reserve i, k, m, n for Nat,
  r, s for Real,
  rn for Real,
  x, y , z, X for set,
  T, T1, T2 for non empty TopSpace,
  p, q for Point of T,
  A, B, C for Subset of T,
  A9 for non empty Subset of T,
  pq for Element of [:the carrier of T,the carrier of T:],
  pq9 for Point of [:T,T:],
  pmet,pmet1 for Function of [:the carrier of T,the carrier of T:],REAL,
  pmet9,pmet19 for RealMap of [:T,T:] ,
  f,f1 for RealMap of T,
  FS2 for Functional_Sequence of [:the carrier of T,the carrier of T:],REAL,
  seq for Real_Sequence;

theorem Th19:
  for T holds (T is regular & T is T_1 & ex Bn being
  FamilySequence of T st Bn is Basis_sigma_locally_finite) iff T is metrizable
proof
  let T;
  thus (T is regular & T is T_1 & ex Bn being FamilySequence of T st Bn is
  Basis_sigma_locally_finite) implies T is metrizable
  proof
    set cTT=the carrier of [:T,T:];
    set bcT = bool the carrier of T;
    set cT = the carrier of T;
    assume that
A1: T is regular and
A2: T is T_1 and
A3: ex Bn be FamilySequence of T st Bn is Basis_sigma_locally_finite;
    set Fun=Funcs(cTT,REAL);
    consider Bn be FamilySequence of T such that
A4: Bn is Basis_sigma_locally_finite by A3;
    defpred NN[object,object,RealMap of [:T,T:]] means
     ex pmet st $3=pmet & $3 is
continuous & pmet is_a_pseudometric_of cT & for p,q holds pmet.[p,q]<=1 & for p
,q st ex A,B st A is open & B is open & A in Bn.$2 & B in Bn.$1 & p in A & Cl A
    c= B & not q in B holds pmet.[p,q] =1;
    defpred N[object,object] means
    ex F be RealMap of [:T,T:] st F = $2 & for n,m st
    PairFunc".$1=[n,m] holds NN[n,m,F];
A5: Union Bn is Basis of T by A4,NAGATA_1:def 6;
A6: Bn is sigma_locally_finite by A4,NAGATA_1:def 6;
A7: for n,m ex pmet9 st NN[n,m,pmet9]
    proof
      defpred add9[Element of Fun,Element of Fun,set] means $1+$2=$3;
      defpred funcdist[RealMap of T,Function] means for p,q holds $2.(p,q)=
      |.$1.p-$1.q.|;
      let n,m;
      deffunc V(object)=
        union{Q where Q is Subset of T:
           ex D1 being set st D1 = $1 & Q in Bn.m & Cl Q c= D1};
      set Bnn = Bn.n;
      deffunc s(Subset of T)={Q where Q is Subset of T: Q in Bn.n& Q meets $1};
      defpred S[set,Subset of T] means $1 in $2 & $2 is open & s($2) is finite;
A8:   for A be object st A in bcT holds V(A) in bcT
      proof
        let A be object such that
        A in bcT;
        set Um={Q where Q is Subset of T:
          ex D1 being set st D1 = A &  Q in Bn.m & Cl Q c=D1};
        now
          let uv be object;
          assume uv in V(A);

          then consider v be set such that
A9:       uv in v and
A10:      v in Um by TARSKI:def 4;
          ex B st v=B & ex D1 being set st D1 = A & B in Bn.m & Cl B c=D1
               by A10;
          hence uv in cT by A9;
        end;
        then V(A) c= cT;
        hence thesis;
      end;
      consider Vm be Function of bcT,bcT such that
A11:  for A be object st A in bcT holds Vm.A=V(A) from FUNCT_2:sch 2(A8
      );
      defpred F[object,object] means
     ex F be RealMap of T, S be Subset of T st S =
$1 & F = $2 & F is continuous & (for p holds 0 <= F.p & F.p <= 1 & (p in S`
      implies F.p = 0) & (p in Cl (Vm.S) implies F.p = 1));
A12: m in NAT by ORDINAL1:def 12;
A13:  Bn.m is locally_finite by A6,NAGATA_1:def 3,A12;
A14:  for A holds Cl (Vm.A) c=A
      proof
        let A;
        set VmA={Q where Q is Subset of T:
          ex D1 being set st D1 = A & Q in Bn.m & Cl Q c=D1};
        VmA c= bcT
        proof
          let B be object;
          assume B in VmA;
          then ex C st B=C &
             ex D1 being set st D1 = A & C in Bn.m & Cl C c=D1;
          hence thesis;
        end;
        then reconsider VmA as Subset-Family of T;
A15:    union clf VmA c= A
        proof
          let ClBu be object;
          assume ClBu in union clf VmA;
          then consider ClB be set such that
A16:      ClBu in ClB and
A17:      ClB in clf VmA by TARSKI:def 4;
          reconsider ClB as Subset of T by A17;
          consider B such that
A18:      Cl B =ClB and
A19:      B in VmA by A17,PCOMPS_1:def 2;
          ex C st B=C &
           ex D1 being set st D1 = A & C in Bn.m & Cl C c=D1 by A19;
          hence thesis by A16,A18;
        end;
        VmA c=Bn.m
        proof
          let B be object;
          assume B in VmA;
          then ex C st B=C &
           ex D1 being set st D1 = A & C in Bn.m & Cl C c=D1;
          hence thesis;
        end;
        then
A20:      Cl union VmA = union clf VmA by A13,PCOMPS_1:9,20;
         Vm.A=V(A) by A11;
        hence Cl (Vm.A) c=A by A15,A20;
      end;
A21:  for A be object st A in Bnn
ex f be object st f in Funcs(cT,REAL) & F[A,f ]
      proof
        let A be object;
        assume A in Bnn;
        then
A22:    A in Union Bn by PROB_1:12;
        Union Bn c= the topology of T by A5,TOPS_2:64;
        then reconsider A as open Subset of T by A22,PRE_TOPC:def 2;
        A` misses A by XBOOLE_1:79;
        then
A23:    A` misses Cl (Vm.A) by A14,XBOOLE_1:63;
        T is normal by A1,A4,NAGATA_1:20;
        then consider f be Function of T,R^1 such that
A24:    f is continuous & for p holds 0 <= f.p & f.p <= 1 & (p in A`
        implies f.p=0)& (p in Cl (Vm.A) implies f.p=1) by A23,URYSOHN3:20;
        reconsider f9=f as RealMap of T by TOPMETR:17;
A25:    ex F be RealMap of T, S be Subset of T st S = A & F = f9 & F is
continuous & for p holds 0 <= F.p & F.p <= 1 & (p in S` implies F.p = 0) & (p
        in Cl (Vm.S) implies F.p = 1)
        proof
          take f9, A;
          thus thesis by A24,JORDAN5A:27;
        end;
        f9 in Funcs(cT,REAL) by FUNCT_2:8;
        hence thesis by A25;
      end;
      consider Fn be Function of Bnn,Funcs(cT,REAL) such that
A26:  for A be object st A in Bnn holds F[A,Fn.A] from FUNCT_2:sch 1(A21
