequentially)) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x)),ap(V0f,V3a)),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CAUCHY__CONTINUOUS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) => ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Ecauchy,V2x)) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V3n)),V1s)) ) ) => p(ap(c_2Ereal__topology_2Ecauchy,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IMP__CAUCHY__CONTINUOUS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(c_2Ereal__topology_2EClosed,V1s)) ) => ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Ecauchy,V2x)) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V3n)),V1s)) ) ) => p(ap(c_2Ereal__topology_2Ecauchy,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__UNIQUENESS__LEMMA,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Ecauchy,V2x)) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V3n)),V1s)) ) ) => p(ap(c_2Ereal__topology_2Ecauchy,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x))) ) ) ) => ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5x] : ( mem(V5x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( ( ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V5x,V6n)),V1s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V5x),V4a),c_2Ereal__topology_2Esequentially)) ) => ? [V7l] : ( mem(V7l,ty_2Erealax_2Ereal) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5x)),V7l),c_2Ereal__topology_2Esequentially)) & ! [V8y] : ( mem(V8y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( ( ! [V9n] : ( mem(V9n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V8y,V9n)),V1s)) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V8y),V4a),c_2Ereal__topology_2Esequentially)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V8y)),V7l),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__EXTENDS__TO__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Ecauchy,V2x)) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V3n)),V1s)) ) ) => p(ap(c_2Ereal__topology_2Ecauchy,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x))) ) ) ) => ? [V4g] : ( mem(V4g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4g),ap(c_2Ereal__topology_2Eclosure,V1s))) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V1s)) => ( ap(V4g,V5x) = ap(V0f,V5x) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__EXTENDS__TO__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) => ? [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V2g),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) => ( ap(V2g,V3x) = ap(V0f,V3x) ) ) ) ) & ! [V4h] : ( mem(V4h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4h),ap(c_2Ereal__topology_2Eclosure,V1s))) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V1s)) => ( ap(V4h,V5x) = ap(V0f,V5x) ) ) ) ) => ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),ap(c_2Ereal__topology_2Eclosure,V1s))) => ( ap(V4h,V6x) = ap(V2g,V6x) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__IMP__CONTINUOUS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2x] : ( mem(V2x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Ecauchy,V2x)) & ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2x,V3n)),V1s)) ) ) => p(ap(c_2Ereal__topology_2Ecauchy,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2x))) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__UNIFORMLY__CONTINUOUS__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) ) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3h] : ( mem(V3h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V2g),ap(c_2Ereal__topology_2Eat,V0x))) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V3h),ap(c_2Ereal__topology_2Eat,ap(V2g,V0x)))) & ( ( ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ap(V2g,ap(V3h,V4y)) = V4y ) ) ) & ( ap(V3h,ap(V2g,V0x)) = V0x ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V1f),ap(c_2Ereal__topology_2Eat,ap(V2g,V0x)))) <=> p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),f3904(V2g,V1f)),ap(c_2Ereal__topology_2Eat,V0x))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1z] : ( mem(V1z,ty_2Erealax_2Ereal) => ! [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V2f),ap(c_2Ereal__topology_2Eat,ap(ap(c_2Erealax_2Ereal__add,V0a),V1z)))) <=> p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),f3905(V0a,V2f)),ap(c_2Ereal__topology_2Eat,V1z))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__IMAGE__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V2x))) ) ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V3x) = ap(V0f,V4y) ) => ( V3x = V4y ) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Einterior,V1s)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__WITHIN__AVOID,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V1x)),V2s))) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V2s)) & ( ~~ ( ap(V0f,V1x) = V3a ) ) ) ) => ? [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) & ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V2s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V5y))),V4e)) ) => ( ~~ ( ap(V0f,V5y) = V3a ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__AVOID,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1x))) & ( ~~ ( ap(V0f,V1x) = V2a ) ) ) => ? [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) & ! [V4y] : ( mem(V4y,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),V1x),V4y))),V3e)) => ( ~~ ( ap(V0f,V4y) = V2a ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__AVOID,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V2s)) & ( ~~ ( ap(V0f,V1x) = V3a ) ) ) ) => ? [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) & ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V2s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V5y))),V4e)) ) => ( ~~ ( ap(V0f,V5y) = V3a ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__AVOID,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( p(ap(c_2Ereal__topology_2EOpen,V2s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V2s)) & ( ~~ ( ap(V0f,V1x) = V3a ) ) ) ) ) => ? [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) & ! [V5y] : ( mem(V5y,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),V1x),V5y))),V4e)) => ( ~~ ( ap(V0f,V5y) = V3a ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN__CASES,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3902(V1s,V0f,V2a)))) ) ) => ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) => ( ~~ ( ap(V0f,V4x) = V2a ) ) ) ) ) | ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V1s)) => ( ap(V0f,V5x) = V2a ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3902(V1s,V0f,V2a)))) & ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) & ( ap(V0f,V4x) = V2a ) ) ) ) ) ) => ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V1s)) => ( ap(V0f,V5x) = V2a ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3902(V1s,V0f,V2a)))) & ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) & ( ap(V0f,V4x) = V2a ) ) ) ) ) ) => ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V1s)) => ( ap(V0f,V5x) = V2a ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__SCALING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V1c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) & p(ap(c_2Ereal__topology_2EOpen,V0s)) ) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__NEGATIONS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__TRANSLATION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3641(V1a)),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__TRANSLATION__EQ,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s))) <=> p(ap(c_2Ereal__topology_2EOpen,V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__AFFINITY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & ( ~~ ( V2c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3907(V2c,V1a)),V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3908(V0a)),ap(c_2Ereal__topology_2Einterior,V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) | p(ap(c_2Ereal__topology_2EOpen,V1t)) ) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUPPER__HEMICONTINUOUS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0f,V3x)),V1t)) ) ) ) => ( ( ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3909(V2s,V0f,V4u)))) ) ) ) <=> ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V6u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3910(V2s,V6u,V0f)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELOWER__HEMICONTINUOUS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0f,V3x)),V1t)) ) ) ) => ( ( ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V4u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3909(V2s,V0f,V4u)))) ) ) ) <=> ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V6u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3910(V2s,V6u,V0f)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__MAP__IFF__LOWER__HEMICONTINUOUS__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( ( ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3u))) ) ) ) <=> ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3911(V2t,V0f,V1s,V4u)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__MAP__IFF__UPPER__HEMICONTINUOUS__PREIMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) => ( ( ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3u))) ) ) ) <=> ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3911(V2t,V0f,V1s,V4u)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS__EXPLICIT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0f,V3x)),V1t)) ) ) ) & ( ( ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3909(V2s,V0f,V4u)))) ) ) ) & ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V6u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3912(V2s,V0f,V6u)))) ) ) ) ) => ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ! [V9e] : ( mem(V9e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),V2s)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V9e)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(V0f,V8x))) & ( ~~ ( ap(V0f,V8x) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ? [V10d] : ( mem(V10d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V10d)) & ! [V11x_27] : ( mem(V11x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V11x_27),V2s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8x),V11x_27))),V10d)) ) => ( ( ! [V12y] : ( mem(V12y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V12y),ap(V0f,V8x))) => ? [V13y_27] : ( mem(V13y_27,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V13y_27),ap(V0f,V11x_27))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12y),V13y_27))),V9e)) ) ) ) ) ) & ! [V14y_27] : ( mem(V14y_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V14y_27),ap(V0f,V11x_27))) => ? [V15y] : ( mem(V15y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V15y),ap(V0f,V8x))) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V14y_27),V15y))),V9e)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Econnected__component,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x),V2y)) <=> ? [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2Econnected,V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V3t)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V3t)) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__IN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x),V2y)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x),V1x)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x),V1x)) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x),V2y)) => p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2y),V1x)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__TRANS,lemma,(! [V0z] : ( mem(V0z,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),V2x),V3y)) & p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),V3y),V0z)) ) => p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),V2x),V0z)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__OF__SUBSET,lemma,(! [V0y] : ( mem(V0y,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1s),V2t)) & p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),V3x),V0y)) ) => p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V2t),V3x),V0y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3914(V0s,V1x)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__BIGUNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),f3915(V1x,V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x)),V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT__SET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) = V0s ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIV,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ap(ap(c_2Ereal__topology_2Econnected__component,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0x) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__UNIV,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) <=> ( V0s = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__SELF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) = V0s ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTED__COMPONENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) ) => p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x),V2y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__MAXIMAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1t)) & ( p(ap(c_2Ereal__topology_2Econnected,V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2x))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__MONO,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2x)),ap(ap(c_2Ereal__topology_2Econnected__component,V1t),V2x))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EMPTY,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( ap(ap(c_2Ereal__topology_2Econnected__component,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),V0x) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x))) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2y) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__CONNECTED__COMPONENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__DISJOINT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1a)),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2b))) <=> ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2b))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__NONOVERLAP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1a)),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) ) | ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2b),V0s)) ) | ( ~~ ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1a) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2b) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__OVERLAP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1a)),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) <=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2b),V0s)) & ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1a) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2b) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x),V2y)) <=> p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2y),V1x)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2y) ) <=> ( ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) ) ) | ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) & p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x),V2y)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENT__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) ) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2y) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__IDEMP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ap(ap(c_2Ereal__topology_2Econnected__component,ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x)),V1x) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIQUE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1c)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1c),V0s)) & ( p(ap(c_2Ereal__topology_2Econnected,V1c)) & ! [V3c_27] : ( mem(V3c_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V3c_27)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3c_27),V0s)) & p(ap(c_2Ereal__topology_2Econnected,V3c_27)) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3c_27),V1c)) ) ) ) ) ) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2x) = V1c ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EJOINABLE__CONNECTED__COMPONENT__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & ( ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2x)),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V3y)),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V2x) = ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V3y) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBIGUNION__CONNECTED__COMPONENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal),f3916(V0s))) = V0s ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPLEMENT__CONNECTED__COMPONENT__BIGUNION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x)) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EDELETE(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal),f3917(V0s))),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__CONNECTED__COMPONENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x))) ) ))).
fof(conj_thm_2Ereal__topology_2EBIGUNION__DIFF,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(arr(A_27a,bool),bool)) => ! [V1t] : ( mem(V1t,arr(A_27a,bool)) => ( ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s)),V1t) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f3918(A_27a,V1t,V0s))) ) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN__IN__CONNECTED__COMPONENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal),f3919(V0s)))) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),ap(ap(c_2Ereal__topology_2Econnected__component,V0s),V1x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQUIVALENCE__RELATION,lemma,(! [V0R] : ( mem(V0R,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(V0R,V2x),V3y)) => p(ap(ap(V0R,V3y),V2x)) ) ) ) ) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ! [V6z] : ( mem(V6z,ty_2Erealax_2Ereal) => ( ( p(ap(ap(V0R,V4x),V5y)) & p(ap(ap(V0R,V5y),V6z)) ) => p(ap(ap(V0R,V4x),V6z)) ) ) ) ) ) & ! [V7a] : ( mem(V7a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7a),V1s)) => ? [V8t] : ( mem(V8t,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V8t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7a),V8t)) & ! [V9x] : ( mem(V9x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9x),V8t)) => p(ap(ap(V0R,V7a),V9x)) ) ) ) ) ) ) ) ) ) => ! [V10a] : ( mem(V10a,ty_2Erealax_2Ereal) => ! [V11b] : ( mem(V11b,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Econnected__component,V1s),V10a),V11b)) => p(ap(ap(V0R,V10a),V11b)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__COMPONENT__INTERMEDIATE__SUBSET,lemma,(! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ! [V1u] : ( mem(V1u,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Econnected__component,V1u),V2a)),V0t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0t),V1u)) ) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0t),V2a) = ap(ap(c_2Ereal__topology_2Econnected__component,V1u),V2a) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Ecomponents,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ecomponents,V0s) = ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal),f3916(V0s)) ) )).
fof(conj_thm_2Ereal__topology_2EIN__COMPONENTS,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),ap(c_2Ereal__topology_2Ecomponents,V0u))) <=> ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0u)) & ( V1s = ap(ap(c_2Ereal__topology_2Econnected__component,V0u),V2x) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBIGUNION__COMPONENTS,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ( V0u = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecomponents,V0u)) ) ))).
fof(conj_thm_2Ereal__topology_2EPAIRWISE__DISJOINT__COMPONENTS,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Epairwise(arr(ty_2Erealax_2Ereal,bool)),c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2Ecomponents,V0u))) ))).
