
theorem Th3:
  for Z be non empty MetrSpace,
      F be non empty Subset of Z
   st Z is complete holds Z | Cl(F) is complete
  proof
    let Z be non empty MetrSpace,
        F be non empty Subset of Z;
    assume A1: Z is complete;
    set N = Z | Cl(F);
A2: the carrier of N = Cl(F) by TOPMETR:def 2;
    let S2 be sequence of N;
    assume A3: S2 is Cauchy;
A4: rng S2 c= Cl(F) by A2;
    rng S2 c= the carrier of Z by A2,XBOOLE_1:1; then
    reconsider S1 = S2 as sequence of Z by FUNCT_2:6;
    for r being Real st r > 0 holds
      ex k being Nat st
    for n, m being Nat st n >= k & m >= k holds
      dist(S1.n,S1.m) < r
    proof
      let r be Real;
      assume r > 0; then
      consider p being Nat such that
A6:   for n, m being Nat st p <= n & p <= m holds
        dist (S2.n,S2.m) < r by A3;
      take p;
      let n, m be Nat;
      assume p <= n & p <= m; then
      dist (S2.n,S2.m) < r by A6;
      hence dist (S1.n,S1.m) < r by TOPMETR:def 1;
    end; then
a7: S1 is Cauchy; then
A8: S1 is_convergent_in_metrspace_to lim S1 by A1,METRIC_6:12;
    consider H be Subset of TopSpaceMetr Z such that
A9: H = F & Cl F = Cl H by ASCOLI:def 1;
    for n being Nat holds S1.n in Cl H
    proof
      let n be Nat;
      S1.n in rng S1 by FUNCT_2:4,ORDINAL1:def 12;
      hence S1.n in Cl H by A4,A9;
    end; then
    lim S1 in Cl(F) by TOPMETR4:6,A9,a7,A1; then
    reconsider L = lim S1 as Point of N by TOPMETR:def 2;
    reconsider L0 = L as Point of Z;
    for r being Real st 0 < r holds
    ex m being Nat st for n being Nat st m <= n holds
      dist (S2.n,L) < r
    proof
      let r be Real;
      assume 0 < r; then
      consider m being Nat such that
  A10:for n being Nat st m <= n holds
      dist (S1.n,L0) < r by METRIC_6:def 2,A8;
      take m;
      let n be Nat;
      assume m <= n; then
      dist (S1.n,L0) < r by A10;
      hence dist (S2.n,L) < r by TOPMETR:def 1;
    end;
    hence S2 is convergent;
  end;