      );
A27: n in NAT by ORDINAL1:def 12;
      Bn.n is locally_finite by A6,NAGATA_1:def 3,A27;
      then
A28:  for p being Element of cT ex A be Element of bcT st S[p,A] by
PCOMPS_1:def 1;
      consider Sx be Function of cT,bcT such that
A29:  for p be Element of cT holds S[p,Sx.p] from FUNCT_2:sch 3(A28);
      defpred ss[object,object] means
     for x,y be Element of T st $1=[x,y] holds $2=
      [:Sx.x,Sx.y:];
A30:  for pq9 be object st pq9 in cTT
ex SS be object st SS in bool cTT & ss[ pq9,SS]
      proof
        let pq9 be object;
        assume pq9 in cTT;
        then pq9 in [:cT,cT:] by BORSUK_1:def 2;
        then consider p,q be object such that
A31:    p in cT & q in cT and
A32:    pq9=[p,q] by ZFMISC_1:def 2;
        reconsider p,q as Point of T by A31;
        now
          let p1,q1 be Point of T;
          assume
A33:      pq9=[p1,q1];
          then p1=p by A32,XTUPLE_0:1;
          hence [:Sx.p,Sx.q:]=[:Sx.p1,Sx.q1:] by A32,A33,XTUPLE_0:1;
        end;
        hence thesis;
      end;
      consider SS be Function of cTT,bool cTT such that
A34:  for pq be object st pq in cTT holds ss[pq,SS.pq] from FUNCT_2:sch
      1(A30);
A35:  for pq9 holds pq9 in SS.pq9 & SS.pq9 is open
      proof
        let pq9 be Point of [:T,T:];
        pq9 in cTT;
        then pq9 in [:cT,cT:] by BORSUK_1:def 2;
        then consider p,q be object such that
A36:    p in cT & q in cT and
A37:    pq9=[p,q] by ZFMISC_1:def 2;
        reconsider p,q as Point of T by A36;
A38:    p in Sx.p & q in Sx.q by A29;
A39:    Sx.p is open & Sx.q is open by A29;
        SS.pq9=[:Sx.p,Sx.q:] by A34,A37;
        hence thesis by A37,A38,A39,BORSUK_1:6,ZFMISC_1:def 2;
      end;
A40:  for f,g be Element of Fun ex fg be Element of Fun st add9[f,g,fg]
      proof
        let f,g be Element of Fun;
        set f9=f,g9=g;
        f9+g9 in Fun by FUNCT_2:8;
        hence thesis;
      end;
      consider ADD be BinOp of Funcs(the carrier of [:T,T:],REAL) such that
A41:  for f,g be Element of Fun holds add9[f,g,ADD.(f,g)] from
      BINOP_1:sch 3(A40);
A42:  for f be Element of Funcs(cT,REAL) ex fxy be Element of Fun st
      funcdist[f,fxy]
      proof
        let f be Element of Funcs(cT,REAL);
        defpred fd[Element of T,Element of T,object] means $3=|.f.$1-f.$2.|;
A43:    for x,y be Element of cT ex r be Element of REAL st fd[x,y,r]
         proof let x,y be Element of cT;
           consider r be Real such that
A44:          fd[x,y,r];
           reconsider r as Element of REAL by XREAL_0:def 1;
           fd[x,y,r] by A44;
          hence thesis;
         end;
        consider Fd be Function of [:cT,cT:],REAL such that
A45:    for x,y be Element of cT holds fd[x,y,Fd.(x,y)] from BINOP_1:
        sch 3(A43);
        [:cT,cT:]= cTT by BORSUK_1:def 2;
        then Fd in Fun by FUNCT_2:8;
        hence thesis by A45;
      end;
      consider Fdist be Function of Funcs(cT,REAL),Fun such that
A46:  for fd be Element of Funcs(cT,REAL) holds funcdist[fd,Fdist.fd]
      from FUNCT_2:sch 3(A42);
      deffunc Fx(Element of T)= {Fn.A where A is Subset of T:A in Bn.n & A in
      s(Sx.$1)};
      deffunc RNG(set)={Fdist.fd where fd is RealMap of T:fd in $1};
      defpred gxy[set,Function] means $2 is one-to-one & ex p,q st [p,q]=$1 &
      rng $2=RNG(Fx(p)\/Fx(q));
A47:  for p holds Fx(p) is finite
      proof
        deffunc Fxx(Subset of T)=Fn.$1;
        let p;
        set SFxx={Fxx(A) where A is Subset of T:A in s(Sx.p)};
A48:    Fx(p) c= SFxx
        proof
          let FA be object;
          assume FA in Fx(p);
          then ex A st FA=Fn.A & A in Bn.n & A in s(Sx.p);
          hence thesis;
        end;
A49:    s(Sx.p) is finite by A29;
        SFxx is finite from FRAENKEL:sch 21(A49);
        hence thesis by A48;
      end;
A50:  for p,q holds RNG(Fx(p)\/Fx(q)) is finite & RNG(Fx(p)\/Fx(q))c=Fun
      proof
        deffunc FD(RealMap of T)=Fdist.$1;
        let p,q;
A51:    RNG(Fx(p)\/Fx(q)) c=Fun
        proof
          let FDm be object;
          assume FDm in RNG(Fx(p)\/Fx(q));
          then consider fd be RealMap of T such that
A52:      FDm=Fdist.fd and
          fd in Fx(p)\/Fx(q);
          fd in Funcs(cT,REAL) by FUNCT_2:8;
          hence thesis by A52,FUNCT_2:5;
        end;
        set RNG9={FD(fd) where fd is Element of Funcs(cT,REAL): fd in Fx(p)\/
        Fx(q)};
A53:    RNG(Fx(p)\/Fx(q))c=RNG9
        proof
          let Fdistfd be object;
          assume Fdistfd in RNG(Fx(p)\/Fx(q));
          then consider fd be RealMap of T such that
A54:      Fdistfd =Fdist.fd & fd in Fx(p)\/Fx(q);
          fd in Funcs(cT,REAL) by FUNCT_2:8;
          hence thesis by A54;
        end;
        Fx(p) is finite & Fx(q) is finite by A47;
        then
A55:    Fx(p)\/Fx(q) is finite;
        RNG9 is finite from FRAENKEL:sch 21(A55);
        hence thesis by A51,A53;
      end;
A56:  for pq be Element of cTT ex G be Element of Fun* st gxy[pq,G]
      proof
        let pq be Element of cTT;
        pq in the carrier of [:T,T:];
        then pq in [:cT,cT:] by BORSUK_1:def 2;
        then consider p,q be object such that
A57:    p in cT & q in cT and
A58:    pq=[p,q] by ZFMISC_1:def 2;
        reconsider p,q as Point of T by A57;
        consider SF be FinSequence such that
A59:    rng SF =RNG(Fx(p)\/Fx(q)) and
A60:    SF is one-to-one by A50,FINSEQ_4:58;
        rng SF c=Fun by A50,A59;
        then reconsider SF as FinSequence of Fun by FINSEQ_1:def 4;
        SF in Fun* by FINSEQ_1:def 11;
        hence thesis by A58,A59,A60;
      end;
      consider SFdist be Function of cTT,Fun* such that
A61:  for pq be Element of cTT holds gxy[pq,SFdist.pq] from FUNCT_2:
      sch 3(A56);
      defpred SFdist[object,object] means
for FS be FinSequence of Fun st FS=SFdist.