fof(conj_thm_2Ereal__topology_2EIN__COMPONENTS__NONEMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) => ( ~~ ( V1c = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__COMPONENTS__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1c),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__COMPONENTS__CONNECTED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) => p(ap(c_2Ereal__topology_2Econnected,V1c)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__COMPONENTS__MAXIMAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) <=> ( ( ~~ ( V1c = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1c),V0s)) & ( p(ap(c_2Ereal__topology_2Econnected,V1c)) & ! [V2c_27] : ( mem(V2c_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V2c_27 = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1c),V2c_27)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2c_27),V0s)) & p(ap(c_2Ereal__topology_2Econnected,V2c_27)) ) ) ) => ( V2c_27 = V1c ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EJOINABLE__COMPONENTS__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2c1] : ( mem(V2c1,arr(ty_2Erealax_2Ereal,bool)) => ! [V3c2] : ( mem(V3c2,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c1),ap(c_2Ereal__topology_2Ecomponents,V0s))) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3c2),ap(c_2Ereal__topology_2Ecomponents,V0s))) & ( ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2c1),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3c2),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) => ( V2c1 = V3c2 ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__IN__COMPONENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0s)),V1c)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__COMPONENTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) ) => p(ap(c_2Ereal__topology_2EClosed,V1c)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__COMPONENTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) ) => p(ap(c_2Ereal__topology_2Ecompact,V1c)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__GEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V2c] : ( mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),ap(c_2Ereal__topology_2Ecomponents,V1s))) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V2c)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2c)) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__FINITE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Ecomponents,V1s))) & ! [V2c] : ( mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),ap(c_2Ereal__topology_2Ecomponents,V1s))) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2c)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__NONOVERLAP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ! [V2c_27] : ( mem(V2c_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c_27),ap(c_2Ereal__topology_2Ecomponents,V0s))) ) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1c),V2c_27) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( ~~ ( V1c = V2c_27 ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ! [V2c_27] : ( mem(V2c_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c_27),ap(c_2Ereal__topology_2Ecomponents,V0s))) ) => ( ( V1c = V2c_27 ) <=> ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1c),V2c_27) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__EMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Ecomponents,V0s) = c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool)) ) <=> ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__EMPTY,lemma,(( ap(c_2Ereal__topology_2Ecomponents,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool)) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENTS__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ! [V2c_27] : ( mem(V2c_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c_27),ap(c_2Ereal__topology_2Ecomponents,V0s))) ) => ( V1c = V2c_27 ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__N__EXISTS,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Ecomponents,V0s) = ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),V0s),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool))) ) <=> ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) & ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ? [V2a] : ( mem(V2a,arr(ty_2Erealax_2Ereal,bool)) & ( ap(c_2Ereal__topology_2Ecomponents,V1s) = ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),V2a),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool))) ) ) ) <=> ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ( ~~ ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ap(c_2Ereal__topology_2Ecomponents,V0s) = ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),V0s),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool))) ) <=> ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__EXISTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ? [V1a] : ( mem(V1a,arr(ty_2Erealax_2Ereal,bool)) & ( ap(c_2Ereal__topology_2Ecomponents,V0s) = ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),V1a),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool))) ) ) ) <=> ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIV,lemma,(( ap(c_2Ereal__topology_2Ecomponents,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)) = ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool))) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Ecomponents,V0s)),ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),V0s),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool))))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING__EXISTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ? [V1a] : ( mem(V1a,arr(ty_2Erealax_2Ereal,bool)) & p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Ecomponents,V0s)),ap(ap(c_2Epred__set_2EINSERT(arr(ty_2Erealax_2Ereal,bool)),V1a),c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__COMPONENTS__SELF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V0s),ap(c_2Ereal__topology_2Ecomponents,V0s))) <=> ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__MAXIMAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2c] : ( mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),ap(c_2Ereal__topology_2Ecomponents,V0s))) & ( p(ap(c_2Ereal__topology_2Econnected,V1t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2c),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2c)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIQUE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1k] : ( mem(V1k,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1k) = V0s ) & ! [V2c] : ( mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),V1k)) => ( p(ap(c_2Ereal__topology_2Econnected,V2c)) & ( ( ~~ ( V2c = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V3c_27] : ( mem(V3c_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V3c_27)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2c),V3c_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3c_27),V0s)) ) ) => ( V3c_27 = V2c ) ) ) ) ) ) ) ) => ( ap(c_2Ereal__topology_2Ecomponents,V0s) = V1k ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__UNIQUE__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1k] : ( mem(V1k,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ap(c_2Ereal__topology_2Ecomponents,V0s) = V1k ) <=> ( ( ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1k) = V0s ) & ! [V2c] : ( mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),V1k)) => ( p(ap(c_2Ereal__topology_2Econnected,V2c)) & ( ( ~~ ( V2c = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V3c_27] : ( mem(V3c_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V3c_27)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2c),V3c_27)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3c_27),V0s)) ) ) => ( V3c_27 = V2c ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEXISTS__COMPONENT__SUPERSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & p(ap(c_2Ereal__topology_2Econnected,V1t)) ) ) => ? [V2c] : ( mem(V2c,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),ap(c_2Ereal__topology_2Ecomponents,V0s))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2c)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPONENTS__INTERMEDIATE__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V0s),ap(c_2Ereal__topology_2Ecomponents,V2u))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V0s),ap(c_2Ereal__topology_2Ecomponents,V1t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__COMPONENTS__BIGUNION__COMPLEMENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1c),ap(c_2Ereal__topology_2Ecomponents,V0s))) => ( ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V0s),V1c) = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EDELETE(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Ecomponents,V0s)),V1c)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__SUBSET__CLOPEN,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2c] : ( mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & ( p(ap(c_2Ereal__topology_2Econnected,V2c)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2c),V0u)) & ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2c),V1s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2c),V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOPEN__BIGUNION__COMPONENTS,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) ) => ? [V2k] : ( mem(V2k,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(arr(ty_2Erealax_2Ereal,bool)),V2k),ap(c_2Ereal__topology_2Ecomponents,V0u))) & ( V1s = ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V2k) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOPEN__IN__COMPONENTS,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ( ~~ ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),ap(c_2Ereal__topology_2Ecomponents,V0u))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__EQUICONTINUOUS,lemma,(! [V0fs] : ( mem(V0fs,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) ) => ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ! [V5f] : ( mem(V5f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V6x_27] : ( mem(V6x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V5f),V0fs)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x_27),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6x_27),V2x))),V4d)) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V5f,V6x_27)),ap(V5f,V2x)))),V3e)) ) ) ) ) ) ) ) ) ) & p(ap(c_2Ereal__topology_2Ecompact,V1s)) ) => ! [V7e] : ( mem(V7e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V7e)) => ? [V8d] : ( mem(V8d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V8d)) & ! [V9f] : ( mem(V9f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V10x] : ( mem(V10x,ty_2Erealax_2Ereal) => ! [V11x_27] : ( mem(V11x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V9f),V0fs)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V11x_27),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V11x_27),V10x))),V8d)) ) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V9f,V11x_27)),ap(V9f,V10x)))),V7e)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__CONTINUOUS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(c_2Ereal__topology_2Ecompact,V1s)) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,V0f),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EABS__TRIANGLE__LE,lemma,(! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Eabs,V1x)),ap(c_2Ereal_2Eabs,V2y))),V0e)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Erealax_2Ereal__add,V1x),V2y))),V0e)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__UNIFORM__LIMIT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & ( p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3920(A_27a,V1f,V3s)),V0net)) & ! [V5e] : ( mem(V5e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3922(A_27a,V3s,V2g,V1f,V5e)),V0net)) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V3s)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EOPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ? [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & ! [V3x_27] : ( mem(V3x_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3x_27),V1x))),V2e)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x_27),V0s)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3x_27] : ( mem(V3x_27,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x_27),V0s)) & ( ( ~~ ( V3x_27 = V1x ) ) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3x_27),V1x))),V2e)) ) ) ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__RANGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1x))) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & ! [V4x_27] : ( mem(V4x_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V4x_27),V1x))),V3d)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V4x_27)),ap(V0f,V1x)))),V2e)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__RANGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) <=> ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ! [V5x_27] : ( mem(V5x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x_27),V1s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V5x_27),V2x))),V4d)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V5x_27)),ap(V0f,V2x)))),V3e)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ABS__COMPOSE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3734(A_27a,V1f)),V0net)) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__ABS__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3814(V0f)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__ABS,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),c_2Ereal_2Eabs),ap(c_2Ereal__topology_2Eat,V0x))) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__DIST,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),f3923(V0a)),ap(c_2Ereal__topology_2Eat,V1x))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DIST,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3923(V0a)),V1s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__ATTAINS__SUP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2y),V1x)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__ATTAINS__INF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V1x),V2y)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__SUP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V1s)) & ( ( ~~ ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3y)),ap(V0f,V2x))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__INF,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V1s)) & ( ( ~~ ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) ) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2x)),ap(V0f,V3y))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDISTANCE__ATTAINS__SUP,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V3y))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2x)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDISTANCE__ATTAINS__INF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2x))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V3y)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__MUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3c] : ( mem(V3c,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V3c),V4d),V0net)) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3924(A_27a,V3c,V1f)),ap(ap(c_2Erealax_2Ereal__mul,V4d),V2l)),V0net)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__VMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1c] : ( mem(V1c,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) => ! [V3v] : ( mem(V3v,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1c),V2d),V0net)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),f3925(A_27a,V1c,V3v)),ap(ap(c_2Erealax_2Ereal__mul,V2d),V3v)),V0net)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__VMUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1c] : ( mem(V1c,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2v] : ( mem(V2v,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1c),V0net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3926(A_27a,V1c,V2v)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__MUL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2c] : ( mem(V2c,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V2c),V0net)) & p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3927(A_27a,V2c,V1f)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__VMUL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2v] : ( mem(V2v,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1c),V0s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3820(V1c,V2v)),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__MUL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1c),V0s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V0s)) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3928(V1c,V2f)),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__POW,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3825(A_27a,V1f)),V0net)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3929(A_27a,V1f,V2n)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__POW,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3930(V0f)),V1s)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3931(V0f,V2n)),V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__PRODUCT,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal))) => ! [V2t] : ( mem(V2t,arr(A_27b,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) & ! [V3i] : ( mem(V3i,A_27b) => ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3i),V2t)) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3932(A_27a,A_27b,V1f,V3i)),V0net)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),f3933(A_27a,A_27b,V1f,V2t)),V0net)) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__PRODUCT,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,arr(A_27a,ty_2Erealax_2Ereal))) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(A_27a,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2t)) & ! [V3i] : ( mem(V3i,A_27a) => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3i),V2t)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3934(A_27a,V0f,V3i)),V1s)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3935(A_27a,V0f,V2t)),V1s)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__INV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) & ( ~~ ( V2l = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Erealax_2Einv),V1f)),ap(c_2Erealax_2Einv,V2l)),V0net)) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__INV,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),V1f),V0net)) & ( ~~ ( ap(V1f,ap(c_2Ereal__topology_2Enetlimit(A_27a),V0net)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Erealax_2Einv),V1f)),V0net)) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN__INV,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V1s))) & ( ~~ ( ap(V0f,V2a) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Erealax_2Einv),V0f)),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2a)),V1s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__INV,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eat,V1a))) & ( ~~ ( ap(V0f,V1a) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Erealax_2Einv),V0f)),ap(c_2Ereal__topology_2Eat,V1a))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INV,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => ( ~~ ( ap(V0f,V2x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Erealax_2Einv),V0f)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__SCALING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__NEGATIONS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) => p(ap(c_2Ereal__topology_2Econnected,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__SCALING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__NEGATIONS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__AFFINITY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) => p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3907(V2c,V1a)),V0s))) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Ediameter,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ediameter,V0s) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),V0s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Esup,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3937(V0s))))) ) )).
fof(conj_thm_2Ereal__topology_2EDIAMETER__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1x),V2y))),ap(c_2Ereal__topology_2Ediameter,V0s))) ) ) ) ) & ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & p(ap(ap(c_2Erealax_2Ereal__lt,V3d),ap(c_2Ereal__topology_2Ediameter,V0s))) ) => ? [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) & ? [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V0s)) & p(ap(ap(c_2Ereal_2Ereal__gt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V4x),V5y))),V3d)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__BOUNDED__BOUND,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1x),V2y))),ap(c_2Ereal__topology_2Ediameter,V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal_2Eabs,ap(V0f,V2x)) = ap(c_2Ereal_2Eabs,V2x) ) ) ) => ( ap(c_2Ereal__topology_2Ediameter,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)) = ap(c_2Ereal__topology_2Ediameter,V1s) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__EMPTY,lemma,(( ap(c_2Ereal__topology_2Ediameter,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ediameter,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__POS__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2Ediameter,V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ediameter,V0s)),ap(c_2Ereal__topology_2Ediameter,V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ( ap(c_2Ereal__topology_2Ediameter,ap(c_2Ereal__topology_2Eclosure,V0s)) = ap(c_2Ereal__topology_2Ediameter,V0s) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL__NONEMPTY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V1z] : ( mem(V1z,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1z),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1z),ap(c_2Ereal__topology_2Ediameter,V0s))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ? [V1z] : ( mem(V1z,ty_2Erealax_2Ereal) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1z),ap(c_2Ereal__topology_2Ediameter,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__EQ__0,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ( ( ap(c_2Ereal__topology_2Ediameter,V0s) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & ( V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1d] : ( mem(V1d,ty_2Erealax_2Ereal) => ( ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) | p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1d)) ) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V2x),V3y))),V1d)) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ediameter,V0s)),V1d)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__CBALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ediameter,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__lt,V1r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V1r)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__BALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ediameter,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__lt,V1r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V1r)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ediameter,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s))))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2Ediameter,V0s)),ap(c_2Ereal__topology_2Ediameter,V1t)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELEBESGUE__COVERING__LEMMA,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( ( ~~ ( V1c = c_2Epred__set_2EEMPTY(arr(ty_2Erealax_2Ereal,bool)) ) ) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1c))) & ! [V2b] : ( mem(V2b,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2b),V1c)) => p(ap(c_2Ereal__topology_2EOpen,V2b)) ) ) ) ) ) => ? [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V0s)) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ediameter,V4t)),V3d)) ) => ? [V5b] : ( mem(V5b,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V5b),V1c)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V5b)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__SCALING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__NEGATIONS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__CLOSED__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__COMPACT__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) | p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3612(V1t,V0s)))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3939(V1t,V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__CLOSED__DIFFERENCES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3791(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__COMPACT__DIFFERENCES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3791(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ETRANSLATION__DIFF,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3908(V0a)),ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V1s),V2t)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3940(V0a)),V1s)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3941(V0a)),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATE__POINT__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) ) ) => ? [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V2d),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V3x)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATE__COMPACT__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2d),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V4y)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATE__CLOSED__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( p(ap(c_2Ereal__topology_2Ecompact,V1t)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2d),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V4y)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => ? [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) & ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(c_2Ereal__topology_2Ecompact,ap(V1f,V2n))) ) ) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V1f,V3n)),V0s)) ) ) & ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V1f,V4n)),ap(V1f,ap(ap(c_2Earithmetic_2E_2B,V4n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) ) ) & ( ( ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum),f3942(V1f))) = V0s ) & ! [V6k] : ( mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V6k)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6k),V0s)) ) => ? [V7N] : ( mem(V7N,ty_2Enum_2Enum) & ! [V8n] : ( mem(V8n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V8n),V7N)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6k),ap(V1f,V8n))) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__UNION__COMPACT__SUBSETS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ? [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) & ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(c_2Ereal__topology_2Ecompact,ap(V1f,V2n))) ) ) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V1f,V3n)),V0s)) ) ) & ( ( ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V1f,V4n)),ap(c_2Ereal__topology_2Einterior,ap(V1f,ap(ap(c_2Earithmetic_2E_2B,V4n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))))) ) ) & ( ( ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum),f3942(V1f))) = V0s ) & ! [V6k] : ( mem(V6k,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V6k)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6k),V0s)) ) => ? [V7N] : ( mem(V7N,ty_2Enum_2Enum) & ! [V8n] : ( mem(V8n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V8n),V7N)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6k),ap(V1f,V8n))) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2EOPEN__interval,axiom, ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3943(V1b,V0a)) ) ) )).
fof(ax_thm_2Ereal__topology_2ECLOSED__interval,axiom, ! [V0l] : ( mem(V0l,ty_2Elist_2Elist(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ( ap(c_2Ereal__topology_2ECLOSED__interval,V0l) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3944(V0l)) ) )).
fof(conj_thm_2Ereal__topology_2Einterval,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3943(V1b,V0a)) ) & ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3945(V1b,V0a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__INTERVAL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)))) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V0x)) & p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V2b)) ) ) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V0x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V2b)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__INTERVAL__REFLECT,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V2x)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V1b)),ap(c_2Erealax_2Ereal__neg,V0a))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) ) ) & ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V5x)),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V4b)),ap(c_2Erealax_2Ereal__neg,V3a))))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3a),V4b)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREFLECT__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3946),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V1b)),ap(c_2Erealax_2Ereal__neg,V0a))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ) & ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3947),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3a),V4b))) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V4b)),ap(c_2Erealax_2Ereal__neg,V3a))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__EQ__EMPTY,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1b),V0a)) <=> ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V0a)) <=> ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__NE__EMPTY,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V1b)) ) & ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) <=> p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V1b)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSET__INTERVAL__IMP,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V1c)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2d),V3b)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1c),V2d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) & ( ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V1c)) & p(ap(ap(c_2Erealax_2Ereal__lt,V2d),V3b)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1c),V2d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V3b)))) ) & ( ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V1c)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2d),V3b)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1c),V2d))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) & ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V1c)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2d),V3b)) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1c),V2d))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V3b)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V0a)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) & ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V0a)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSET__INTERVAL,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ! [V1d] : ( mem(V1d,ty_2Erealax_2Ereal) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0c),V1d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V0c),V1d)) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V2a),V0c)) & p(ap(ap(c_2Ereal_2Ereal__lte,V1d),V3b)) ) ) ) & ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0c),V1d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V0c),V1d)) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V2a),V0c)) & p(ap(ap(c_2Erealax_2Ereal__lt,V1d),V3b)) ) ) ) & ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0c),V1d))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V0c),V1d)) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V2a),V0c)) & p(ap(ap(c_2Ereal_2Ereal__lte,V1d),V3b)) ) ) ) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0c),V1d))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V0c),V1d)) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V2a),V0c)) & p(ap(ap(c_2Ereal_2Ereal__lte,V1d),V3b)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDISJOINT__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1b),V0a)) | ( p(ap(ap(c_2Erealax_2Ereal__lt,V3d),V2c)) | ( p(ap(ap(c_2Erealax_2Ereal__lt,V1b),V2c)) | p(ap(ap(c_2Erealax_2Ereal__lt,V3d),V0a)) ) ) ) ) & ( ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1b),V0a)) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V3d),V2c)) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2c)) | p(ap(ap(c_2Ereal_2Ereal__lte,V3d),V0a)) ) ) ) ) & ( ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V0a)) | ( p(ap(ap(c_2Erealax_2Ereal__lt,V3d),V2c)) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2c)) | p(ap(ap(c_2Ereal_2Ereal__lte,V3d),V0a)) ) ) ) ) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V0a)) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V3d),V2c)) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2c)) | p(ap(ap(c_2Ereal_2Ereal__lte,V3d),V0a)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EENDS__IN__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> ( ~~ ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) & ( ( ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3b),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> ( ~~ ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) & ( ( ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5b] : ( mem(V5b,ty_2Erealax_2Ereal) => ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4a),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V5b)))) ) ) ) ) & ! [V6a] : ( mem(V6a,ty_2Erealax_2Ereal) => ! [V7b] : ( mem(V7b,ty_2Erealax_2Ereal) => ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7b),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6a),V7b)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EENDS__IN__UNIT__INTERVAL,lemma,(( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))))) ) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTER__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Emax,V0a),V2c)),ap(ap(c_2Ereal_2Emin,V1b),V3d))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__OPEN__SUBSET__CLOSED,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LT__MIN,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ! [V2z] : ( mem(V2z,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V2z),ap(ap(c_2Ereal_2Emin,V0x),V1y))) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V2z),V0x)) & p(ap(ap(c_2Erealax_2Ereal__lt,V2z),V1y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__LEMMA,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V2x)) & p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1b)) ) => ? [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3d)) & ! [V4x_27] : ( mem(V4x_27,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V4x_27),V2x))),V3d)) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V4x_27)) & p(ap(ap(c_2Erealax_2Ereal__lt,V4x_27),V1b)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__CLOSED__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Einterior,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__CLOSED__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ENOT__INTERVAL__UNIV,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ~~ ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__MIDPOINT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(ap(c_2Erealax_2Ereal__add,V0a),V1b))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL__CONVEX,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ! [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) & p(ap(ap(c_2Ereal_2Ereal__lte,V4e),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V4e),V2x)),ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V4e)),V3y))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LE__INV2,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Einv,V1y)),ap(c_2Erealax_2Einv,V0x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__INV__LE__1,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Einv,V0x)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__OPEN__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosure,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL__SYMMETRIC,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V1a)),V1a)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & ? [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL__SYMMETRIC,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V1a)),V1a)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) => ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & ? [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__CLOSED__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFRONTIER__OPEN__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Efrontier,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTER__INTERVAL__MIXED__EQ__EMPTY,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) <=> ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__TRANSLATION,lemma,(( ( ! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V0c),V1a)),ap(ap(c_2Erealax_2Ereal__add,V0c),V2b))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3689(V0c)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) ) ) ) ) ) & ! [V4c] : ( mem(V4c,ty_2Erealax_2Ereal) => ! [V5a] : ( mem(V5a,ty_2Erealax_2Ereal) => ! [V6b] : ( mem(V6b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V4c),V5a)),ap(ap(c_2Erealax_2Ereal__add,V4c),V6b))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3690(V4c)),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5a),V6b))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEMPTY__AS__INTERVAL,lemma,(( c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ))).
fof(conj_thm_2Ereal__topology_2EUNIT__INTERVAL__NONEMPTY,lemma,(( ( ~~ ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIMAGE__STRETCH__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2m] : ( mem(V2m,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3949(V2m)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3950(V1b,V2m,V0a))),ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3951(V1b,V2m,V0a)))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__IMAGE__STRETCH__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2m] : ( mem(V2m,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ? [V3u] : ( mem(V3u,ty_2Erealax_2Ereal) & ? [V4v] : ( mem(V4v,ty_2Erealax_2Ereal) & ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3953(V2m)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3u),V4v)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__IMAGE__UNIT__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3955(V1b,V0a)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__INTERVALS,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( ( ~~ ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3957(V3d,V2c,V1b,V0a))) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V0a),V2c)),ap(ap(c_2Erealax_2Ereal__add,V1b),V3d))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ) ) ) ) & ! [V6a] : ( mem(V6a,ty_2Erealax_2Ereal) => ! [V7b] : ( mem(V7b,ty_2Erealax_2Ereal) => ! [V8c] : ( mem(V8c,ty_2Erealax_2Ereal) => ! [V9d] : ( mem(V9d,ty_2Erealax_2Ereal) => ( ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6a),V7b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8c),V9d)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3959(V9d,V8c,V7b,V6a))) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V6a),V8c)),ap(ap(c_2Erealax_2Ereal__add,V7b),V9d))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL__OPEN__INTERVAL,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ? [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),V0s)) ) ) ) ) ) ) ) ) & ! [V4s] : ( mem(V4s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V4s)) <=> ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V4s)) => ? [V6a] : ( mem(V6a,ty_2Erealax_2Ereal) & ? [V7b] : ( mem(V7b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6a),V7b)))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6a),V7b))),V4s)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ? [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),V0s)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CONTAINS__OPEN__INTERVAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) <=> ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ? [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),V0s)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIAMETER__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ediameter,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1b),V0a))) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ediameter,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3b),V2a))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIMAGE__TWIZZLE__INTERVAL,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0p] : ( mem(V0p,A_27a) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),i(ty_2Erealax_2Ereal)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EEQ__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) <=> ( ( ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) | ( ( V0a = V2c ) & ( V1b = V3d ) ) ) ) ) ) ) ) ) & ( ( ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5b] : ( mem(V5b,ty_2Erealax_2Ereal) => ! [V6c] : ( mem(V6c,ty_2Erealax_2Ereal) => ! [V7d] : ( mem(V7d,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V5b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6c),V7d)) ) <=> ( ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V5b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6c),V7d)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) & ( ( ! [V8a] : ( mem(V8a,ty_2Erealax_2Ereal) => ! [V9b] : ( mem(V9b,ty_2Erealax_2Ereal) => ! [V10c] : ( mem(V10c,ty_2Erealax_2Ereal) => ! [V11d] : ( mem(V11d,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V9b)) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V10c),V11d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) <=> ( ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V9b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V10c),V11d)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) & ! [V12a] : ( mem(V12a,ty_2Erealax_2Ereal) => ! [V13b] : ( mem(V13b,ty_2Erealax_2Ereal) => ! [V14c] : ( mem(V14c,ty_2Erealax_2Ereal) => ! [V15d] : ( mem(V15d,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V13b)) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V14c),V15d)) ) <=> ( ( ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V13b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V14c),V15d)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) | ( ( V12a = V14c ) & ( V13b = V15d ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INTERVAL__EQ,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2EClosed,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) <=> ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__INTERVAL__EQ,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2EOpen,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPACT__INTERVAL__EQ,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) <=> ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEQ__BALLS,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1a_27] : ( mem(V1a_27,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ! [V3r_27] : ( mem(V3r_27,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2r)) = ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a_27),V3r_27)) ) <=> ( ( ( V0a = V1a_27 ) & ( V2r = V3r_27 ) ) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V2r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) & p(ap(ap(c_2Ereal_2Ereal__lte,V3r_27),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) ) & ( ( ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5a_27] : ( mem(V5a_27,ty_2Erealax_2Ereal) => ! [V6r] : ( mem(V6r,ty_2Erealax_2Ereal) => ! [V7r_27] : ( mem(V7r_27,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V6r)) = ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5a_27),V7r_27)) ) <=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V6r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) & p(ap(ap(c_2Erealax_2Ereal__lt,V7r_27),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) & ( ( ! [V8a] : ( mem(V8a,ty_2Erealax_2Ereal) => ! [V9a_27] : ( mem(V9a_27,ty_2Erealax_2Ereal) => ! [V10r] : ( mem(V10r,ty_2Erealax_2Ereal) => ! [V11r_27] : ( mem(V11r_27,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V10r)) = ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V9a_27),V11r_27)) ) <=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V10r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) & p(ap(ap(c_2Ereal_2Ereal__lte,V11r_27),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) & ! [V12a] : ( mem(V12a,ty_2Erealax_2Ereal) => ! [V13a_27] : ( mem(V13a_27,ty_2Erealax_2Ereal) => ! [V14r] : ( mem(V14r,ty_2Erealax_2Ereal) => ! [V15r_27] : ( mem(V15r_27,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V14r)) = ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V13a_27),V15r_27)) ) <=> ( ( ( V12a = V13a_27 ) & ( V14r = V15r_27 ) ) | ( p(ap(ap(c_2Erealax_2Ereal__lt,V14r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) & p(ap(ap(c_2Erealax_2Ereal__lt,V15r_27),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__CASES,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) | ( ( V2x = V0a ) | ( V2x = V1b ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__OPEN__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V1b)) => ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1r)) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1r)),ap(ap(c_2Erealax_2Ereal__add,V0x),V1r))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) & ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1r)) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1r)),ap(ap(c_2Erealax_2Ereal__add,V0x),V1r))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESPHERE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Erealax_2Ereal__lt,V1r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V0a),V1r)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,V0a),V1r)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__SPHERE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)))) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V0a)) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V3b),V2a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__INTERVAL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1e)),ap(ap(c_2Erealax_2Ereal__add,V0x),V1e))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__INTERVAL,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1e)),ap(ap(c_2Erealax_2Ereal__add,V0x),V1e))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBALL__INTERVAL__0,lemma,(! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0e)) = ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V0e)),V0e)) ) ))).
fof(conj_thm_2Ereal__topology_2ECBALL__INTERVAL__0,lemma,(! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0e)) = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V0e)),V0e)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__DIFF__OPEN__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Ereal_2Ereal__lte,V0a),V1b)),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Erealax_2Ereal__lt,V2a),V3b)),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Emidpoint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEGMENT,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Ereal_2Ereal__lte,V0a),V1b)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V0a)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Ereal_2Ereal__lte,V2a),V3b)),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3b),V2a))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__SEGMENT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2EOpen,ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) ) ))).
fof(conj_thm_2Ereal__topology_2ESEGMENT__SCALAR__MULTIPLE,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2v] : ( mem(V2v,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,V0a),V2v)),ap(ap(c_2Erealax_2Ereal__mul,V1b),V2v))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3960(V0a,V1b,V2v)) ) ) ) ) ) & ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5b] : ( mem(V5b,ty_2Erealax_2Ereal) => ! [V6v] : ( mem(V6v,ty_2Erealax_2Ereal) => ( ( ~~ ( V6v = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__mul,V4a),V6v)),ap(ap(c_2Erealax_2Ereal__mul,V5b),V6v))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3961(V4a,V5b,V6v)) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Eis__interval,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) <=> ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2b),V0s)) ) => ( ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) | ( p(ap(ap(c_2Ereal_2Ereal__lte,V2b),V3x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V1a)) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Eis__interval,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) & p(ap(c_2Ereal__topology_2Eis__interval,ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__EMPTY,lemma,(p(ap(c_2Ereal__topology_2Eis__interval,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__UNIV,lemma,(p(ap(c_2Ereal__topology_2Eis__interval,c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__POINTWISE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) => ( ( ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2a),V0s)) & ( V2a = V1x ) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V0s)) ) <=> ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & ? [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) <=> ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2b),V0s)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x)) & p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__CASES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ( V0s = c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal) ) | ( ( ? [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3962(V1a)) ) ) ) | ( ( ? [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3963(V3a)) ) ) ) | ( ( ? [V5b] : ( mem(V5b,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3964(V5b)) ) ) ) | ( ( ? [V7b] : ( mem(V7b,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3965(V7b)) ) ) ) | ( ( ? [V9a] : ( mem(V9a,ty_2Erealax_2Ereal) & ? [V10b] : ( mem(V10b,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3966(V10b,V9a)) ) ) ) ) | ( ( ? [V12a] : ( mem(V12a,ty_2Erealax_2Ereal) & ? [V13b] : ( mem(V13b,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3967(V13b,V12a)) ) ) ) ) | ( ( ? [V15a] : ( mem(V15a,ty_2Erealax_2Ereal) & ? [V16b] : ( mem(V16b,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3968(V16b,V15a)) ) ) ) ) | ? [V18a] : ( mem(V18a,ty_2Erealax_2Ereal) & ? [V19b] : ( mem(V19b,ty_2Erealax_2Ereal) & ( V0s = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3969(V19b,V18a)) ) ) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__INTER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) & p(ap(c_2Ereal__topology_2Eis__interval,V1t)) ) => p(ap(c_2Ereal__topology_2Eis__interval,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__SUBSET__IS__INTERVAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),V0s)) <=> ( ( ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1a),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2b),V0s)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERVAL__CONTAINS__COMPACT__NEIGHBOURHOOD,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ? [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) & ? [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4d)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),V0s)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V4d))),V0s)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => p(ap(c_2Ereal__topology_2Eis__interval,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__SCALING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) => p(ap(c_2Ereal__topology_2Eis__interval,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIS__INTERVAL__SCALING__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2Eis__interval,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))) <=> ( ( V1c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) | p(ap(c_2Ereal__topology_2Eis__interval,V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECARD__FRONTIER__INTERVAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Eis__interval,V0s)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s))) & p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Efrontier,V0s))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__COMPONENT__UBOUND,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ! [V4k] : ( mem(V4k,A_27b) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3970(A_27a,V1f,V3b)),V0net)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2l),V3b)) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ELIM__COMPONENT__LBOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3971(A_27a,V1f,V3b)),V0net)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3b),V2l)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__COMPONENT__EQ,lemma,( ! [A_27a]: ( ne(A_27a) =>  ! [A_27b]: ( ne(A_27b) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2i] : ( mem(V2i,A_27b) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) & ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3972(A_27a,V1f,V4b)),V0net)) ) ) => ( V3l = V4b ) ) ) ) ) ) ))))).
fof(conj_thm_2Ereal__topology_2ELIM__COMPONENT__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4m] : ( mem(V4m,ty_2Erealax_2Ereal) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V4m),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3973(A_27a,V1f,V2g)),V0net)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3l),V4m)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__DROP__LE,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4m] : ( mem(V4m,ty_2Erealax_2Ereal) => ( ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V3l),V0net)) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V2g),V4m),V0net)) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3973(A_27a,V1f,V2g)),V0net)) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3l),V4m)) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__DROP__UBOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) & ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3974(A_27a,V1f,V3b)),V0net)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2l),V3b)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ELIM__DROP__LBOUND,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0net] : ( mem(V0net,ty_2Ereal__topology_2Enet(A_27a)) => ! [V1f] : ( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(A_27a),V1f),V2l),V0net)) & ( ( ~~ p(ap(c_2Ereal__topology_2Etrivial__limit(A_27a),V0net)) ) & p(ap(ap(c_2Ereal__topology_2Eeventually(A_27a),f3971(A_27a,V1f,V3b)),V0net)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3b),V2l)) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2EIMAGE__CLOSURE__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( p(ap(c_2Ereal__topology_2EClosed,V2t)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosure,V1s))),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__IMAGE__CLOSURE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) = ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__IMAGE__BOUNDED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosure,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__ABS__LE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( ( ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V4y))),V3b)) ) ) ) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosure,V1s))) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V2x))),V3b)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__LE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( ( ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4y)),V3b)) ) ) ) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosure,V1s))) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2x)),V3b)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__GE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Ereal__topology_2Eclosure,V1s))) & ( ( ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,V3b),ap(V0f,V4y))) ) ) ) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosure,V1s))) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3b),ap(V0f,V2x))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__MAP__CLOSURES,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) <=> ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosure,V1s))),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIM__WITHIN__UNION,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t)))) <=> ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V3s))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Erealax_2Ereal),V0f),V1l),ap(ap(c_2Ereal__topology_2Ewithin(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eat,V2x)),V4t))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V1s)) & ( p(ap(c_2Ereal__topology_2EClosed,V2t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2t)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V1s),V2t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V3s)) & ( p(ap(c_2Ereal__topology_2EClosed,V4t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V3s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V4t)) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V3s)) & ( ~~ p(ap(V0P,V5x)) ) ) | ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V4t)) & p(ap(V0P,V5x)) ) ) => ( ap(V1f,V5x) = ap(V2g,V5x) ) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3874(V1f,V0P,V2g)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL,lemma,(! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V2s),V0t))),V2s)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V2s),V0t))),V0t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V2s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0t)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V2s),V0t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL,lemma,(! [V0P] : ( mem(V0P,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))),V3s)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))),V4t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V3s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2g),V4t)) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V3s)) & ( ~~ p(ap(V0P,V5x)) ) ) | ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V4t)) & p(ap(V0P,V5x)) ) ) => ( ap(V1f,V5x) = ap(V2g,V5x) ) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3874(V1f,V0P,V2g)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V3s),V4t))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3975(V3s,V2h,V4a)))) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3976(V3s,V2h,V4a)))) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2h),V3s)) & ! [V7t] : ( mem(V7t,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7t),V3s)) & ( ap(V2h,V7t) = V4a ) ) => ( ap(V0f,V7t) = ap(V1g,V7t) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3977(V0f,V4a,V2h,V1g)),V3s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__1,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3978(V3a,V2s)))) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3979(V3a,V2s)))) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3a),V2s)) => ( ap(V0f,V3a) = ap(V1g,V3a) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f3980(V0f,V3a,V1g)),V2s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEXTENSION__FROM__CLOPEN,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V2t)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V2t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2t)),V3u)) & ( ( V3u = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) => ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) => ? [V4g] : ( mem(V4g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4g),V1s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4g),V1s)),V3u)) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V2t)) => ( ap(V4g,V5x) = ap(V0f,V5x) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__IVT__HYPERPLANE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,V3a),V1x)),V4b)) & p(ap(ap(c_2Ereal_2Ereal__lte,V4b),ap(ap(c_2Erealax_2Ereal__mul,V3a),V2y))) ) ) ) ) => ? [V5z] : ( mem(V5z,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5z),V0s)) & ( ap(ap(c_2Erealax_2Ereal__mul,V3a),V5z) = V4b ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__IVT__COMPONENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) & ( p(ap(ap(c_2Ereal_2Ereal__lte,V1x),V3a)) & p(ap(ap(c_2Ereal_2Ereal__lte,V3a),V2y)) ) ) ) ) => ? [V4z] : ( mem(V4z,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4z),V0s)) & ( V4z = V3a ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EMAPPING__CONNECTED__ONTO__SEGMENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Econnected,V0s)) & ( ~~ ? [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ) => ? [V4f] : ( mem(V4f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4f),V0s)) & ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4f),V0s) = ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__INCREASING__CONVERGENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f3981(V0s)))) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,V2n)),ap(V0s,ap(c_2Enum_2ESUC,V2n)))) ) ) => ? [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),V3l),c_2Ereal__topology_2Esequentially)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__DECREASING__CONVERGENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f3981(V0s)))) & ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0s,ap(c_2Enum_2ESUC,V2n))),ap(V0s,V2n))) ) ) => ? [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0s),V3l),c_2Ereal__topology_2Esequentially)) ) ) ))).