      $1 holds $2= ADD "**" FS;
A62:  for pq be object st pq in cTT
ex S be object st S in Fun & SFdist[pq,S]
      proof
        let pq be object;
        assume pq in cTT;
        then SFdist.pq in Fun* by FUNCT_2:5;
        then reconsider SF=SFdist.pq as FinSequence of Fun by FINSEQ_1:def 11;
        for FS be FinSequence of Funcs(cTT,REAL) st FS=SFdist.pq holds
        ADD "**" FS = ADD "**" SF;
        hence thesis;
      end;
      consider SumFdist be Function of cTT,Funcs(cTT,REAL) such that
A63:  for xy be object st xy in cTT holds SFdist[xy,SumFdist.xy] from
      FUNCT_2:sch 1(A62);
      reconsider SumFdist9=SumFdist as Function of cTT,Funcs(cTT,the carrier
      of R^1) by TOPMETR:17;
      reconsider SumFTS9=SumFdist9 Toler as RealMap of [:T,T:] by TOPMETR:17;
      cTT=[:cT,cT:] by BORSUK_1:def 2;
      then reconsider
      SumFTS = SumFdist9 Toler as Function of [:cT,cT:],REAL by TOPMETR:17;
A64:  for f1,f2 be RealMap of [:T,T:] holds ADD.(f1,f2)=f1+f2
      proof
        let f1,f2 be RealMap of [:T,T:];
        reconsider f19=f1,f29=f2 as Element of Fun by FUNCT_2:8;
        ADD.(f19,f29)=f19+f29 by A41;
        hence thesis;
      end;
A65:  for p,q st ex A,B st A is open & B is open & A in Bn.m & B in Bn.n
      & p in A & Cl A c= B & not q in B holds SumFTS9.[p,q] >= 1
      proof
        let p,q;
        assume ex A,B st A is open & B is open & A in Bn.m & B in Bn.n & p
        in A & Cl A c= B & not q in B;
        then consider A,B such that
        A is open and
        B is open and
A66:    A in Bn.m and
A67:    B in Bn.n and
A68:    p in A and
A69:    Cl A c= B and
A70:    not q in B;
A71:    F[B,Fn.B] by A26,A67;
        A in {Q where Q is Subset of T:
          ex D1 being set st D1 = B & Q in Bn.m & Cl Q c=D1} by A66,A69;
        then A c=V(B) by ZFMISC_1:74;
        then
A72:    A c= Vm.B by A11;
        Vm.B c= Cl(Vm.B) by PRE_TOPC:18;
        then
A73:    p in Cl(Vm.B) by A68,A72;
        Cl(Vm.B)c=B & p in Sx.p by A14,A29;
        then Sx.p meets B by A73,XBOOLE_0:3;
        then
A74:    B in s(Sx.p) by A67;
        reconsider pq=[p,q] as Point of [:T,T:] by BORSUK_1:def 2;
        reconsider SF=SFdist.pq as FinSequence of Fun by FINSEQ_1:def 11;
        consider x,y be Point of T such that
A75:    [x,y]=pq and
A76:    rng SF=RNG(Fx(x)\/Fx(y)) by A61;
        reconsider ASF=ADD"**"SF as RealMap of [:T,T:] by FUNCT_2:66;
        assume
A77:    SumFTS9.[p,q] < 1;
        reconsider FnB=Fn.B as RealMap of T by A67,FUNCT_2:5,66;
A78:    FnB in Funcs(cT,REAL) by A67,FUNCT_2:5;
        q in B` by A70,XBOOLE_0:def 5;
        then
A79:    FnB.q=0 by A71;
        x=p by A75,XTUPLE_0:1;
        then FnB in Fx(x) by A67,A74;
        then FnB in Fx(x)\/Fx(y) by XBOOLE_0:def 3;
        then
A80:    Fdist.FnB in rng SF by A76;
        then reconsider FdistFnB=Fdist.FnB as RealMap of [:T,T:] by FUNCT_2:66;
        SF<>{} by A80,RELAT_1:38;
        then len SF >=1 by NAT_1:14;
        then consider F be sequence of Fun such that
A81:    F.1 = SF.1 and
A82:    for n being Nat st 0 <> n & n < len SF holds F.(n+1)=ADD.(F.n,SF.(n+1))
           and
A83:    ADD"**"SF=F.(len SF) by FINSOP_1:def 1;
A84:    SumFTS.pq=SumFdist.pq.pq & SumFdist.pq=ASF by A63,NAGATA_1:def 8;
        defpred P[Nat] means
         for k st k<>0 & k<=$1 & $1<=len SF for
        fk,Fn be RealMap of [:T,T:] st fk=SF.k & Fn=F.$1 holds fk.pq<=Fn.pq;
A85:    for k st k <>0 & k <=len SF for f be RealMap of [:T,T:] st f=SF.
        k holds f.pq>=0
        proof
          let k such that
A86:      k <>0 and
A87:      k <=len SF;
          k >=1 by A86,NAT_1:14;
          then k in dom SF by A87,FINSEQ_3:25;
          then SF.k in RNG(Fx(x)\/Fx(y)) by A76,FUNCT_1:def 3;
          then consider fd be RealMap of T such that
A88:      Fdist.fd =SF.k and
          fd in Fx(x)\/Fx(y);
          let f be RealMap of [:T,T:] such that
A89:      f=SF.k;
          fd in Funcs(cT,REAL) by FUNCT_2:8;
          then f.(p,q)=|.fd.p-fd.q.| by A46,A89,A88;
          hence thesis by COMPLEX1:46;
        end;
A90:    for i holds P[i] implies P[i+1]
        proof
          let i;
          assume
A91:      P[i];
          let k such that
A92:      k<>0 and
A93:      k<=i+1 and
A94:      i+1<=len SF;
          now
            1<=i+1 by NAT_1:14;
            then i+1 in dom SF by A94,FINSEQ_3:25;
            then SF.(i+1) in rng SF by FUNCT_1:def 3;
            then reconsider SFi1=SF.(i+1) as RealMap of [:T,T:] by FUNCT_2:66;
            reconsider Fi=F.i as RealMap of [:T,T:] by FUNCT_2:66;
            let fk,Fn be RealMap of [:T,T:] such that
A95:        fk=SF.k and
A96:        Fn=F.(i+1);
            per cases by A93,XXREAL_0:1;
            suppose
A97:          k<i+1;
A98:          i<len SF by A94,NAT_1:13;
              i<>0 by A92,A97,NAT_1:13;
              then F.(i+1)=ADD.(F.i,SF.(i+1)) by A82,A98;
              then
A99:          Fn =Fi+SFi1 by A64,A96;
              SFi1.pq>=0 by A85,A94;
              then Fi.pq +0 <= Fi.pq + SFi1.pq by XREAL_1:7;
              then
A100:          Fn.pq>=Fi.pq by A99,NAGATA_1:def 7;
A101:          i<=len SF by A94,NAT_1:13;
              k<=i by A97,NAT_1:13;
              then fk.pq<=Fi.pq by A91,A92,A95,A101;
              hence fk.pq<=Fn.pq by A100,XXREAL_0:2;
            end;
            suppose
A102:          k=i+1;
              per cases;
              suppose
                i=0;
                hence fk.pq<=Fn.pq by A81,A95,A96,A102;
              end;
              suppose
A103:            i<>0;
                i+0<i+1 by XREAL_1:8;
                then
A104:           i<len SF by A94,XXREAL_0:2;
                1<=i by A103,NAT_1:14;
                then i in dom SF by A104,FINSEQ_3:25;
                then SF.i in rng SF by FUNCT_1:def 3;
                then reconsider SFi=SF.i as RealMap of [:T,T:] by FUNCT_2:66;
                0<=SFi.pq by A85,A103,A104;