fof(ax_thm_2Ereal__topology_2Ehomeomorphism,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3g] : ( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2f),V3g))) <=> ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) => ( ap(V3g,ap(V2f,V4x)) = V4x ) ) ) ) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V0s) = V1t ) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V0s)) & ( ( ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V1t)) => ( ap(V2f,ap(V3g,V5y)) = V5y ) ) ) ) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3g),V1t) = V0s ) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V3g),V1t)) ) ) ) ) ) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Ehomeomorphic,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) <=> ? [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ? [V3g] : ( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2f),V3g))) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3g] : ( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2f),V3g))) <=> ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V0s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V0s)),V1t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V3g),V1t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3g),V1t)),V0s)) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) => ( ap(V3g,ap(V2f,V4x)) = V4x ) ) ) ) & ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V1t)) => ( ap(V2f,ap(V3g,V5y)) = V5y ) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__OF__SUBSETS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4s_27] : ( mem(V4s_27,arr(ty_2Erealax_2Ereal,bool)) => ! [V5t_27] : ( mem(V5t_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4s_27),V2s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5t_27),V3t)) & ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4s_27) = V5t_27 ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V4s_27),V5t_27)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__ID,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V0s)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),i(ty_2Erealax_2Ereal)),i(ty_2Erealax_2Ereal)))) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__REFL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V0s)) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__SYM,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V3t),V2s)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V1g),V0f))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SYM,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1t),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__COMPOSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3k] : ( mem(V3k,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V4s] : ( mem(V4s,arr(ty_2Erealax_2Ereal,bool)) => ! [V5t] : ( mem(V5t,arr(ty_2Erealax_2Ereal,bool)) => ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V4s),V5t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V5t),V6u)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2h),V3k))) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V4s),V6u)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2h),V0f)),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3k)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1t),V2u)) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V2u)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__IMP__CARD__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0s),V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITENESS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) <=> p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__EMPTY,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) <=> ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) & ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),V1s)) <=> ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__MINIMAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) <=> ? [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ? [V3g] : ( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V2f,V4x)),V1t)) & ( ap(V3g,ap(V2f,V4x)) = V4x ) ) ) ) ) & ( ( ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V1t)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V3g,V5y)),V0s)) & ( ap(V2f,ap(V3g,V5y)) = V5y ) ) ) ) ) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V0s)) & p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V3g),V1t)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__SELF,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__LEFT__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V3x) = ap(V0f,V4y) ) => ( V3x = V4y ) ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V2t)) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__RIGHT__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V3x) = ap(V0f,V4y) ) => ( V3x = V4y ) ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2t))) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__SELF,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3789(V0a)),V1s)),V1s)) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__LEFT__EQ,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3908(V0a)),V1s)),V2t)) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__RIGHT__EQ,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3908(V0a)),V2t))) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__QUOTIENT__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4u),V3t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3834(V2s,V0f,V4u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V4u)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__LEFT,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0c)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3684(V0c)),V1s)),V2t)) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__RIGHT,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0c)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3684(V0c)),V2t))) <=> p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V1s),V2t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) ) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) <=> ( ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V0s) = ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V1t) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE__STRONG,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) | p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) ) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) <=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1t)) & ( ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V0s) = ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),V1t) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SING,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1b),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ))).
fof(conj_thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE__UNIQUE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s) = V3t ) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V2s) = V4u ) & ( ( ! [V5v] : ( mem(V5v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V3t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3835(V2s,V0f,V5v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),V5v)) ) ) ) ) & ( ( ! [V7v] : ( mem(V7v,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V7v),V4u)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3982(V2s,V1g,V7v)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V4u)),V7v)) ) ) ) ) & ! [V9x] : ( mem(V9x,ty_2Erealax_2Ereal) => ! [V10y] : ( mem(V10y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V9x),V2s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10y),V2s)) ) => ( ( ap(V0f,V9x) = ap(V0f,V10y) ) <=> ( ap(V1g,V9x) = ap(V1g,V10y) ) ) ) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V3t),V4u)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__OPEN__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s) = V3t ) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V2s)) => ( ap(V1g,ap(V0f,V4x)) = V4x ) ) ) ) & ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V5u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__CLOSED__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s) = V3t ) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V2s)) => ( ap(V1g,ap(V0f,V4x)) = V4x ) ) ) ) & ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V5u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) & ( ap(V0f,V3x) = ap(V0f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) & ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))) ) ) ) ) ) => ? [V6g] : ( mem(V6g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V6g))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) & ( ap(V0f,V3x) = ap(V0f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) & ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))) ) ) ) ) ) => ? [V6g] : ( mem(V6g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V6g))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__OPEN__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V4u)) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4u))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__CLOSED__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V1g))) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),V4u)) ) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V4u))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) & ( ap(V0f,V3x) = ap(V0f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) ) => ( ( ? [V5g] : ( mem(V5g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V5g))) ) ) <=> ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) & ( ap(V0f,V3x) = ap(V0f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) ) => ( ( ? [V5g] : ( mem(V5g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V5g))) ) ) <=> ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINJECTIVE__MAP__OPEN__IFF__CLOSED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) & ( ap(V0f,V3x) = ap(V0f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) ) => ( ( ! [V5u] : ( mem(V5u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V5u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V5u))) ) ) ) <=> ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V6u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V6u))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__IMP__CLOSED__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & p(ap(c_2Ereal__topology_2Ecompact,V1s)) ) ) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3u))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__IMP__QUOTIENT__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & p(ap(c_2Ereal__topology_2Ecompact,V1s)) ) ) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3u),V2t)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3821(V1s,V0f,V3u)))) <=> p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),V3u)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( p(ap(c_2Ereal__topology_2Ecompact,V2s)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) => ( ap(V1g,ap(V0f,V3x)) = V3x ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V0s) = V2t ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V0s)) & ( ap(V1f,V3x) = ap(V1f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) ) ) => ? [V5g] : ( mem(V5g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V2t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V1f),V5g))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V0s) = V2t ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V0s)) & ( ap(V1f,V3x) = ap(V1f,V4y) ) ) ) => ( V3x = V4y ) ) ) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__SURJECTIVE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s) = V3t ) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ? [V5h] : ( mem(V5h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V4u)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V5h))) ) ) ) ) ) => ( ( ? [V6f_27] : ( mem(V6f_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V6f_27))) ) ) & ? [V7g_27] : ( mem(V7g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V3t),V4u)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V1g),V7g_27))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__INJECTIVE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V2s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)),V3t)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V3t)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V3t)),V4u)) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ! [V6y] : ( mem(V6y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V3t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6y),V3t)) & ( ap(V1g,V5x) = ap(V1g,V6y) ) ) ) => ( V5x = V6y ) ) ) ) ) & ? [V7h] : ( mem(V7h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V4u)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1g),V0f)),V7h))) ) ) ) ) ) ) => ( ( ? [V8f_27] : ( mem(V8f_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V8f_27))) ) ) & ? [V9g_27] : ( mem(V9g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V3t),V4u)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V1g),V9g_27))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACTNESS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) => ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) <=> p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__CONNECTEDNESS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> p(ap(c_2Ereal__topology_2Econnected,V1t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( ( ~~ ( V1c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3906(V1c)),V0s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3641(V1a)),V0s))) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__AFFINITY,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( ~~ ( V2c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3907(V2c,V1a)),V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS__CBALL__SPHERE,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) => ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2d))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V3e)))) ) ) ) ) ) ) & ( ( ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5b] : ( mem(V5b,ty_2Erealax_2Ereal) => ! [V6d] : ( mem(V6d,ty_2Erealax_2Ereal) => ! [V7e] : ( mem(V7e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V6d)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V7e)) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V6d))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5b),V7e)))) ) ) ) ) ) ) & ! [V8a] : ( mem(V8a,ty_2Erealax_2Ereal) => ! [V9b] : ( mem(V9b,ty_2Erealax_2Ereal) => ! [V10d] : ( mem(V10d,ty_2Erealax_2Ereal) => ! [V11e] : ( mem(V11e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V10d)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V11e)) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V10d))),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V9b),V11e)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) => ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2d))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V3e)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__CBALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) => ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2d))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V3e)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__SPHERE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) => ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2d))),ap(c_2Ereal__topology_2Esphere,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V3e)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__ONE__POINT__COMPACTIFICATIONS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( p(ap(c_2Ereal__topology_2Ecompact,V1t)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2a),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3b),V1t)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V0s),V2a)),ap(ap(c_2Epred__set_2EDELETE(ty_2Erealax_2Ereal),V1t),V3b))) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V0s),V1t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVALS,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V1b)) & p(ap(ap(c_2Erealax_2Ereal__lt,V2c),V3d)) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2c),V3d)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVAL__UNIV,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V1b)) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECARD__EQ__REAL,lemma,(p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,arr(ty_2Enum_2Enum,bool)),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),c_2Epred__set_2EUNIV(arr(ty_2Enum_2Enum,bool)))))).
fof(conj_thm_2Ereal__topology_2EUNCOUNTABLE__REAL,lemma,(( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ))).
fof(conj_thm_2Ereal__topology_2ECARD__EQ__REAL__IMP__UNCOUNTABLE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0s),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECARD__EQ__EUCLIDEAN,lemma,(p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))))).
fof(conj_thm_2Ereal__topology_2EUNCOUNTABLE__EUCLIDEAN,lemma,(( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ))).
fof(conj_thm_2Ereal__topology_2ECARD__EQ__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNCOUNTABLE__INTERVAL,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) ) ) & ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( ~~ ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOUNTABLE__OPEN__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)))) <=> ( ap(c_2Ereal__topology_2EOPEN__interval,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECARD__EQ__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0s),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ) ))).
fof(conj_thm_2Ereal__topology_2EUNCOUNTABLE__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EOpen,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ~~ p(ap(c_2Epred__set_2Ecountable(ty_2Erealax_2Ereal),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECARD__EQ__BALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1r)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECARD__EQ__CBALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1r)) => p(ap(ap(c_2Ecardinal_2Ecardeq(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r))),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__IMP__NOT__OPEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ~~ p(ap(c_2Ereal__topology_2EOpen,V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__IMP__INFINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V0s)) => ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ~~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EEMPTY__INTERIOR__FINITE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s)) => ( ap(c_2Ereal__topology_2Einterior,V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__CBALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V1r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__BALL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1r] : ( mem(V1r,ty_2Erealax_2Ereal) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1r)))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V1r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__DISCRETE__FINITE__RANGE__CONSTANT__EQ,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V0s)),V2t)) & ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V2t)) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V2t),V3y) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) ) => ? [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s)) => ( ap(V1f,V5x) = V4a ) ) ) ) ) ) ) ) ) ) & ( ( ! [V6s] : ( mem(V6s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V6s)) <=> ! [V7f] : ( mem(V7f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V7f),V6s)) & ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),V6s)) => ? [V9e] : ( mem(V9e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V9e)) & ! [V10y] : ( mem(V10y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V10y),V6s)) & ( ~~ ( ap(V7f,V10y) = ap(V7f,V8x) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V9e),ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V7f,V10y)),ap(V7f,V8x))))) ) ) ) ) ) ) ) => ? [V11a] : ( mem(V11a,ty_2Erealax_2Ereal) & ! [V12x] : ( mem(V12x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V12x),V6s)) => ( ap(V7f,V12x) = V11a ) ) ) ) ) ) ) ) ) & ! [V13s] : ( mem(V13s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V13s)) <=> ! [V14f] : ( mem(V14f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V14f),V13s)) & p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V14f),V13s))) ) => ? [V15a] : ( mem(V15a,ty_2Erealax_2Ereal) & ! [V16x] : ( mem(V16x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V16x),V13s)) => ( ap(V14f,V16x) = V15a ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V0s)),V2t)) & ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V2t)) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V2t),V3y) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) ) => ? [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s)) => ( ap(V1f,V5x) = V4a ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => ? [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V0s)) & ( ~~ ( ap(V1f,V4y) = ap(V1f,V2x) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3e),ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V1f,V4y)),ap(V1f,V2x))))) ) ) ) ) ) ) ) => ? [V5a] : ( mem(V5a,ty_2Erealax_2Ereal) & ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V0s)) => ( ap(V1f,V6x) = V5a ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Econnected,V0s)) <=> ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V0s)) & p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V0s))) ) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => ( ap(V1f,V3x) = V2a ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT,lemma,(! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V2s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1f),V2s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V2s)),V0t)) & ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0t)) => ( ap(ap(c_2Ereal__topology_2Econnected__component,V0t),V3y) = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) ) ) => ? [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V2s)) => ( ap(V1f,V5x) = V4a ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) => ? [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) & ( ~~ ( ap(V0f,V4y) = ap(V0f,V2x) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3e),ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V4y)),ap(V0f,V2x))))) ) ) ) ) ) ) ) ) => ? [V5a] : ( mem(V5a,ty_2Erealax_2Ereal) & ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V1s)) => ( ap(V0f,V6x) = V5a ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Econnected,V1s)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V0f),V1s)) & p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) => ? [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) => ( ap(V0f,V3x) = V2a ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANES,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) & ( ~~ ( V2c = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3983(V1b,V0a))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3984(V3d,V2c)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANE__STANDARD__HYPERPLANE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3882(V1b,V0a))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3985(V2c)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__STANDARD__HYPERPLANE__HYPERPLANE,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( ~~ ( V0a = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3986(V2c))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3983(V1b,V0a)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECAUCHY__ISOMETRIC,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ! [V3x] : ( mem(V3x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & ( p(ap(c_2Ereal__topology_2Esubspace,V1s)) & ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,ap(V0f,V4x))),ap(ap(c_2Erealax_2Ereal__mul,V2e),ap(c_2Ereal_2Eabs,V4x)))) ) ) ) & ( ( ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V3x,V5n)),V1s)) ) ) & p(ap(c_2Ereal__topology_2Ecauchy,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3x))) ) ) ) ) ) => p(ap(c_2Ereal__topology_2Ecauchy,V3x)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPLETE__ISOMETRIC__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) & ( p(ap(c_2Ereal__topology_2Esubspace,V1s)) & ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,ap(V0f,V3x))),ap(ap(c_2Erealax_2Ereal__mul,V2e),ap(c_2Ereal_2Eabs,V3x)))) ) ) ) & p(ap(c_2Ereal__topology_2Ecomplete,V1s)) ) ) ) ) => p(ap(c_2Ereal__topology_2Ecomplete,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINJECTIVE__IMP__ISOMETRIC,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V1s)) & ( p(ap(c_2Ereal__topology_2Esubspace,V1s)) & ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & ( ap(V0f,V2x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( V2x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) => ? [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Eabs,ap(V0f,V4x))),ap(ap(c_2Erealax_2Ereal__mul,V3e),ap(c_2Ereal_2Eabs,V4x)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__IMAGE__SUBSPACE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Esubspace,V1s)) & ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & ( ap(V0f,V2x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( V2x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) & p(ap(c_2Ereal__topology_2EClosed,V1s)) ) ) ) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__SURJECTIVE__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ( ap(V0f,V2x) = V1y ) ) ) ) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V3s)) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EOPEN__BIJECTIVE__LINEAR__IMAGE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ? [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) & ( ap(V0f,V5x) = V4y ) ) ) ) ) => ( p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) <=> p(ap(c_2Ereal__topology_2EOpen,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) => ( V1x = V2y ) ) ) ) ) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V3s)) => p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) => ( p(ap(c_2Ereal__topology_2EClosed,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) <=> p(ap(c_2Ereal__topology_2EClosed,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__LINEAR__IMAGE__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Elinear,V0f)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosure,V1s))),ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__INJECTIVE__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosure,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__BOUNDED__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1s)) ) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Eclosure,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__INTERIOR__IMAGE__SUBSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Einterior,V1s)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__IMAGE__SUBSET__INTERIOR,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ? [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) & ( ap(V0f,V3x) = V2y ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Einterior,V1s))),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__BIJECTIVE__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( ( ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ? [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) & ( ap(V0f,V5x) = V4y ) ) ) ) ) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Einterior,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIN__INTERIOR__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( p(ap(c_2Ereal__topology_2Elinear,V1g)) & ( ( ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1g) = c_2Ecombin_2EI(ty_2Erealax_2Ereal) ) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),ap(c_2Ereal__topology_2Einterior,V2s))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V0f,V3x)),ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELINEAR__OPEN__MAPPING,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ( p(ap(c_2Ereal__topology_2Elinear,V1g)) & ( ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1g) = c_2Ecombin_2EI(ty_2Erealax_2Ereal) ) ) ) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EOpen,V2s)) => p(ap(c_2Ereal__topology_2EOpen,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__INJECTIVE__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),ap(c_2Ereal__topology_2Einterior,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V1x) = ap(V0f,V2y) ) => ( V1x = V2y ) ) ) ) ) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Ecomplete,V3s)) => p(ap(c_2Ereal__topology_2Ecomplete,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ap(V0f,V2x) = ap(V0f,V3y) ) => ( V2x = V3y ) ) ) ) ) => ( p(ap(c_2Ereal__topology_2Ecomplete,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s))) <=> p(ap(c_2Ereal__topology_2Ecomplete,V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELIMPT__INJECTIVE__LINEAR__IMAGE__EQ,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V1f)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( ap(V1f,V3x) = ap(V1f,V4y) ) => ( V3x = V4y ) ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Elimit__point__of,ap(V1f,V0x)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1f),V2s))) <=> p(ap(ap(c_2Ereal__topology_2Elimit__point__of,V0x),V2s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINTERIOR__NEGATIONS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Einterior,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3946),ap(c_2Ereal__topology_2Einterior,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ESYMMETRIC__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V1x)),V0s)) ) ) ) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Einterior,V0s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V2x)),ap(c_2Ereal__topology_2Einterior,V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSURE__NEGATIONS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Eclosure,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3685),V0s)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3946),ap(c_2Ereal__topology_2Eclosure,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ESYMMETRIC__CLOSURE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V1x)),V0s)) ) ) ) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),ap(c_2Ereal__topology_2Eclosure,V0s))) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,V2x)),ap(c_2Ereal__topology_2Eclosure,V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUBSPACE__SUBSTANDARD,lemma,(p(ap(c_2Ereal__topology_2Esubspace,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3987))))).
fof(conj_thm_2Ereal__topology_2ECLOSED__SUBSTANDARD,lemma,(p(ap(c_2Ereal__topology_2EClosed,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3987))))).
fof(conj_thm_2Ereal__topology_2EDIM__SUBSTANDARD,lemma,(( ap(c_2Ereal__topology_2Edim,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3987)) = c_2Enum_2E0 ))).
fof(conj_thm_2Ereal__topology_2EAFFINITY__INVERSES,lemma,(! [V0m] : ( mem(V0m,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ( ( ~~ ( V0m = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ( ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3988(V0m,V1c)),f3989(V1c,V0m)) = i(ty_2Erealax_2Ereal) ) & ( ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3990(V1c,V0m)),f3991(V0m,V1c)) = i(ty_2Erealax_2Ereal) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__AFFINITY__LE,lemma,(! [V0m] : ( mem(V0m,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0m)) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0m),V2x)),V1c)),V3y)) <=> p(ap(ap(c_2Ereal_2Ereal__lte,V2x),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Einv,V0m)),V3y)),ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2E_2F,V1c),V0m))))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LE__AFFINITY,lemma,(! [V0m] : ( mem(V0m,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0m)) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V3y),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0m),V2x)),V1c))) <=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Einv,V0m)),V3y)),ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2E_2F,V1c),V0m)))),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__AFFINITY__LT,lemma,(! [V0m] : ( mem(V0m,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0m)) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0m),V2x)),V1c)),V3y)) <=> p(ap(ap(c_2Erealax_2Ereal__lt,V2x),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Einv,V0m)),V3y)),ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2E_2F,V1c),V0m))))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LT__AFFINITY,lemma,(! [V0m] : ( mem(V0m,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0m)) => ( p(ap(ap(c_2Erealax_2Ereal__lt,V3y),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0m),V2x)),V1c))) <=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Einv,V0m)),V3y)),ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2E_2F,V1c),V0m)))),V2x)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__AFFINITY__EQ,lemma,(! [V0m] : ( mem(V0m,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ~~ ( V0m = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ( ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0m),V2x)),V1c) = V3y ) <=> ( V2x = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Einv,V0m)),V3y)),ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2E_2F,V1c),V0m))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__EQ__AFFINITY,lemma,(! [V0m] : ( mem(V0m,ty_2Erealax_2Ereal) => ! [V1c] : ( mem(V1c,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( ~~ ( V0m = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) => ( ( V3y = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0m),V2x)),V1c) ) <=> ( ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Einv,V0m)),V3y)),ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal_2E_2F,V1c),V0m))) = V2x ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EIMAGE__AFFINITY__INTERVAL,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2m] : ( mem(V2m,ty_2Erealax_2Ereal) => ! [V3c] : ( mem(V3c,ty_2Erealax_2Ereal) => ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3992(V2m,V3c)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2m)),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V2m),V0a)),V3c)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V2m),V1b)),V3c))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2ECLOSED__interval,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V2m),V1b)),V3c)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V2m),V0a)),V3c))),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Esums,axiom, ! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),V2s)) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3993(V2s,V0f)),V1l),c_2Ereal__topology_2Esequentially)) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Einfsum,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ap(ap(c_2Ereal__topology_2Einfsum,V0s),V1f) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3994(V1f,V0s)) ) ) )).
fof(ax_thm_2Ereal__topology_2Esummable,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V0s),V1f)) <=> ? [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V1f),V2l),V0s)) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ESUMS__SUMMABLE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),V2s)) => p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0f)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__INFSUM,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Ereal__topology_2Einfsum,V1s),V0f)),V1s)) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__LIM,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Ereal__topology_2Elim__def(ty_2Enum_2Enum),c_2Ereal__topology_2Esequentially),f3995(V1s,V0f))),V1s)) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EFINITE__INTER__NUMSEG,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V0s),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__FROM,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2k] : ( mem(V2k,ty_2Enum_2Enum) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),ap(c_2Ereal__topology_2Efrom,V2k))) <=> p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f3996(V2k,V0f)),V1l),c_2Ereal__topology_2Esequentially)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__UNIQUE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2l_27] : ( mem(V2l_27,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),V3s)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V2l_27),V3s)) ) => ( V1l = V2l_27 ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__UNIQUE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),V2s)) => ( ap(ap(c_2Ereal__topology_2Einfsum,V2s),V0f) = V1l ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__TERMS__TOZERO,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),ap(c_2Ereal__topology_2Efrom,V2n))) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__FINITE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),V1s),V0f)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__LINEAR,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1h] : ( mem(V1h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V2l),V3s)) & p(ap(c_2Ereal__topology_2Elinear,V1h)) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f3997(V0f,V1h)),ap(V1h,V2l)),V3s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__0,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,k(ty_2Enum_2Enum,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0s)) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__ADD,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1x0] : ( mem(V1x0,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3y0] : ( mem(V3y0,ty_2Erealax_2Ereal) => ! [V4s] : ( mem(V4s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),V1x0),V4s)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V2y),V3y0),V4s)) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f3998(V0x,V2y)),ap(ap(c_2Erealax_2Ereal__add,V1x0),V3y0)),V4s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__SUB,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1x0] : ( mem(V1x0,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3y0] : ( mem(V3y0,ty_2Erealax_2Ereal) => ! [V4s] : ( mem(V4s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),V1x0),V4s)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V2y),V3y0),V4s)) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f3999(V0x,V2y)),ap(ap(c_2Ereal_2Ereal__sub,V1x0),V3y0)),V4s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__CMUL,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1x0] : ( mem(V1x0,ty_2Erealax_2Ereal) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),V1x0),V3s)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f4000(V0x,V2c)),ap(ap(c_2Erealax_2Ereal__mul,V2c),V1x0)),V3s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__NEG,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1x0] : ( mem(V1x0,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),V1x0),V2s)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f4001(V0x)),ap(c_2Erealax_2Ereal__neg,V1x0)),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__IFF,lemma,(! [V0l] : ( mem(V0l,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3k] : ( mem(V3k,arr(ty_2Enum_2Enum,bool)) => ( ( ! [V4x] : ( mem(V4x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4x),V3k)) => ( ap(V1f,V4x) = ap(V2g,V4x) ) ) ) ) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V1f),V0l),V3k)) <=> p(ap(ap(ap(c_2Ereal__topology_2Esums,V2g),V0l),V3k)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__EQ,lemma,(! [V0l] : ( mem(V0l,ty_2Erealax_2Ereal) => ! [V1f] : ( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2g] : ( mem(V2g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V3k] : ( mem(V3k,arr(ty_2Enum_2Enum,bool)) => ( ( ( ! [V4x] : ( mem(V4x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4x),V3k)) => ( ap(V1f,V4x) = ap(V2g,V4x) ) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V1f),V0l),V3k)) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V2g),V0l),V3k)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__0,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( ( ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2n),V1s)) => ( ap(V0f,V2n) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__FINITE__SUPPORT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2k] : ( mem(V2k,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),V2k))) & ! [V3x] : ( mem(V3x,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3x),V2k)) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3x),V1s)) ) ) => ( ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),V2k)),V0f)),V2k)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__COMPONENT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V2l),V1s)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f4002(V0f)),V2l),V1s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__DIFFS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,ty_2Enum_2Enum) => ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,f4003(V0f)),ap(V0f,V1k)),ap(c_2Ereal__topology_2Efrom,V1k))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__TRIVIAL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__RESTRICT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,arr(ty_2Enum_2Enum,bool)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,f4004(V0f,V1k)),V2l),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) <=> p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V2l),V1k)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__SUM,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2k] : ( mem(V2k,arr(ty_2Enum_2Enum,bool)) => ! [V3s] : ( mem(V3s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V3s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V3s),V2k)) & ( ( ! [V4x] : ( mem(V4x,ty_2Enum_2Enum) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4x),V3s)) ) => ( ap(V0f,V4x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) & ( ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),V3s),V0f) = V1l ) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),V2k)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__REINDEX,lemma,(! [V0k] : ( mem(V0k,ty_2Enum_2Enum) => ! [V1a] : ( mem(V1a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,f4005(V0k,V1a)),V2l),ap(c_2Ereal__topology_2Efrom,V3n))) <=> p(ap(ap(ap(c_2Ereal__topology_2Esums,V1a),V2l),ap(c_2Ereal__topology_2Efrom,ap(ap(c_2Earithmetic_2E_2B,V3n),V0k)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__REINDEX__GEN,lemma,(! [V0k] : ( mem(V0k,ty_2Enum_2Enum) => ! [V1a] : ( mem(V1a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,f4005(V0k,V1a)),V2l),V3s)) <=> p(ap(ap(ap(c_2Ereal__topology_2Esums,V1a),V2l),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),f4006(V0k)),V3s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__LINEAR,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1h] : ( mem(V1h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0f)) & p(ap(c_2Ereal__topology_2Elinear,V1h)) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V2s),f4007(V0f,V1h))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__0,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => p(ap(ap(c_2Ereal__topology_2Esummable,V0s),k(ty_2Enum_2Enum,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__ADD,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1y] : ( mem(V1y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0x)) & p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V1y)) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V2s),f4008(V0x,V1y))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__SUB,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1y] : ( mem(V1y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0x)) & p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V1y)) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V2s),f4009(V0x,V1y))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__CMUL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1x] : ( mem(V1x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V0s),V1x)) => p(ap(ap(c_2Ereal__topology_2Esummable,V0s),f4010(V1x,V2c))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__NEG,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0x)) => p(ap(ap(c_2Ereal__topology_2Esummable,V1s),f2866(V0x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__IFF,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2k] : ( mem(V2k,arr(ty_2Enum_2Enum,bool)) => ( ( ! [V3x] : ( mem(V3x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3x),V2k)) => ( ap(V0f,V3x) = ap(V1g,V3x) ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V0f)) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V1g)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2k] : ( mem(V2k,arr(ty_2Enum_2Enum,bool)) => ( ( ( ! [V3x] : ( mem(V3x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3x),V2k)) => ( ap(V0f,V3x) = ap(V1g,V3x) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V0f)) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V1g)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__COMPONENT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) => p(ap(ap(c_2Ereal__topology_2Esummable,V1s),f4011(V0f))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__SUBSET,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Enum_2Enum,bool)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V1s),V2t)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,f4012(V0x,V1s)),V3l),V2t)) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),V3l),V1s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__SUBSET,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V1s),V2t)) & p(ap(ap(c_2Ereal__topology_2Esummable,V2t),f4013(V0x,V1s))) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0x)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__TRIVIAL,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => p(ap(ap(c_2Ereal__topology_2Esummable,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)),V0f)) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__RESTRICT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),f4014(V0f,V1k))) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V1k),V0f)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__FINITE__DIFF,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1t] : ( mem(V1t,arr(ty_2Enum_2Enum,bool)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V1t),V2s)) & ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1t)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V3l),V2s)) ) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Ereal_2Ereal__sub,V3l),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),V1t),V0f))),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V2s),V1t))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__FINITE__UNION,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Enum_2Enum,bool)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V2t)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V3l),V1s)) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Erealax_2Ereal__add,V3l),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V2t),V1s)),V0f))),ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V1s),V2t))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__OFFSET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),ap(c_2Ereal__topology_2Efrom,V2m))) & ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V3n)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V3n)) ) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Ereal_2Ereal__sub,V1l),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),ap(ap(c_2Earithmetic_2E_2D,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f))),ap(c_2Ereal__topology_2Efrom,V3n))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMS__OFFSET__REV,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1l] : ( mem(V1l,ty_2Erealax_2Ereal) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V1l),ap(c_2Ereal__topology_2Efrom,V2m))) & ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2m)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V3n),V2m)) ) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),ap(ap(c_2Erealax_2Ereal__add,V1l),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V3n),ap(ap(c_2Earithmetic_2E_2D,V2m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f))),ap(c_2Ereal__topology_2Efrom,V3n))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__REINDEX,lemma,(! [V0k] : ( mem(V0k,ty_2Enum_2Enum) => ! [V1a] : ( mem(V1a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V2n)),f4015(V0k,V1a))) <=> p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,ap(ap(c_2Earithmetic_2E_2B,V2n),V0k))),V1a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__DROP__LE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V3a),V2s)) & ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V1g),V4b),V2s)) & ! [V5x] : ( mem(V5x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5x),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V5x)),ap(V1g,V5x))) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V3a),V4b)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__DROP__POS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V2a),V1s)) & ! [V3x] : ( mem(V3x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3x),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V3x))) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2a)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__BOUND,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V3a),V2s)) & ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V1g),V4b),V2s)) & ! [V5i] : ( mem(V5i,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5i),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V5i))),ap(V1g,V5i))) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V3a)),V4b)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__LINEAR,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1h] : ( mem(V1h,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0f)) & p(ap(c_2Ereal__topology_2Elinear,V1h)) ) => ( ap(ap(c_2Ereal__topology_2Einfsum,V2s),f4007(V0f,V1h)) = ap(V1h,ap(ap(c_2Ereal__topology_2Einfsum,V2s),V0f)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__0,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ( ap(ap(c_2Ereal__topology_2Einfsum,V0s),k(ty_2Enum_2Enum,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__ADD,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1y] : ( mem(V1y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0x)) & p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V1y)) ) => ( ap(ap(c_2Ereal__topology_2Einfsum,V2s),f4016(V0x,V1y)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal__topology_2Einfsum,V2s),V0x)),ap(ap(c_2Ereal__topology_2Einfsum,V2s),V1y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__SUB,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1y] : ( mem(V1y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0x)) & p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V1y)) ) => ( ap(ap(c_2Ereal__topology_2Einfsum,V2s),f4017(V0x,V1y)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal__topology_2Einfsum,V2s),V0x)),ap(ap(c_2Ereal__topology_2Einfsum,V2s),V1y)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__CMUL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1x] : ( mem(V1x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V0s),V1x)) => ( ap(ap(c_2Ereal__topology_2Einfsum,V0s),f4010(V1x,V2c)) = ap(ap(c_2Erealax_2Ereal__mul,V2c),ap(ap(c_2Ereal__topology_2Einfsum,V0s),V1x)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__NEG,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1x] : ( mem(V1x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V0s),V1x)) => ( ap(ap(c_2Ereal__topology_2Einfsum,V0s),f4018(V1x)) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal__topology_2Einfsum,V0s),V1x)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__EQ,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2k] : ( mem(V2k,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V0f)) & ( p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V1g)) & ! [V3x] : ( mem(V3x,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3x),V2k)) => ( ap(V0f,V3x) = ap(V1g,V3x) ) ) ) ) ) => ( ap(ap(c_2Ereal__topology_2Einfsum,V2k),V0f) = ap(ap(c_2Ereal__topology_2Einfsum,V2k),V1g) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EINFSUM__RESTRICT,lemma,(! [V0k] : ( mem(V0k,arr(ty_2Enum_2Enum,bool)) => ! [V1a] : ( mem(V1a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ap(ap(c_2Ereal__topology_2Einfsum,c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),f4019(V1a,V0k)) = ap(ap(c_2Ereal__topology_2Einfsum,V0k),V1a) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPARTIAL__SUMS__COMPONENT__LE__INFSUM,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( ( ! [V3i] : ( mem(V3i,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3i),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V3i))) ) ) ) & p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V2n))),V0f)),ap(ap(c_2Ereal__topology_2Einfsum,V1s),V0f))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EPARTIAL__SUMS__DROP__LE__INFSUM,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( ( ( ! [V3i] : ( mem(V3i,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3i),V1s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V3i))) ) ) ) & p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V2n))),V0f)),ap(ap(c_2Ereal__topology_2Einfsum,V1s),V0f))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEQUENCE__CAUCHY__WLOG,lemma,(! [V0e] : ( mem(V0e,ty_2Erealax_2Ereal) => ! [V1P] : ( mem(V1P,arr(ty_2Enum_2Enum,bool)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( ( ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( p(ap(V1P,V3m)) & p(ap(V1P,V4n)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V2s,V3m)),ap(V2s,V4n)))),V0e)) ) ) ) ) <=> ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( ( p(ap(V1P,V5m)) & ( p(ap(V1P,V6n)) & p(ap(ap(c_2Earithmetic_2E_3C_3D,V5m),V6n)) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V2s,V5m)),ap(V2s,V6n)))),V0e)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUM__DIFF__LEMMA,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,arr(ty_2Enum_2Enum,bool)) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V3n)) => ( ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1k),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V3n))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1k),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V2m))),V0f)) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1k),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V2m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V3n))),V0f) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EABS__SUM__TRIVIAL__LEMMA,lemma,(! [V0P] : ( mem(V0P,bool) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ! [V4f] : ( mem(V4f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V5e] : ( mem(V5e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5e)) => ( ( p(V0P) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n))),V4f))),V5e)) ) <=> ( p(V0P) => ( p(ap(ap(c_2Eprim__rec_2E_3C,V3n),V2m)) | p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n))),V4f))),V5e)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__CAUCHY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( ( ? [V2l] : ( mem(V2l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V2l),V1s)) ) ) <=> ! [V3e] : ( mem(V3e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)) => ? [V4N] : ( mem(V4N,ty_2Enum_2Enum) & ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V5m),V4N)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,V5m),V6n))),V0f))),V3e)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__CAUCHY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) <=> ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => ? [V3N] : ( mem(V3N,ty_2Enum_2Enum) & ! [V4m] : ( mem(V4m,ty_2Enum_2Enum) => ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V4m),V3N)) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V1s),ap(ap(c_2Eiterate_2E_2E_2E,V4m),V5n))),V0f))),V2e)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__IFF__EVENTUALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2k] : ( mem(V2k,arr(ty_2Enum_2Enum,bool)) => ( ( ? [V3N] : ( mem(V3N,ty_2Enum_2Enum) & ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3N),V4n)) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4n),V2k)) ) => ( ap(V0f,V4n) = ap(V1g,V4n) ) ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V0f)) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V1g)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__EQ__EVENTUALLY,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2k] : ( mem(V2k,arr(ty_2Enum_2Enum,bool)) => ( ( ( ? [V3N] : ( mem(V3N,ty_2Enum_2Enum) & ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3N),V4n)) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4n),V2k)) ) => ( ap(V0f,V4n) = ap(V1g,V4n) ) ) ) ) ) & p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V0f)) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V2k),V1g)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__IFF__COFINITE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Enum_2Enum,bool)) => ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V1s),V2t)),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V2t),V1s)))) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) <=> p(ap(ap(c_2Ereal__topology_2Esummable,V2t),V0f)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__EQ__COFINITE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V1s),V2t)),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),V2t),V1s)))) & p(ap(ap(c_2Ereal__topology_2Esummable,V1s),V0f)) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V2t),V0f)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__FROM__ELSEWHERE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1m] : ( mem(V1m,ty_2Enum_2Enum) => ! [V2n] : ( mem(V2n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V1m)),V0f)) => p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V2n)),V0f)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__CAUCHY__UNIFORM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0P] : ( mem(V0P,arr(A_27a,bool)) => ! [V1f] : ( mem(V1f,arr(A_27a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))) => ! [V2k] : ( mem(V2k,arr(ty_2Enum_2Enum,bool)) => ( ( ? [V3l] : ( mem(V3l,arr(A_27a,ty_2Erealax_2Ereal)) & ! [V4e] : ( mem(V4e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e)) => ? [V5N] : ( mem(V5N,ty_2Enum_2Enum) & ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ! [V7x] : ( mem(V7x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V5N),V6n)) & p(ap(V0P,V7x)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V2k),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V6n))),ap(V1f,V7x))),ap(V3l,V7x)))),V4e)) ) ) ) ) ) ) ) ) <=> ! [V8e] : ( mem(V8e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V8e)) => ? [V9N] : ( mem(V9N,ty_2Enum_2Enum) & ! [V10m] : ( mem(V10m,ty_2Enum_2Enum) => ! [V11n] : ( mem(V11n,ty_2Enum_2Enum) => ! [V12x] : ( mem(V12x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V9N),V10m)) & p(ap(V0P,V12x)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V2k),ap(ap(c_2Eiterate_2E_2E_2E,V10m),V11n))),ap(V1f,V12x)))),V8e)) ) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ESERIES__GOESTOZERO,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,bool)) => ! [V1x] : ( mem(V1x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V0s),V1x)) => ! [V2e] : ( mem(V2e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Enum_2Enum),f4020(V2e,V1x,V0s)),c_2Ereal__topology_2Esequentially)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__TOZERO,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V1k),V0f)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f4014(V0f,V1k)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__BOUNDED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V1k),V0f)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Erealax_2Ereal),V0f),V1k))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__IMP__SUMS__BOUNDED,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V1k)),V0f)) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f4021(V1k,V0f)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__COMPARISON,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ( ( ? [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V1g),V3l),V2s)) ) ) & ? [V4N] : ( mem(V4N,ty_2Enum_2Enum) & ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V5n),V4N)) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5n),V2s)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V5n))),ap(V1g,V5n))) ) ) ) ) => ? [V6l] : ( mem(V6l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V6l),V2s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__COMPARISON,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V1g)) & ? [V3N] : ( mem(V3N,ty_2Enum_2Enum) & ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V4n),V3N)) & p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4n),V2s)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V4n))),ap(V1g,V4n))) ) ) ) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V2s),V0f)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__ABSCONV__IMP__CONV,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,arr(ty_2Enum_2Enum,bool)) => ( p(ap(ap(c_2Ereal__topology_2Esummable,V1k),f4022(V0x))) => p(ap(ap(c_2Ereal__topology_2Esummable,V1k),V0x)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__SUBSET__ABSCONV,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Enum_2Enum,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V1s),f4023(V0x))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),V2t),V1s)) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V2t),f4024(V0x))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__COMPARISON__BOUND,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ( ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V1g),V3a),V2s)) & ! [V4i] : ( mem(V4i,ty_2Enum_2Enum) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4i),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V4i))),ap(V1g,V4i))) ) ) ) => ? [V5l] : ( mem(V5l,ty_2Erealax_2Ereal) & ( p(ap(ap(ap(c_2Ereal__topology_2Esums,V0f),V5l),V2s)) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V5l)),V3a)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__COMPARISON__UNIFORM,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0f] : ( mem(V0f,arr(A_27a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2P] : ( mem(V2P,arr(A_27a,bool)) => ! [V3s] : ( mem(V3s,arr(ty_2Enum_2Enum,bool)) => ( ( ( ? [V4l] : ( mem(V4l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V1g),V4l),V3s)) ) ) & ? [V5N] : ( mem(V5N,ty_2Enum_2Enum) & ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ! [V7x] : ( mem(V7x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V5N),V6n)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V6n),V3s)) & p(ap(V2P,V7x)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(V0f,V7x),V6n))),ap(V1g,V6n))) ) ) ) ) ) => ? [V8l] : ( mem(V8l,arr(A_27a,ty_2Erealax_2Ereal)) & ! [V9e] : ( mem(V9e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V9e)) => ? [V10N] : ( mem(V10N,ty_2Enum_2Enum) & ! [V11n] : ( mem(V11n,ty_2Enum_2Enum) => ! [V12x] : ( mem(V12x,A_27a) => ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V10N),V11n)) & p(ap(V2P,V12x)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EINTER(ty_2Enum_2Enum),V3s),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V11n))),ap(V0f,V12x))),ap(V8l,V12x)))),V9e)) ) ) ) ) ) ) ) ) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ESERIES__RATIO,lemma,(! [V0c] : ( mem(V0c,ty_2Erealax_2Ereal) => ! [V1a] : ( mem(V1a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Enum_2Enum,bool)) => ! [V3N] : ( mem(V3N,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0c),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) & ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V4n),V3N)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V1a,ap(c_2Enum_2ESUC,V4n)))),ap(ap(c_2Erealax_2Ereal__mul,V0c),ap(c_2Ereal_2Eabs,ap(V1a,V4n))))) ) ) ) => ? [V5l] : ( mem(V5l,ty_2Erealax_2Ereal) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V1a),V5l),V2s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBOUNDED__PARTIAL__SUMS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1k] : ( mem(V1k,ty_2Enum_2Enum) => ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f4021(V1k,V0f)))) => p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),ap(c_2Epair_2EUNCURRY(ty_2Enum_2Enum,ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f4026(V0f))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__BILINEAR__PARTIAL__PRE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ! [V4k] : ( mem(V4k,ty_2Enum_2Enum) => ( ( p(ap(c_2Ereal__topology_2Ebilinear,V2h)) & ( p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f4027(V2h,V0f,V1g)),V3l),c_2Ereal__topology_2Esequentially)) & p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V4k)),f4028(V2h,V0f,V1g))) ) ) => p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V4k)),f4029(V2h,V0f,V1g))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__DIRICHLET__BILINEAR,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2h] : ( mem(V2h,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V3k] : ( mem(V3k,ty_2Enum_2Enum) => ! [V4m] : ( mem(V4m,ty_2Enum_2Enum) => ! [V5p] : ( mem(V5p,ty_2Enum_2Enum) => ! [V6l] : ( mem(V6l,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ebilinear,V2h)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f4030(V4m,V0f)))) & ( p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V5p)),f4031(V1g))) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f4032(V2h,V1g,V0f)),V6l),c_2Ereal__topology_2Esequentially)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V3k)),f4033(V2h,V1g,V0f))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__DIRICHLET,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1g] : ( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V2N] : ( mem(V2N,ty_2Enum_2Enum) => ! [V3k] : ( mem(V3k,ty_2Enum_2Enum) => ! [V4m] : ( mem(V4m,ty_2Enum_2Enum) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Enum_2Enum),f4034(V4m,V0f)))) & ( ( ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2N),V6n)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1g,ap(ap(c_2Earithmetic_2E_2B,V6n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V1g,V6n))) ) ) ) & p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),V1g),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)) ) ) => p(ap(ap(c_2Ereal__topology_2Esummable,ap(c_2Ereal__topology_2Efrom,V3k)),f4035(V1g,V0f))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE__STRONG,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),V2f),V1s)),f4023(V0x))) & ! [V4m] : ( mem(V4m,ty_2Enum_2Enum) => ! [V5n] : ( mem(V5n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4m),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5n),V1s)) & ( ap(V2f,V4m) = ap(V2f,V5n) ) ) ) => ( V4m = V5n ) ) ) ) ) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f4036(V1s,V0x,V2f)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Ereal__topology_2Esequentially)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),V2f),V1s)),f4024(V0x))) & ! [V5m] : ( mem(V5m,ty_2Enum_2Enum) => ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5m),V1s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V6n),V1s)) & ( ap(V2f,V5m) = ap(V2f,V6n) ) ) ) => ( V5m = V6n ) ) ) ) ) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Enum_2Enum),V0x),V2f)),V3l),V1s)) <=> p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),V3l),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),V2f),V1s))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__REARRANGE__EQ,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2p] : ( mem(V2p,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V1s),f4024(V0x))) & p(ap(ap(c_2Ereal__topology_2Epermutes(ty_2Enum_2Enum),V2p),V1s)) ) => ( p(ap(ap(ap(c_2Ereal__topology_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Enum_2Enum),V0x),V2p)),V3l),V1s)) <=> p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),V3l),V1s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESERIES__REARRANGE,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2p] : ( mem(V2p,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ! [V3l] : ( mem(V3l,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V1s),f4024(V0x))) & ( p(ap(ap(c_2Ereal__topology_2Epermutes(ty_2Enum_2Enum),V2p),V1s)) & p(ap(ap(ap(c_2Ereal__topology_2Esums,V0x),V3l),V1s)) ) ) => p(ap(ap(ap(c_2Ereal__topology_2Esums,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Enum_2Enum),V0x),V2p)),V3l),V1s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESUMMABLE__REARRANGE,lemma,(! [V0x] : ( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Enum_2Enum,bool)) => ! [V2p] : ( mem(V2p,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) => ( ( p(ap(ap(c_2Ereal__topology_2Esummable,V1s),f4023(V0x))) & p(ap(ap(c_2Ereal__topology_2Epermutes(ty_2Enum_2Enum),V2p),V1s)) ) => p(ap(ap(c_2Ereal__topology_2Esummable,V1s),ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,ty_2Erealax_2Ereal,ty_2Enum_2Enum),V0x),V2p))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EBANACH__FIX,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2c] : ( mem(V2c,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ecomplete,V1s)) & ( ( ~~ ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2c)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,V2c),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V1s)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V3x)),ap(V0f,V4y)))),ap(ap(c_2Erealax_2Ereal__mul,V2c),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V4y))))) ) ) ) ) ) ) ) ) => p(ap(c_2Ebool_2E_3F_21(ty_2Erealax_2Ereal),f4037(V1s,V0f))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDINI,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) => ! [V1g] : ( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V2s)) & ( ( ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,ap(V0f,V3n)),V2s)) ) ) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V1g),V2s)) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V2s)) => p(ap(ap(ap(c_2Ereal__topology_2E_2D_2D_3E(ty_2Enum_2Enum),f4038(V0f,V4x)),ap(V1g,V4x)),c_2Ereal__topology_2Esequentially)) ) ) ) & ! [V6n] : ( mem(V6n,ty_2Enum_2Enum) => ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V2s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(V0f,V6n),V7x)),ap(ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V6n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V7x))) ) ) ) ) ) ) ) => ! [V8e] : ( mem(V8e,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V8e)) => p(ap(ap(c_2Ereal__topology_2Eeventually(ty_2Enum_2Enum),f4040(V2s,V1g,V0f,V8e)),c_2Ereal__topology_2Esequentially)) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Eclosest__point,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1a) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f4042(V1a,V0s)) ) ) )).