                then 0<=Fi.pq by A91,A103,A104;
                then
A105:           Fi.pq+fk.pq>=0+fk.pq by XREAL_1:7;
                F.(i+1)=ADD.(F.i,SF.(i+1)) by A82,A103,A104;
                then Fn=Fi+fk by A64,A95,A96,A102;
                hence fk.pq<=Fn.pq by A105,NAGATA_1:def 7;
              end;
            end;
          end;
          hence thesis;
        end;
A106:   P[ 0 ];
A107:   for i holds P[i] from NAT_1:sch 2(A106,A90);
        consider k be object such that
A108:   k in dom SF and
A109:   SF.k=Fdist.FnB by A80,FUNCT_1:def 3;
A110:   k in Seg(len SF) by A108,FINSEQ_1:def 3;
        FnB.p =1 by A73,A71;
        then
A111:   FdistFnB.(p,q)=|.1-0 .| by A46,A78,A79;
        reconsider k as Element of NAT by A108;
A112:   |.1.|=1 by ABSVALUE:def 1;
        k<>0 & k <=len SF by A110,FINSEQ_1:1;
        hence thesis by A77,A84,A83,A107,A111,A112,A109;
      end;
A113: now
        let p,q;
        assume ex A,B st A is open & B is open & A in Bn.m & B in Bn.n & p
        in A & Cl A c= B & not q in B;
        then SumFTS9.[p,q]>=1 by A65;
        then
A114:   1 = min(1,SumFTS9.[p,q]) by XXREAL_0:def 9;
        [:cT,cT:]=cTT by BORSUK_1:def 2;
        hence 1=min(jj,SumFTS9).[p,q] by A114,NAGATA_1:def 9;
      end;
A115: for pq be Element of cTT, map9 be Element of Fun st map9
      is_a_unity_wrt ADD holds map9.pq=0
      proof
        let pq be Element of cTT, map9 be Element of Fun;
        assume map9 is_a_unity_wrt ADD;
        then ADD.(map9,map9)=map9 by BINOP_1:3;
        then (map9+map9).pq=map9.pq by A41;
        then map9.pq+map9.pq=map9.pq by NAGATA_1:def 7;
        hence thesis;
      end;
A116: for pq1,pq2 be Point of [:T,T:] holds (pq1 in SS.pq2 implies
SumFdist.pq1.pq1 = SumFdist.pq2.pq1) & for SumFdist1,SumFdist2 be RealMap of [:
T,T:] st SumFdist1=SumFdist.pq1 & SumFdist2 = SumFdist.pq2 holds SumFdist1.pq1
      >= SumFdist2.pq1
      proof
        deffunc No0(Element of T,Element of T)= {Fn.A where A is Subset of T:A
in Bn.n & for FnA be RealMap of T st Fn.A=FnA holds (FnA.$1 > 0 or FnA.$2 > 0)}
        ;
        let pq1,pq2 be Point of [:T,T:];
        reconsider S1=SFdist.pq1,S2=SFdist.pq2 as FinSequence of Fun by
FINSEQ_1:def 11;
        consider p1,q1 be Point of T such that
A117:   [p1,q1]=pq1 and
A118:   rng S1=RNG(Fx(p1)\/Fx(q1)) by A61;
A119:   for p,q,p1,q1 be Point of T st [p,q] in SS.[p1,q1] holds No0(p,q
        ) c= Fx(p1)\/Fx(q1)
        proof
          let p,q,p1,q1 be Point of T such that
A120:     [p,q] in SS.[p1,q1];
          reconsider pq1=[p1,q1] as Element of cTT by BORSUK_1:def 2;
          [:Sx.p1,Sx.q1:]=SS.pq1 by A34;
          then
A121:     p in Sx.p1 & q in Sx.q1 by A120,ZFMISC_1:87;
          let no0 be object;
          assume no0 in No0(p,q);
          then consider A be Subset of T such that
A122:     no0=Fn.A and
A123:     A in Bn.n and
A124:     for FnA be RealMap of T st Fn.A=FnA holds (FnA.p > 0 or FnA.q > 0);
          reconsider FnA=Fn.A as RealMap of T by A123,FUNCT_2:5,66;
A125:     FnA.p > 0 or FnA.q > 0 by A124;
          F[A,Fn.A] by A26,A123;
          then not p in cT\A or not q in cT\A by A125;
          then p in A or q in A by XBOOLE_0:def 5;
          then A meets Sx.p1 or A meets Sx.q1 by A121,XBOOLE_0:3;
          then A in s(Sx.p1) or A in s(Sx.q1) by A123;
          then FnA in Fx(p1) or FnA in Fx(q1) by A123;
          hence thesis by A122,XBOOLE_0:def 3;
        end;
A126:   RNG(No0(p1,q1))c=RNG(Fx(p1)\/Fx(q1))
        proof
          p1 in Sx.p1 & q1 in Sx.q1 by A29;
          then [p1,q1] in [:Sx.p1,Sx.q1:] by ZFMISC_1:87;
          then [p1,q1] in SS.[p1,q1] by A34;
          then
A127:     No0(p1,q1)c= Fx(p1)\/Fx(q1) by A119;
          let FD be object;
          assume FD in RNG(No0(p1,q1));
          then ex fd be RealMap of T st FD=Fdist.fd & fd in (No0(p1,q1) );
          hence thesis by A127;
        end;
A128:   for f be FinSequence of Funcs(cTT,REAL),p,q,p1,q1 be Point of T
        st rng f= RNG(Fx(p1)\/Fx(q1))\RNG(No0(p,q)) holds (ADD "**" f).[p,q]=0
        proof
          let f be FinSequence of Funcs(cTT,REAL),p,q,p1,q1 be Point of T such
          that
A129:     rng f= RNG(Fx(p1)\/Fx(q1))\RNG(No0(p,q));
          reconsider pq=[p,q] as Element of cTT by BORSUK_1:def 2;
          now
            per cases;
            suppose
A130:         len f = 0;
A131:         ADD is having_a_unity by A64,NAGATA_1:23;
              then
A132:         ex f be Element of Fun st f is_a_unity_wrt ADD by SETWISEO:def 2;
              ADD "**" f=the_unity_wrt ADD by A130,A131,FINSOP_1:def 1;
              then ADD"**"f is_a_unity_wrt ADD by A132,BINOP_1:def 8;
              hence (ADD"**"f).pq=0 by A115;
            end;
            suppose
A133:         len f<>0;
              then len f >=1 by NAT_1:14;
              then consider F be sequence of Fun such that
A134:         F.1 = f.1 and
A135:         for n being Nat st 0 <> n & n < len f
               holds F.(n + 1) = ADD.(F.n,f.(n + 1)) and
A136:         ADD"**"f = F.(len f) by FINSOP_1:def 1;
              defpred R[Nat] means
                 $1<>0 & $1<=len f implies F.$1.pq=0;
A137:         for k holds R[k] implies R[k+1]
              proof
                let k;
                assume
A138:           R[k];
                assume that
                k+1<>0 and
A139:           k+1 <=len f;
A140:           k< len f by A139,NAT_1:13;
                k+1>=1 by NAT_1:14;
                then k+1 in dom f by A139,FINSEQ_3:25;
                then
A141:           f.(k+1) in RNG(Fx(p1)\/Fx(q1))\RNG(No0(p,q)) by A129,
FUNCT_1:def 3;
                then f.(k+1) in RNG(Fx(p1)\/Fx(q1));
                then consider fd be RealMap of T such that
A142:           Fdist.fd =f.(k+1) and
A143:           fd in Fx(p1)\/Fx(q1);
                fd in Funcs(cT,REAL) by FUNCT_2:8;
                then reconsider
                Fdistfd=Fdist.fd,Fk1=F.(k+1), Fk=F.k,fk1=f.(k+1) as