fof(conj_thm_2Ereal__topology_2ECLOSEST__POINT__EXISTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1a)),V0s)) & ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1a)))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2y)))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__SET,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1a)),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSEST__POINT__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1a)))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2x)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSEST__POINT__SELF,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) => ( ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1x) = V1x ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSEST__POINT__REFL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ( ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1x) = V1x ) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EDIST__CLOSEST__POINT__LIPSCHITZ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1x)))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2y),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V2y)))))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2y)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__DIST__CLOSEST__POINT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),f4043(V0s)),ap(c_2Ereal__topology_2Eat,V1x))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__DIST__CLOSEST__POINT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f4043(V0s)),V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__DIST__CLOSEST__POINT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f4043(V0s)),V1t)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEGMENT__TO__CLOSEST__POINT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1a)))),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEGMENT__TO__POINT__EXISTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2b),V0s)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eopen__segment,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b))),V0s) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__INTERIOR,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1x)),ap(c_2Ereal__topology_2Einterior,V0s))) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2Einterior,V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSEST__POINT__IN__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2Einterior,V0s))) ) ) ) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Eclosest__point,V0s),V1x)),ap(c_2Ereal__topology_2Efrontier,V0s))) ) ) ))).
fof(ax_thm_2Ereal__topology_2Esetdist,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),V0s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),V1t),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Eiterate_2Einf,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f4045(V1t,V0s))))) ) ) )).
fof(conj_thm_2Ereal__topology_2ESETDIST__EMPTY,lemma,(( ( ! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),V0t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) & ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__POS__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SUBSETS__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2s_27] : ( mem(V2s_27,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2s_27),V0s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t_27),V1t)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V1t)) ) => ? [V6x_27] : ( mem(V6x_27,ty_2Erealax_2Ereal) & ? [V7y_27] : ( mem(V7y_27,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x_27),V2s_27)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7y_27),V3t_27)) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6x_27),V7y_27))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V5y)))) ) ) ) ) ) ) ) ) ) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s_27),V3t_27)) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LE__SETDIST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2d),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V4y)))) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2d),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__LE__DIST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3y)))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LE__SETDIST__EQ,lemma,(! [V0d] : ( mem(V0d,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal_2Ereal__lte,V0d),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)))) <=> ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V2t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V0d),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V4y)))) ) ) ) ) & ( ( ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( V2t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V0d),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__SETDIST__LT__EXISTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),V2b)) ) ) => ? [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) & ? [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1t)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V4y))),V2b)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__REFL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V0s)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SYM,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1t),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__TRIANGLE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V2t))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))))),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V2t))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SINGS,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__LIPSCHITZ,lemma,( ! [A_27a]: ( ne(A_27a) => ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,A_27a) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s))),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s))))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3y)))) ) ) ) )))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__AT__SETDIST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal__topology_2Econtinuous(ty_2Erealax_2Ereal),f4046(V0s)),ap(c_2Ereal__topology_2Eat,V1x))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__ON__SETDIST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Econtinuous__on,f4046(V0s)),V1t)) ) ))).
fof(conj_thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SETDIST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal__topology_2Euniformly__continuous__on,f4046(V0s)),V1t)) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__DIFFERENCES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f3791(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SUBSET__RIGHT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V2u)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V2u))),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SUBSET__LEFT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1t),V2u))),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V2u)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__CLOSURE,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Eclosure,V0s)),V1t)) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) ) ) ) ) & ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),ap(c_2Ereal__topology_2Eclosure,V3t))) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__FRONTIER,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal),V0s),V1t)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Efrontier,V0s)),V1t)) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) ) ) ) ) ) & ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal),V2s),V3t)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),ap(c_2Ereal__topology_2Efrontier,V3t))) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__COMPACT__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ? [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1t)) & ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3y)) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__CLOSED__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( p(ap(c_2Ereal__topology_2Ecompact,V1t)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ? [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) & ? [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1t)) & ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3y)) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__COMPACT__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & p(ap(c_2Ereal__topology_2EClosed,V1t)) ) => ( ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & p(ap(c_2Ereal__topology_2Ecompact,V1t)) ) => ( ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__BOUNDED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) | p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => ( ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ~~ ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3908(V0a)),V1s)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3940(V0a)),V2t))) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal_2Eabs,ap(V0f,V3x)) = ap(c_2Ereal_2Eabs,V3x) ) ) ) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2t))) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__UNIQUE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ! [V4d] : ( mem(V4d,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2a),V0s)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3b),V1t)) & ( ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)) = V4d ) & ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ! [V6y] : ( mem(V6y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6y),V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5x),V6y)))) ) ) ) ) ) ) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = V4d ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__UNIV,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) & ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V1t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__ZERO,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ~~ p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal),V0s),V1t)) ) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__ZERO__STRONG,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ~~ p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eclosure,V0s)),ap(c_2Ereal__topology_2Eclosure,V1t))) ) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__FRONTIERS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal),V0s),V1t)),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Efrontier,V0s)),ap(c_2Ereal__topology_2Efrontier,V1t)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SING__FRONTIER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(c_2Ereal__topology_2Efrontier,V0s))) = ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__CLOSEST__POINT,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V1s)) & ( ~~ ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1s)) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),ap(ap(c_2Ereal__topology_2Eclosest__point,V1s),V0a))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__SING,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),ap(c_2Ereal__topology_2Eclosure,V0s))) ) ) ) ) ) & ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V2s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),ap(c_2Ereal__topology_2Eclosure,V2s))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => ( ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__IN,lemma,(! [V0u] : ( mem(V0u,arr(ty_2Erealax_2Ereal,bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V0u)),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0u)) ) => ( ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1s)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V1s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SING__IN__SET,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V1s)) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1s)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SING__FRONTIER__CASES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(c_2Ereal__topology_2Efrontier,V0s)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SING__TRIANGLE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1x] : ( mem(V1x,ty_2Erealax_2Ereal) => ! [V2y] : ( mem(V2y,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s))),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s))))),ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2y)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__LE__SING,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__BALLS,lemma,(( ( ! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1b] : ( mem(V1b,ty_2Erealax_2Ereal) => ! [V2r] : ( mem(V2r,ty_2Erealax_2Ereal) => ! [V3s] : ( mem(V3s,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1b),V3s)))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Ereal_2Ereal__lte,V2r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(ap(c_2Ereal_2Ereal__lte,V3s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Emax,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1b))),ap(ap(c_2Erealax_2Ereal__add,V2r),V3s)))) ) ) ) ) ) ) & ( ( ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5b] : ( mem(V5b,ty_2Erealax_2Ereal) => ! [V6r] : ( mem(V6r,ty_2Erealax_2Ereal) => ! [V7s] : ( mem(V7s,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V6r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5b),V7s)))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Ereal_2Ereal__lte,V6r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(ap(c_2Erealax_2Ereal__lt,V7s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Emax,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V5b))),ap(ap(c_2Erealax_2Ereal__add,V6r),V7s)))) ) ) ) ) ) ) & ( ( ! [V8a] : ( mem(V8a,ty_2Erealax_2Ereal) => ! [V9b] : ( mem(V9b,ty_2Erealax_2Ereal) => ! [V10r] : ( mem(V10r,ty_2Erealax_2Ereal) => ! [V11s] : ( mem(V11s,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V10r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V9b),V11s)))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Erealax_2Ereal__lt,V10r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(ap(c_2Ereal_2Ereal__lte,V11s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Emax,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V9b))),ap(ap(c_2Erealax_2Ereal__add,V10r),V11s)))) ) ) ) ) ) ) & ! [V12a] : ( mem(V12a,ty_2Erealax_2Ereal) => ! [V13b] : ( mem(V13b,ty_2Erealax_2Ereal) => ! [V14r] : ( mem(V14r,ty_2Erealax_2Ereal) => ! [V15s] : ( mem(V15s,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V14r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V13b),V15s)))) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2E_5C_2F,ap(ap(c_2Erealax_2Ereal__lt,V14r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),ap(ap(c_2Erealax_2Ereal__lt,V15s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Emax,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V13b))),ap(ap(c_2Erealax_2Ereal__add,V14r),V15s)))) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATION__CLOSURES,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Eclosure,V1t)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1t),ap(c_2Ereal__topology_2Eclosure,V0s)) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) => ? [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & ? [V3v] : ( mem(V3v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Erealax_2Ereal),V2u),V3v)) & ( p(ap(c_2Ereal__topology_2EOpen,V2u)) & ( p(ap(c_2Ereal__topology_2EOpen,V3v)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V2u)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V3v)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & ? [V3v] : ( mem(V3v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2u)) & ( p(ap(c_2Ereal__topology_2EOpen,V3v)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V2u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V3v)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2u),V3v) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL__LOCAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V1t)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V3s_27] : ( mem(V3s_27,arr(ty_2Erealax_2Ereal,bool)) & ? [V4t_27] : ( mem(V4t_27,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V3s_27)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V4t_27)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V3s_27)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V4t_27)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3s_27),V4t_27) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATION__NORMAL__COMPACT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & ? [V3v] : ( mem(V3v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2u)) & ( p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Eclosure,V2u))) & ( p(ap(c_2Ereal__topology_2EOpen,V3v)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V2u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V3v)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2u),V3v) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATION__HAUSDORFF,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ~~ ( V0x = V1y ) ) => ? [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & ? [V3v] : ( mem(V3v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2u)) & ( p(ap(c_2Ereal__topology_2EOpen,V3v)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2u)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),V3v)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2u),V3v) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATION__T2,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ~~ ( V0x = V1y ) ) <=> ? [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & ? [V3v] : ( mem(V3v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2u)) & ( p(ap(c_2Ereal__topology_2EOpen,V3v)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2u)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),V3v)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V2u),V3v) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATION__T1,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ~~ ( V0x = V1y ) ) <=> ? [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & ? [V3v] : ( mem(V3v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2u)) & ( p(ap(c_2Ereal__topology_2EOpen,V3v)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2u)) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),V2u)) ) & ( ( ~~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V3v)) ) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),V3v)) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESEPARATION__T0,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ( ~~ ( V0x = V1y ) ) <=> ? [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(c_2Ereal__topology_2EOpen,V2u)) & ( ~~ ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),V2u)) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1y),V2u)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CHAIN,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => ( p(ap(c_2Ereal__topology_2Ecompact,V1s)) & p(ap(c_2Ereal__topology_2Econnected,V1s)) ) ) ) ) & ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2s),V0f)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3t),V0f)) ) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V2s),V3t)) | p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3t),V2s)) ) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__CHAIN__GEN,lemma,(! [V0f] : ( mem(V0f,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V1s),V0f)) => ( p(ap(c_2Ereal__topology_2EClosed,V1s)) & p(ap(c_2Ereal__topology_2Econnected,V1s)) ) ) ) ) & ( ( ? [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2s),V0f)) & p(ap(c_2Ereal__topology_2Ecompact,V2s)) ) ) ) & ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V3s),V0f)) & p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V4t),V0f)) ) => ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V3s),V4t)) | p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4t),V3s)) ) ) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V0f))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__NEST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(c_2Ereal__topology_2Ecompact,ap(V0s,V1n))) & p(ap(c_2Ereal__topology_2Econnected,ap(V0s,V1n))) ) ) ) & ! [V2m] : ( mem(V2m,ty_2Enum_2Enum) => ! [V3n] : ( mem(V3n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V3n)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0s,V3n)),ap(V0s,V2m))) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum),f3804(V0s))))) ) ))).
fof(conj_thm_2Ereal__topology_2ECONNECTED__NEST__GEN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool))) => ( ( ( ! [V1n] : ( mem(V1n,ty_2Enum_2Enum) => ( p(ap(c_2Ereal__topology_2EClosed,ap(V0s,V1n))) & p(ap(c_2Ereal__topology_2Econnected,ap(V0s,V1n))) ) ) ) & ( ( ? [V2n] : ( mem(V2n,ty_2Enum_2Enum) & p(ap(c_2Ereal__topology_2Ecompact,ap(V0s,V2n))) ) ) & ! [V3m] : ( mem(V3m,ty_2Enum_2Enum) => ! [V4n] : ( mem(V4n,ty_2Enum_2Enum) => ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0s,V4n)),ap(V0s,V3m))) ) ) ) ) ) => p(ap(c_2Ereal__topology_2Econnected,ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum),f4047(V0s))))) ) ))).
fof(ax_thm_2Ereal__topology_2Ehausdist,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4048(V1t,V0s))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4049(V0s,V1t)))),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))),ap(c_2Ebool_2E_3F(ty_2Erealax_2Ereal),f4053(V1t,V0s)))),ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4054(V1t,V0s))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4055(V0s,V1t))))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) )).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__POS__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__REFL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V0s)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__SYM,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1t),V0s)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__EMPTY,lemma,(( ( ! [V0t] : ( mem(V0t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)),V0t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) & ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__SINGS,lemma,(! [V0x] : ( mem(V0x,ty_2Erealax_2Ereal) => ! [V1y] : ( mem(V1y,ty_2Erealax_2Ereal) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2s_27] : ( mem(V2s_27,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t_27] : ( mem(V3t_27,arr(ty_2Erealax_2Ereal,bool)) => ( ( ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V5x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1t))),V4b)) ) ) ) & ! [V6y] : ( mem(V6y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6y),V1t)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V6y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s))),V4b)) ) ) ) <=> ( ( ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V2s_27)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V7x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V3t_27))),V4b)) ) ) ) & ! [V8y] : ( mem(V8y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8y),V3t_27)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V8y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V2s_27))),V4b)) ) ) ) ) ) ) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s_27),V3t_27)) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__TRANSLATION,lemma,(! [V0a] : ( mem(V0a,ty_2Erealax_2Ereal) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3908(V0a)),V1s)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3940(V0a)),V2t))) = ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__LINEAR__IMAGE,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V0f)) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ap(c_2Ereal_2Eabs,ap(V0f,V3x)) = ap(c_2Ereal_2Eabs,V3x) ) ) ) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V2t))) = ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__CLOSURE,lemma,(( ( ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(c_2Ereal__topology_2Eclosure,V0s)),V1t)) = ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) ) ) ) ) & ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),ap(c_2Ereal__topology_2Eclosure,V3t))) = ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V2s),V3t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1t))),V2b)) ) ) ) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1t)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V4y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s))),V2b)) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),V2b)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f4057(V2b,V1t))))) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f4059(V2b,V0s))))) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),V2b)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LE__HAUSDIST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ! [V4c] : ( mem(V4c,ty_2Erealax_2Ereal) => ! [V5z] : ( mem(V5z,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V6x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1t))),V3b)) ) ) ) & ( ( ! [V7y] : ( mem(V7y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7y),V1t)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V7y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s))),V4c)) ) ) ) & ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5z),V0s)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2a),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V5z),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1t)))) ) | ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5z),V1t)) & p(ap(ap(c_2Ereal_2Ereal__lte,V2a),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V5z),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s)))) ) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,V2a),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__LE__HAUSDIST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__SING__LE__HAUSDIST,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1t))),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ESETDIST__HAUSDIST__TRIANGLE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & p(ap(c_2Ereal__topology_2Ebounded__def,V2u)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V2u))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1t),V2u))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__SETDIST__TRIANGLE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V2u))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1t),V2u))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__LT__HAUSDIST__POINT__EXISTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3d] : ( mem(V3d,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),V3d)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) ) ) ) ) => ? [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1t)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V4y))),V3d)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2s)) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V0f,V3x)),V1t)) ) ) ) & ( ( ! [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V4u)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3909(V2s,V0f,V4u)))) ) ) ) & ! [V6u] : ( mem(V6u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1t)),V6u)) => p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2s)),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3912(V2s,V0f,V6u)))) ) ) ) ) => ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ! [V9e] : ( mem(V9e,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),V2s)) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V9e)) & p(ap(c_2Ereal__topology_2Ebounded__def,ap(V0f,V8x))) ) ) => ? [V10d] : ( mem(V10d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V10d)) & ! [V11x_27] : ( mem(V11x_27,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V11x_27),V2s)) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8x),V11x_27))),V10d)) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(V0f,V8x)),ap(V0f,V11x_27)))),V9e)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4048(V1t,V0s))),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4049(V0s,V1t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL__ALT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(ap(c_2Ereal_2Emax,ap(c_2Ereal_2Esup,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4048(V1t,V0s)))),ap(c_2Ereal_2Esup,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4049(V0s,V1t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EREAL__HAUSDIST__LE__EQ,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2b] : ( mem(V2b,ty_2Erealax_2Ereal) => ( ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) ) ) => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),V2b)) <=> ( ( ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3x),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V1t))),V2b)) ) ) ) & ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1t)) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Esetdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V4y),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0s))),V2b)) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__UNION__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V2u)) & ( ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V2u = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)),ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V2u)))),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1t),V2u)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__INSERT__LE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2a),V0s)),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V2a),V1t)))),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__COMPACT__EXISTS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ecompact,V1t)) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) => ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)) => ? [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1t)) & p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3y))),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)))) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__TRIANGLE,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V2u)) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V2u))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1t),V2u))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__COMPACT__SUMS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ecompact,V1t)) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) => p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Epair_2Eprod(ty_2Erealax_2Ereal,bool)),f4061(V1t,V0s))))) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__TRANS,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V2u)) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V2u))),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t))),ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1t),V2u))))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__EQ__0,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) ) => ( ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) <=> ( ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) | ( ap(c_2Ereal__topology_2Eclosure,V0s) = ap(c_2Ereal__topology_2Eclosure,V1t) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__COMPACT__NONTRIVIAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V0s)) & ( p(ap(c_2Ereal__topology_2Ecompact,V1t)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Eiterate_2Einf,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4066(V0s,V1t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHAUSDIST__ALT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & ( ~~ ( V1t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) ) ) => ( ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V1t)) = ap(c_2Ereal_2Esup,ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4067(V1t,V0s))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ECONTINUOUS__DIAMETER,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1e] : ( mem(V1e,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) & ( ( ~~ ( V0s = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e)) ) ) => ? [V2d] : ( mem(V2d,ty_2Erealax_2Ereal) & ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2d)) & ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V3t)) & ( ( ~~ ( V3t = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal__topology_2Ehausdist,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V0s),V3t))),V2d)) ) ) => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal__topology_2Ediameter,V0s)),ap(c_2Ereal__topology_2Ediameter,V3t)))),V1e)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EISOMETRY__IMP__OPEN__MAP,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ! [V3u] : ( mem(V3u,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ( ( ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5y),V1s)) ) => ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V4x)),ap(V0f,V5y))) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4x),V5y)) ) ) ) ) ) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V3u)) ) ) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2t)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V3u))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EISOMETRY__IMP__EMBEDDING,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s) = V2t ) & ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ! [V4y] : ( mem(V4y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4y),V1s)) ) => ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V3x)),ap(V0f,V4y))) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3x),V4y)) ) ) ) ) ) => ? [V5g] : ( mem(V5g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V2t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V5g))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EISOMETRY__IMP__HOMEOMORPHISM__COMPACT,lemma,(! [V0f] : ( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Ecompact,V1s)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0f),V1s)),V1s)) & ! [V2x] : ( mem(V2x,ty_2Erealax_2Ereal) => ! [V3y] : ( mem(V3y,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V1s)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V1s)) ) => ( ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(V0f,V2x)),ap(V0f,V3y))) = ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2x),V3y)) ) ) ) ) ) ) => ? [V4g] : ( mem(V4g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V1s),V1s)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V0f),V4g))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EURYSOHN__LOCAL__STRONG,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V1t)) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ~~ ( V3a = V4b ) ) ) ) ) => ? [V5f] : ( mem(V5f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V5f),V2u)) & ( ( ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V2u)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V5f,V6x)),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3a),V4b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) & ( ( ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V2u)) => ( ( ap(V5f,V7x) = V3a ) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V0s)) ) ) ) ) & ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),V2u)) => ( ( ap(V5f,V8x) = V4b ) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),V1t)) ) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EURYSOHN__LOCAL,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ! [V3a] : ( mem(V3a,ty_2Erealax_2Ereal) => ! [V4b] : ( mem(V4b,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V0s)) & ( p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V2u)),V1t)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V5f] : ( mem(V5f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V5f),V2u)) & ( ( ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V2u)) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V5f,V6x)),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V3a),V4b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) ) & ( ( ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V0s)) => ( ap(V5f,V7x) = V3a ) ) ) ) & ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V8x),V1t)) => ( ap(V5f,V8x) = V4b ) ) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EURYSOHN__STRONG,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ( ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) & ( ~~ ( V2a = V3b ) ) ) ) ) => ? [V4f] : ( mem(V4f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4f),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V4f,V5x)),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) & ( ( ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( ( ap(V4f,V6x) = V2a ) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V0s)) ) ) ) & ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( ( ap(V4f,V7x) = V3b ) <=> p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V1t)) ) ) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EURYSOHN,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ! [V1t] : ( mem(V1t,arr(ty_2Erealax_2Ereal,bool)) => ! [V2a] : ( mem(V2a,ty_2Erealax_2Ereal) => ! [V3b] : ( mem(V3b,ty_2Erealax_2Ereal) => ( ( p(ap(c_2Ereal__topology_2EClosed,V0s)) & ( p(ap(c_2Ereal__topology_2EClosed,V1t)) & ( ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t) = c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) ) ) ) => ? [V4f] : ( mem(V4f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) & ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V4f),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) & ( ( ! [V5x] : ( mem(V5x,ty_2Erealax_2Ereal) => p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(V4f,V5x)),ap(c_2Ereal__topology_2Eclosed__segment,ap(ap(c_2Elist_2ECONS(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2a),V3b)),c_2Elist_2ENIL(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))))) ) ) & ( ( ! [V6x] : ( mem(V6x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V6x),V0s)) => ( ap(V4f,V6x) = V2a ) ) ) ) & ! [V7x] : ( mem(V7x,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V7x),V1t)) => ( ap(V4f,V7x) = V3b ) ) ) ) ) ) ) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Elocally,axiom, ! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V1s)) <=> ! [V2w] : ( mem(V2w,arr(ty_2Erealax_2Ereal,bool)) => ! [V3x] : ( mem(V3x,ty_2Erealax_2Ereal) => ( ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V2w)) & p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V2w)) ) => ? [V4u] : ( mem(V4u,arr(ty_2Erealax_2Ereal,bool)) & ? [V5v] : ( mem(V5v,arr(ty_2Erealax_2Ereal,bool)) & ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V4u)) & ( p(ap(V0P,V5v)) & ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V4u)) & ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V4u),V5v)) & p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5v),V2w)) ) ) ) ) ) ) ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2ELOCALLY__MONO,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1Q] : ( mem(V1Q,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( ( ( ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(V0P,V3t)) => p(ap(V1Q,V3t)) ) ) ) & p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V2s)) ) => p(ap(ap(c_2Ereal__topology_2Elocally,V1Q),V2s)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELOCALLY__OPEN__SUBSET,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V1s)) & p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V2t)) ) => p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V2t)) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELOCALLY__DIFF__CLOSED,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V1s)) & p(ap(ap(c_2Etopology_2Eclosed__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V1s)),V2t)) ) => p(ap(ap(c_2Ereal__topology_2Elocally,V0P),ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),V1s),V2t))) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELOCALLY__EMPTY,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => p(ap(ap(c_2Ereal__topology_2Elocally,V0P),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) ))).
fof(conj_thm_2Ereal__topology_2ELOCALLY__SING,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) <=> p(ap(V0P,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V1a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELOCALLY__INTER,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ! [V1s] : ( mem(V1s,arr(ty_2Erealax_2Ereal,bool)) => ! [V2t] : ( mem(V2t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(V0P,V1s)) & p(ap(V0P,V2t)) ) => p(ap(V0P,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V1s),V2t))) ) ) ) ) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V3s)) & p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V4t)) ) => p(ap(ap(c_2Ereal__topology_2Elocally,V0P),ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V3s),V4t))) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHISM__LOCALLY,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1Q] : ( mem(V1Q,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3g] : ( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ( ( ! [V4s] : ( mem(V4s,arr(ty_2Erealax_2Ereal,bool)) => ! [V5t] : ( mem(V5t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V4s),V5t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2f),V3g))) => ( p(ap(V0P,V4s)) <=> p(ap(V1Q,V5t)) ) ) ) ) ) => ! [V6s] : ( mem(V6s,arr(ty_2Erealax_2Ereal,bool)) => ! [V7t] : ( mem(V7t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphism,ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,bool),arr(ty_2Erealax_2Ereal,bool)),V6s),V7t)),ap(ap(c_2Epair_2E_2C(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)),V2f),V3g))) => ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V6s)) <=> p(ap(ap(c_2Ereal__topology_2Elocally,V1Q),V7t)) ) ) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2EHOMEOMORPHIC__LOCALLY,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1Q] : ( mem(V1Q,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ! [V3t] : ( mem(V3t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V2s),V3t)) => ( p(ap(V0P,V2s)) <=> p(ap(V1Q,V3t)) ) ) ) ) ) => ! [V4s] : ( mem(V4s,arr(ty_2Erealax_2Ereal,bool)) => ! [V5t] : ( mem(V5t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Ehomeomorphic,V4s),V5t)) => ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V4s)) <=> p(ap(ap(c_2Ereal__topology_2Elocally,V1Q),V5t)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELOCALLY__TRANSLATION,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ! [V1a] : ( mem(V1a,ty_2Erealax_2Ereal) => ! [V2s] : ( mem(V2s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(V0P,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4068(V1a)),V2s))) <=> p(ap(V0P,V2s)) ) ) ) ) => ! [V4a] : ( mem(V4a,ty_2Erealax_2Ereal) => ! [V5s] : ( mem(V5s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f4069(V4a)),V5s))) <=> p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V5s)) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELOCALLY__INJECTIVE__LINEAR__IMAGE,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1Q] : ( mem(V1Q,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ( ( ! [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V2f)) & ! [V4x] : ( mem(V4x,ty_2Erealax_2Ereal) => ! [V5y] : ( mem(V5y,ty_2Erealax_2Ereal) => ( ( ap(V2f,V4x) = ap(V2f,V5y) ) => ( V4x = V5y ) ) ) ) ) => ( p(ap(V0P,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V3s))) <=> p(ap(V1Q,V3s)) ) ) ) ) ) => ! [V6f] : ( mem(V6f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V7s] : ( mem(V7s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(c_2Ereal__topology_2Elinear,V6f)) & ! [V8x] : ( mem(V8x,ty_2Erealax_2Ereal) => ! [V9y] : ( mem(V9y,ty_2Erealax_2Ereal) => ( ( ap(V6f,V8x) = ap(V6f,V9y) ) => ( V8x = V9y ) ) ) ) ) => ( p(ap(ap(c_2Ereal__topology_2Elocally,V0P),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6f),V7s))) <=> p(ap(ap(c_2Ereal__topology_2Elocally,V1Q),V7s)) ) ) ) ) ) ) ))).
fof(conj_thm_2Ereal__topology_2ELOCALLY__OPEN__MAP__IMAGE,lemma,(! [V0P] : ( mem(V0P,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V1Q] : ( mem(V1Q,arr(arr(ty_2Erealax_2Ereal,bool),bool)) => ! [V2f] : ( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) => ! [V3s] : ( mem(V3s,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Ereal__topology_2Econtinuous__on,V2f),V3s)) & ( ( ! [V4t] : ( mem(V4t,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),V3s)),V4t)) => p(ap(ap(c_2Etopology_2Eopen__in(ty_2Erealax_2Ereal),ap(ap(c_2Ereal__topology_2Esubtopology(ty_2Erealax_2Ereal),c_2Ereal__topology_2Eeuclidean),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V3s))),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V4t))) ) ) ) & ( ( ! [V5t] : ( mem(V5t,arr(ty_2Erealax_2Ereal,bool)) => ( ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V5t),V3s)) & p(ap(V0P,V5t)) ) => p(ap(V1Q,ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V5t))) ) ) ) & p(ap(ap(c_2Ereal__topology_2Elocally,V0P),V3s)) ) ) ) => p(ap(ap(c_2Ereal__topology_2Elocally,V1Q),ap(ap(c_2Epred__set_2EIMAGE(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2f),V3s))) ) ) ) ) ))).
fof(ax_thm_2Ereal__topology_2Egdelta,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Egdelta,V0s)) <=> ? [V1g] : ( mem(V1g,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & ( p(ap(c_2Epred__set_2Ecountable(arr(ty_2Erealax_2Ereal,bool)),V1g)) & ( ( ! [V2u] : ( mem(V2u,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2u),V1g)) => p(ap(c_2Ereal__topology_2EOpen,V2u)) ) ) ) & ( ap(c_2Epred__set_2EBIGINTER(ty_2Erealax_2Ereal),V1g) = V0s ) ) ) ) ) )).
fof(ax_thm_2Ereal__topology_2Efsigma,axiom, ! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Efsigma,V0s)) <=> ? [V1g] : ( mem(V1g,arr(arr(ty_2Erealax_2Ereal,bool),bool)) & ( p(ap(c_2Epred__set_2Ecountable(arr(ty_2Erealax_2Ereal,bool)),V1g)) & ( ( ! [V2c] : ( mem(V2c,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Erealax_2Ereal,bool)),V2c),V1g)) => p(ap(c_2Ereal__topology_2EClosed,V2c)) ) ) ) & ( ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),V1g) = V0s ) ) ) ) ) )).
fof(conj_thm_2Ereal__topology_2EGDELTA__COMPLEMENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Egdelta,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) <=> p(ap(c_2Ereal__topology_2Efsigma,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2EFSIGMA__COMPLEMENT,lemma,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2Efsigma,ap(ap(c_2Epred__set_2EDIFF(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal)),V0s))) <=> p(ap(c_2Ereal__topology_2Egdelta,V0s)) ) ))).
fof(conj_thm_2Ereal__topology_2ECLOSED__AS__GDELTA,conjecture,(! [V0s] : ( mem(V0s,arr(ty_2Erealax_2Ereal,bool)) => ( p(ap(c_2Ereal__topology_2EClosed,V0s)) => p(ap(c_2Ereal__topology_2Egdelta,V0s)) ) ))).