                RealMap of [:T,T:] by A142,FUNCT_2:5,66;
                fd in Funcs(cT,REAL) by FUNCT_2:8;
                then
A144:           Fdistfd.(p,q)=|.fd.p- fd.q.| by A46;
A145:           not f.(k+1) in RNG(No0(p,q)) by A141,XBOOLE_0:def 5;
A146:           fd.p=0 & fd.q=0
                proof
                  assume
A147:             fd.p<>0 or fd.q<>0;
                  per cases by A143,XBOOLE_0:def 3;
                  suppose
                    fd in Fx(p1);
                    then consider A be Subset of T such that
A148:               fd=Fn.A and
A149:               A in Bn.n and
                    A in s(Sx.p1);
A150:               F[A,Fn.A] by A26,A149;
                    not fd in No0(p,q) by A145,A142;
                    then ex FnA be RealMap of T st Fn.A=FnA &( not FnA.p > 0)&
                    not FnA.q > 0 by A148,A149;
                    hence contradiction by A147,A148,A150;
                  end;
                  suppose
                    fd in Fx(q1);
                    then consider A be Subset of T such that
A151:               fd=Fn.A and
A152:               A in Bn.n and
                    A in s(Sx.q1);
A153:               F[A,Fn.A] by A26,A152;
                    not fd in No0(p,q) by A145,A142;
                    then ex FnA be RealMap of T st Fn.A=FnA &( not FnA.p > 0)&
                    not FnA.q > 0 by A151,A152;
                    hence contradiction by A147,A151,A153;
                  end;
                end;
                per cases;
                suppose
                  k=0;
                  hence thesis by A134,A142,A146,A144,ABSVALUE:2;
                end;
                suppose
A154:             k>0;
                  then Fk1=ADD.(Fk,fk1) by A135,A140;
                  then Fk1=Fk+fk1 by A64;
                  then Fk1.pq=0+fk1.pq by A138,A139,A154,NAGATA_1:def 7
,NAT_1:13;
                  hence thesis by A142,A146,A144,ABSVALUE:2;
                end;
              end;
A155:         R[ 0 ];
              for n holds R[n] from NAT_1:sch 2(A155,A137);
              hence (ADD"**"f).pq=0 by A133,A136;
            end;
          end;
          hence thesis;
        end;
A156:   RNG(Fx(p1)\/Fx(q1)) is finite by A50;
        then consider No be FinSequence such that
A157:   rng No =RNG(No0(p1,q1)) and
A158:   No is one-to-one by A126,FINSEQ_4:58;
        RNG(Fx(p1)\/Fx(q1))c=Fun by A50;
        then
A159:   RNG(No0(p1,q1)) c= Fun by A126;
        then reconsider No as FinSequence of Fun by A157,FINSEQ_1:def 4;
        consider p2,q2 be Point of T such that
A160:   [p2,q2]=pq2 and
A161:   rng S2=RNG(Fx(p2)\/Fx(q2)) by A61;
        reconsider S1=SFdist.pq1,S2=SFdist.pq2 as FinSequence of Fun by
FINSEQ_1:def 11;
        set RNiS2=RNG(No0(p1,q1))/\RNG(Fx(p2)\/Fx(q2));
A162:   S2 is one-to-one by A61;
A163:   ADD is having_a_unity & ADD is commutative associative by A64,
NAGATA_1:23;
        S1 is one-to-one by A61;
        then consider S1No be FinSequence of Fun such that
        S1No is one-to-one and
A164:   rng S1No=rng S1 \rng No and
A165:   ADD"**"S1 =ADD.(ADD"**"No,ADD"**"S1No) by A118,A126,A157,A158,A163,Th18
;
        consider NoiS2 be FinSequence such that
A166:   rng NoiS2 =RNiS2 and
A167:   NoiS2 is one-to-one by A126,A156,FINSEQ_4:58;
        RNiS2 c=RNG(No0(p1,q1)) by XBOOLE_1:17;
        then RNiS2 c= Fun by A159;
        then reconsider NoiS2 as FinSequence of Fun by A166,FINSEQ_1:def 4;
        rng NoiS2 c= rng No by A157,A166,XBOOLE_1:17;
        then consider NoNoiS2 be FinSequence of Fun such that
        NoNoiS2 is one-to-one and
A168:   rng NoNoiS2=rng No \rng NoiS2 and
A169:   ADD"**"No=ADD.(ADD"**"NoiS2,ADD"**"NoNoiS2) by A158,A163,A167,Th18;
        rng NoiS2 c= rng S2 by A161,A166,XBOOLE_1:17;
        then consider S2No be FinSequence of Fun such that
        S2No is one-to-one and
A170:   rng S2No=rng S2 \rng NoiS2 and
A171:   ADD"**"S2=ADD.(ADD"**"NoiS2,ADD"**"S2No) by A163,A167,A162,Th18;
        reconsider ANoNoiS2=ADD"**"NoNoiS2,ANo=ADD"**"No,AS1No=ADD"**"S1No,
AS2No=ADD"**"S2No,ANoiS2=ADD"**"NoiS2,AS1=ADD"**"S1,AS2=ADD"**"S2 as RealMap of
        [:T,T:] by FUNCT_2:66;
        rng S2No= RNG(Fx(p2)\/Fx(q2))\RNG(No0(p1,q1)) by A161,A166,A170,
XBOOLE_1:47;
        then
A172:   AS2No.pq1=0 by A128,A117;
        ANo=ANoiS2+ANoNoiS2 by A64,A169;
        then
A173:   ANo.pq1=ANoiS2.pq1+ANoNoiS2. pq1 by NAGATA_1:def 7;
        AS1=ANo+AS1No by A64,A165;
        then
A174:   AS1.pq1=ANo.pq1+AS1No.pq1 by NAGATA_1:def 7;
        AS2=ANoiS2+AS2No by A64,A171;
        then
A175:   AS2.pq1=ANoiS2.pq1+AS2No.pq1 by NAGATA_1:def 7;
A176:   AS1No.pq1=0 by A128,A117,A118,A157,A164;
        thus pq1 in SS.pq2 implies SumFdist.pq1.pq1 = SumFdist.pq2.pq1
        proof
A177:     ADD is having_a_unity by A64,NAGATA_1:23;
          then
A178:     ex f be Element of Fun st f is_a_unity_wrt ADD by SETWISEO:def 2;
          assume
A179:     pq1 in SS.pq2;
          now
            let FD be object;
            assume FD in RNG (No0(p1,q1));
            then
A180:       ex fd be RealMap of T st FD=Fdist.fd & fd in (No0(p1,q1) );
            No0(p1,q1) c= Fx(p2)\/Fx(q2) by A119,A117,A160,A179;
            hence FD in RNG(Fx(p2)\/Fx(q2)) by A180;
          end;
          then RNG (No0(p1,q1)) c= RNG(Fx(p2)\/Fx(q2));
          then RNiS2=RNG(No0(p1,q1)) by XBOOLE_1:28;
          then rng NoNoiS2={} by A157,A166,A168,XBOOLE_1:37;
          then NoNoiS2={} by RELAT_1:41;
          then len NoNoiS2 =0;
          then ADD"**"NoNoiS2=the_unity_wrt ADD by A177,FINSOP_1:def 1;
          then ADD"**"NoNoiS2 is_a_unity_wrt ADD by A178,BINOP_1:def 8;
          then
A181:     AS1.pq1=AS2.pq1 by A115,A174,A173,A175,A176,A172;
          SumFdist.pq1=AS1 by A63;
          hence thesis by A63,A181;
        end;
A182:   ANoNoiS2.(p1,q1)>=0
        proof
          set N=NoNoiS2;
          per cases;
          suppose
A183:       len N=0;
A184:       ADD is having_a_unity by A64,NAGATA_1:23;
            then
A185:       ex f be Element of Fun st f is_a_unity_wrt ADD by SETWISEO:def 2;
            ADD "**" N=the_unity_wrt ADD by A183,A184,FINSOP_1:def 1;
            then ADD"**"N is_a_unity_wrt ADD by A185,BINOP_1:def 8;
            hence thesis by A115,A117;
          end;
          suppose
A186:       len N<>0;
            then len N >=1 by NAT_1:14;
            then consider F be sequence of Fun such that
A187:       F.1 = N.1 and
A188:       for n being Nat st 0 <> n & n < len N
             holds F.(n + 1) = ADD.(F.n,N.(n + 1)) and
A189:       ADD "**" N=F.(len N) by FINSOP_1:def 1;
            defpred R[Nat] means
                $1<>0 & $1<=len N implies for Fn
            be RealMap of [:T,T:] st Fn=F.$1 holds Fn.(p1,q1)>=0;
A190:       for k holds R[k] implies R[k+1]
            proof
              let k;
              assume
A191:         R[k];
              assume that
              k+1<>0 and
A192:         k+1 <=len N;
A193:         k< len N by A192,NAT_1:13;
              k+1>=1 by NAT_1:14;
              then k+1 in dom N by A192,FINSEQ_3:25;
              then N.(k+1) in rng No \rng NoiS2 by A168,FUNCT_1:def 3;
              then N.(k+1) in RNG(No0(p1,q1)) by A157,XBOOLE_0:def 5;
              then consider fd be RealMap of T such that
A194:         Fdist.fd =N.(k+1) and
              fd in No0(p1,q1);
              fd in Funcs(cT,REAL) by FUNCT_2:8;
              then reconsider
              Fdistfd=Fdist.fd,Fk1=F.(k+1), Fk=F.k,Nk1=N.(k+1) as
              RealMap of [:T,T:] by A194,FUNCT_2:5,66;
A195:         |.fd.p1-fd.q1.|>=0 by COMPLEX1:46;
A196:         fd in Funcs(cT,REAL) by FUNCT_2:8;
              then
A197:         Fdistfd.(p1,q1)=|.fd. p1-fd.q1.| by A46;
A198:         now
                per cases;
                suppose
                  k=0;
                  hence Fk1.(p1,q1)>=0 by A46,A187,A194,A196,A195;
                end;
                suppose
A199:             k>0;
                  Fk1=ADD.(Fk,Nk1) by A188,A193,A199;
                  then
A200:             Fk1=Fk+Nk1 by A64;
                  Fk.(p1,q1) >= 0 by A191,A192,A199,NAT_1:13;
                  then 0+0<=Fk.(p1,q1)+Nk1.(p1,q1) by A194,A195,A197;
                  hence Fk1.(p1,q1) >=0 by A117,A200,NAGATA_1:def 7;
                end;
              end;
              let Fn be RealMap of [:T,T:];
              assume Fn=F.(k+1);
              hence thesis by A198;
            end;
A201:       R[ 0 ];
            for n holds R[n] from NAT_1:sch 2(A201,A190);
            hence thesis by A186,A189;
          end;
        end;
        now
A202:     AS2.(p1,q1)<=AS1.(p1,q1) by A117,A182,A174,A173,A175,A176,A172,
XREAL_1:7;
          let SumFdist1,SumFdist2 be RealMap of [:T,T:];
          assume that
A203:     SumFdist1=SumFdist.pq1 and
A204:     SumFdist2 = SumFdist.pq2;
          SumFdist2=AS2 by A63,A204;
          hence SumFdist1.pq1 >= SumFdist2.pq1 by A63,A117,A203,A202;
        end;
        hence thesis;
      end;
      now
        let p,q,r be Point of T;
        thus SumFTS.(p,p)=0
        proof
          reconsider pp=[p,p] as Point of [:T,T:] by BORSUK_1:def 2;
          reconsider SF=SFdist.pp as FinSequence of Fun by FINSEQ_1:def 11;
A205:     now
            per cases;
            suppose
A206:         len SF=0;
A207:         ADD is having_a_unity by A64,NAGATA_1:23;
              then
A208:         ex f be Element of Fun st f is_a_unity_wrt ADD by SETWISEO:def 2;
              ADD "**" SF=the_unity_wrt ADD by A206,A207,FINSOP_1:def 1;
              then ADD"**"SF is_a_unity_wrt ADD by A208,BINOP_1:def 8;
              hence (ADD"**"SF).pp=0 by A115;
            end;
            suppose
A209:         len SF <>0;
              then len SF >=1 by NAT_1:14;
              then consider F be sequence of Fun such that
A210:         F.1 = SF.1 and
A211:         for n being Nat
               st 0 <> n & n < len SF holds F.(n + 1) = ADD.(F.
              n,SF.(n + 1 )) and
A212:         ADD"**"SF = F.(len SF) by FINSOP_1:def 1;
              defpred R[Nat] means
              $1<>0 & $1<=len SF implies F.$1.pp=0;
A213:         for k holds R[k] implies R[k+1]
              proof
                let k;
                assume
A214:           R[k];
                consider x,y be Point of T such that
                [x,y]=pp and
A215:           rng SF=RNG(Fx(x)\/Fx(y)) by A61;
                assume that
                k+1<>0 and
A216:           k+1 <=len SF;
A217:           k< len SF by A216,NAT_1:13;
                k+1>=1 by NAT_1:14;
                then k+1 in dom SF by A216,FINSEQ_3:25;
                then SF.(k+1) in RNG(Fx(x)\/Fx(y)) by A215,FUNCT_1:def 3;
                then consider fd be RealMap of T such that
A218:           Fdist.fd =SF.(k+1) and
                fd in Fx(x)\/Fx(y);
                fd in Funcs(cT,REAL) by FUNCT_2:8;
                then reconsider
                Fdistfd=Fdist.fd,Fk1=F.(k+1),Fk=F.k, SFk1=SF.(k+1)
                as RealMap of [:T,T:] by A218,FUNCT_2:5,66;
                fd in Funcs(cT,REAL) by FUNCT_2:8;
                then
A219:           Fdistfd.(p,p)=|.fd.p-fd .p.| by A46;
                per cases;
                suppose
                  k=0;
                  hence thesis by A210,A218,A219,ABSVALUE:2;
                end;
                suppose
A220:             k>0;
                  Fk1=ADD.(Fk,SFk1) by A211,A217,A220;
                  then Fk1=Fk+SFk1 by A64;
                  then Fk1.pp=0+SFk1.pp by A214,A216,A220,NAGATA_1:def 7
,NAT_1:13;
                  hence thesis by A218,A219,ABSVALUE:2;
                end;
              end;
A221:         R[ 0 ];
              for n holds R[n] from NAT_1:sch 2(A221,A213);
              hence (ADD"**"SF).pp=0 by A209,A212;
            end;
          end;
          SumFdist.pp=ADD "**" SF by A63;
          hence thesis by A205,NAGATA_1:def 8;
        end;
        thus SumFTS.(p,r)<=SumFTS.(p,q)+SumFTS.(r,q)
        proof
          reconsider pr=[p,r],pq=[p,q],rq=[r,q] as Point of [:T,T:] by
BORSUK_1:def 2;
          reconsider SFpr=SFdist.pr as FinSequence of Fun by FINSEQ_1:def 11;
          reconsider ASFpr=ADD"**"SFpr as RealMap of [:T,T:] by FUNCT_2:66;
          reconsider SumFpr=SumFdist.pr,SumFpq=SumFdist.pq, SumFrq=SumFdist.rq
          as RealMap of [:T,T:] by FUNCT_2:66;
A222:     SumFTS.pr = SumFpr.pr & SumFTS.pq = SumFpq.pq by NAGATA_1:def 8;
          SumFpr.pq <=SumFpq.pq & SumFpr.rq <=SumFrq.rq by A116;
          then
A223:     SumFpr.pq + SumFpr.rq <= SumFpq.pq+SumFrq.rq by XREAL_1:7;
A224:     now
            per cases;
            suppose
A225:         len SFpr = 0;
A226:         ADD is having_a_unity by A64,NAGATA_1:23;
              then
A227:         ex f being Element of Fun st f is_a_unity_wrt ADD by
SETWISEO:def 2;
              ADD"**"SFpr=the_unity_wrt ADD by A225,A226,FINSOP_1:def 1;
              then
A228:         ADD"**"SFpr is_a_unity_wrt ADD by A227,BINOP_1:def 8;
              then ASFpr.pr=0 & ASFpr.pq=0 by A115;
              hence ASFpr.pr <= ASFpr.pq + ASFpr.rq by A115,A228;
            end;
            suppose
A229:         len SFpr <> 0;
              then len SFpr >=1 by NAT_1:14;
              then consider F be sequence of Fun such that
A230:         F.1 = SFpr.1 and
A231:         for n being Nat
                st 0 <> n & n < len SFpr holds F.(n+1)=ADD.(F.n,
              SFpr.(n+1)) and
A232:         ADD"**"SFpr=F.(len SFpr) by FINSOP_1:def 1;
              defpred T[Nat] means
                $1<>0 & $1<=len SFpr implies for
              F9 be RealMap of [:T,T:] st F9=F.$1 holds F9.pr<=F9.pq+F9.rq;
A233:         for k holds T[k] implies T[k+1]
              proof
                let k;
                assume
A234:           T[k];
                consider x,y be Point of T such that
                [x,y]=pr and
A235:           rng SFpr=RNG(Fx(x)\/Fx(y)) by A61;
                assume that
                k+1<>0 and
A236:           k+1 <=len SFpr;
A237:           k<len SFpr by A236,NAT_1:13;
                k+1>=1 by NAT_1:14;
                then k+1 in dom SFpr by A236,FINSEQ_3:25;
                then SFpr.(k+1) in RNG(Fx(x)\/Fx(y)) by A235,FUNCT_1:def 3;
                then consider fd be RealMap of T such that
A238:           Fdist.fd =SFpr.(k+1) and
                fd in Fx(x)\/Fx(y);
                fd in Funcs(cT,REAL) by FUNCT_2:8;
                then reconsider
                Fdistfd=Fdist.fd, Fk1=F.(k+1), Fk=F.k, SFk1=SFpr.(k
                +1) as RealMap of [:T,T:] by A238,FUNCT_2:5,66;
A239:           fd.p-fd.r=(fd.p-fd.q)+(fd.q-fd.r);
                then
A240:           |.fd.p-fd.r.|<=|.fd.p-fd.q.|+ |.fd.q- fd.r.| by COMPLEX1:56;
A241:           fd in Funcs(cT,REAL) by FUNCT_2:8;
                then
A242:           Fdistfd.(p,r)= |.fd.p-fd.r.| & Fdistfd.(p,q)=|.fd.p-
                fd.q.| by A46;
A243:           |.fd.q-fd.r.|=|.-(fd.q-fd.r).| & Fdistfd.(r,q)=|.fd.
                r-fd.q.| by A46,A241,COMPLEX1:52;
                let F9 be RealMap of [:T,T:] such that
A244:           F9=F.(k+1);
                per cases;
                suppose
                  k=0;
                  hence thesis by A230,A238,A239,A242,A243,A244,COMPLEX1:56;
                end;
                suppose
A245:             k>0;
                  then Fk.pr<=Fk.pq+Fk.rq by A234,A236,NAT_1:13;
                  then
A246:             Fk.pr+SFk1.pr<=(Fk.pq+Fk.rq)+(SFk1.pq+SFk1.rq) by A238,A240
,A242,A243,XREAL_1:7;
                  Fk1=ADD.(Fk,SFk1) by A231,A237,A245;
                  then
A247:             Fk1=Fk+SFk1 by A64;
                  then Fk1.pq=Fk.pq+SFk1.pq & Fk1.rq=Fk.rq+SFk1.rq by
NAGATA_1:def 7;
                  hence thesis by A244,A247,A246,NAGATA_1:def 7;
                end;
              end;
A248:         T[ 0 ];
              for n holds T[n] from NAT_1:sch 2(A248,A233);
              hence ASFpr.pr <= ASFpr.pq + ASFpr.rq by A229,A232;
            end;
          end;
          SumFpr=ADD "**" SFpr by A63;
          then SumFpr.pr<=SumFpq.pq+SumFrq.rq by A224,A223,XXREAL_0:2;
          hence thesis by A222,NAGATA_1:def 8;
        end;
      end;
      then
A249: SumFTS is_a_pseudometric_of cT by NAGATA_1:28;
A250: for p be Point of T,fd be Element of Funcs(cT, REAL) st fd in Fx(p)
      holds funcdist[fd,Fdist.fd] & Fdist.fd is continuous RealMap of [:T,T:]
      proof
        let p be Point of T,fd be Element of Funcs(cT,REAL);
        reconsider FD=Fdist.fd as RealMap of [:T,T:] by FUNCT_2:66;
        assume fd in Fx(p);
        then consider A be Subset of T such that
A251:   fd=Fn.A and
A252:   A in Bn.n and
        A in s(Sx.p);
A253:   funcdist[fd,Fdist.fd] by A46;
        F[A,Fn.A] by A26,A252;
        then FD is continuous by A251,A253,NAGATA_1:21;
        hence thesis by A46;
      end;
A254: for pq9 holds SumFdist.pq9 is continuous RealMap of [:T,T:]
      proof
        let pq9;
        reconsider SF=SFdist.pq9 as FinSequence of Fun by FINSEQ_1:def 11;
        consider p,q such that
        [p,q]=pq9 and
A255:   rng SF=RNG(Fx(p)\/Fx(q)) by A61;
        for n being Element of NAT
          st 0<>n & n<=len SF holds SF.n is continuous RealMap of [:T ,T:]
        proof
          let n be Element of NAT;
          assume that
A256:     0<>n and
A257:     n<=len SF;
          n>=1 by A256,NAT_1:14;
          then n in dom SF by A257,FINSEQ_3:25;
          then SF.n in RNG(Fx(p)\/Fx(q)) by A255,FUNCT_1:def 3;
          then consider fd be RealMap of T such that
A258:     SF.n=Fdist.fd and
A259:     fd in (Fx(p)\/Fx(q));
A260:     fd in Funcs(cT,REAL) by FUNCT_2:8;
          fd in Fx(p) or fd in Fx(q) by A259,XBOOLE_0:def 3;
          hence thesis by A250,A258,A260;
        end;
        then ADD"**"SF is continuous RealMap of [:T,T:] by A64,NAGATA_1:25;
        hence thesis by A63;
      end;
A261: for pq9 holds SumFdist9.pq9 is continuous Function of [:T,T:],R^1
      proof
        let pq9;
        reconsider SF=SumFdist.pq9 as Function of [:T,T:],R^1 by A254,
TOPMETR:17;
        SumFdist.pq9 is continuous RealMap of [:T,T:] by A254;
        then SF is continuous by JORDAN5A:27;
        hence thesis;
      end;
      take min(jj,SumFTS9);
A262: for p,q holds min(jj,SumFTS9).[p,q]<=1
      proof
        let p,q;
        cTT=[:cT,cT:] by BORSUK_1:def 2;
        then min(jj,SumFTS9).[p,q]=min(1,SumFTS9.[p,q]) by NAGATA_1:def 9;
        hence thesis by XXREAL_0:17;
      end;
      for p,q be Point of [:T,T:] st p in SS.q holds SumFdist9.p.p=
      SumFdist9.q.p by A116;
      then SumFdist9 Toler is continuous by A261,A35,NAGATA_1:26;
      then SumFTS9 is continuous by JORDAN5A:27;
      then min(jj,SumFTS9)=min(jj,SumFTS) & min(jj,SumFTS9) is continuous by
BORSUK_1:def 2,NAGATA_1:27;
      hence thesis by A249,A262,A113,NAGATA_1:30;
    end;
A263: for k be object st k in NAT
ex f be object st f in Funcs(cTT,REAL) & N[k,f ]
    proof
A264: NAT =rng PairFunc by Th2,FUNCT_2:def 3;
      let k be object;
      assume k in NAT;
      then consider nm be object such that
A265: nm in dom PairFunc and
A266: k=PairFunc.nm by A264,FUNCT_1:def 3;
      consider n,m be object such that
A267: n in NAT & m in NAT and
A268: nm=[n,m] by A265,ZFMISC_1:def 2;
      consider pmet9 such that
A269: NN[n,m,pmet9] by A7,A267;
      take pmet9;
      thus pmet9 in Funcs(cTT,REAL) by FUNCT_2:8;
      take pm = pmet9;
      thus pm = pmet9;
      let n1,m1 be Nat;
      assume PairFunc".k=[n1,m1];
      then [n1,m1]=[n,m] by A265,A266,A268,Lm2,Th2,FUNCT_1:32;
      then n1=n & m1=m by XTUPLE_0:1;
      hence thesis by A269;
    end;
    consider F be sequence of Funcs(cTT,REAL) such that
A270: for n be object st n in NAT holds N[n,F.n] from FUNCT_2:sch 1(A263);
A271: now
      let n be Nat;
      [:cT,cT:]=cTT by BORSUK_1:def 2;
      hence F.n is PartFunc of [:cT,cT:],REAL by FUNCT_2:66;
    end;
    dom F=NAT by FUNCT_2:def 1;
    then reconsider F9=F as Functional_Sequence of[:cT,cT:],REAL by A271,
SEQFUNC:1;
A272: for p,A9 st not p in A9 & A9 is closed ex k st for pmet st F9.k=pmet
    holds lower_bound(pmet,A9).p>0
    proof

      let p,A9 such that
A273: not p in A9 and
A274: A9 is closed;
      set O=A9`;
      p in O by A273,XBOOLE_0:def 5;
      then consider U be Subset of T such that
A275: p in U and
A276: Cl U c=O and
A277: U in Union Bn by A1,A5,A274,NAGATA_1:19;
      Union Bn c=the topology of T by A5,TOPS_2:64;
      then reconsider U as open Subset of T by A277,PRE_TOPC:def 2;
      consider n such that
A278: U in Bn.n by A277,PROB_1:12;
      consider W be Subset of T such that
A279: p in W & Cl W c=U and
A280: W in Union Bn by A1,A5,A275,NAGATA_1:19;
      Union Bn c=the topology of T by A5,TOPS_2:64;
      then reconsider W as open Subset of T by A280,PRE_TOPC:def 2;
      consider m such that
A281: W in Bn.m by A280,PROB_1:12;
      set k=PairFunc.[n,m];
A282:   k in NAT by ORDINAL1:def 12;
A283:   n in NAT & m in NAT by ORDINAL1:def 12;
      consider G be RealMap of [:T,T:] such that
A284: G = F.k and
A285: for n,m being Nat st PairFunc".k=[n,m]
        holds NN[n,m,G] by A270,A282;
A286:   [n,m] in [:NAT,NAT:] by A283,ZFMISC_1:87;
      reconsider kk=k as Element of NAT by ORDINAL1:def 12;
      dom PairFunc = [:NAT,NAT:] by FUNCT_2:def 1;
      then [n,m]=PairFunc".kk by Lm2,Th2,FUNCT_1:32,A286;
      then consider pmet such that
A287: G=pmet and
      G is continuous and
      pmet is_a_pseudometric_of cT and
A288: for p,q holds pmet.[p,q]<=1 & for p,q st ex A,B st A is open &
B is open & A in Bn.m & B in Bn.n & p in A & Cl A c= B & not q in B holds pmet.
      [p,q] = 1 by A285;
A289: for rn st rn in (dist(pmet,p)).:A9 holds rn>=1
      proof
        let rn;
        assume rn in (dist(pmet,p)).:A9;
        then consider a be object such that
A290:   a in dom dist(pmet,p) and
A291:   a in A9 and
A292:   rn=(dist(pmet,p)).a by FUNCT_1:def 6;
        reconsider a as Point of T by A290;
A293:   pmet.(p,a)=dist(pmet,p).a by Def2;
        U c= Cl U by PRE_TOPC:18;
        then U c= O by A276;
        then U misses A9 by SUBSET_1:23;
        then not a in U by A291,XBOOLE_0:3;
        hence thesis by A279,A278,A281,A288,A292,A293;
      end;
      take k;
      cT=dom dist(pmet,p) by FUNCT_2:def 1;
      then lower_bound ((dist(pmet,p)).:A9) >= 1 by A289,SEQ_4:43;
      hence thesis by A284,A287,Def3;
    end;
    for k ex pmet st F9.k=pmet & pmet is_a_pseudometric_of cT & (for pq
    holds pmet.pq<=1) & for pmet9 st pmet=pmet9 holds pmet9 is continuous
    proof
      let k;
A294:   k in NAT by ORDINAL1:def 12;
      then consider Fk be RealMap of [:T,T:] such that
A295: Fk = F.k and
A296: for n,m st PairFunc".k=[n,m] holds NN[n,m,Fk] by A270;
      NAT =rng PairFunc by Th2,FUNCT_2:def 3;
      then consider nm be object such that
A297: nm in dom PairFunc and
A298: k=PairFunc.nm by FUNCT_1:def 3,A294;
      consider n,m be object such that
A299: n in NAT & m in NAT and
A300: nm=[n,m] by A297,ZFMISC_1:def 2;
      [n,m]=PairFunc".k by A297,A298,A300,Lm2,Th2,FUNCT_1:32;
      then consider pmet such that
A301: Fk=pmet and
A302: Fk is continuous and
A303: pmet is_a_pseudometric_of cT and
A304: for p,q holds pmet.[p,q]<=1 & for p,q st ex A,B st A is open &
B is open & A in Bn.m & B in Bn.n & p in A & Cl A c= B & not q in B holds pmet.
      [p,q] = 1 by A299,A296;
      take pmet;
      thus F9.k=pmet & pmet is_a_pseudometric_of cT by A295,A301,A303;
      thus for pq holds pmet.pq<=1
      proof
        let pq;
        ex p,q be object st p in cT & q in cT & pq=[p,q] by ZFMISC_1:def 2;
        hence thesis by A304;
      end;
      thus thesis by A301,A302;
    end;
    hence thesis by A2,A272,Th17;
  end;
  thus thesis by NAGATA_1:15,16;
end;
