 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);
 reserve S,Z for RealNormSpace;
 reserve T for RealBanachSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem Th7:
for Z be RealNormSpace,
    H be non empty Subset of MetricSpaceNorm Z
   st Z is complete
holds
(MetricSpaceNorm Z) | Cl(H) is complete
proof
let Z be RealNormSpace,
    H be non empty Subset of MetricSpaceNorm Z;
assume A1: Z is complete;
 reconsider F=H as non empty Subset of Z;
A2: Cl(F) = Cl(H) by Th1;
set N = (MetricSpaceNorm Z) | Cl(H);
A3: the carrier of N = Cl(H) by TOPMETR:def 2;
for S2 being sequence of N st S2 is Cauchy holds
S2 is convergent
proof
  let S2 be sequence of N;
  assume A4: S2 is Cauchy;
A5:rng S2 c= Cl(H) by A3;
   rng S2 c= the carrier of MetricSpaceNorm Z by A3,XBOOLE_1:1; then
   reconsider S1 = S2 as sequence of MetricSpaceNorm Z by FUNCT_2:6;
  reconsider seq2 = S1 as sequence of Z;
A6: rng seq2 c= Cl(F) by Th1,A5;
for r being Real st r > 0 holds
     ex k being Nat st
     for n, m being Nat st n >= k & m >= k holds
      ||.((seq2 . n) - (seq2 . m)).|| < r
proof
  let r be Real;
  assume r > 0; then
  consider p being Nat such that
  A7:for n, m being Nat st p <= n & p <= m holds
    dist ((S2 . n),(S2 . m)) < r by A4;
  take p;
  let n, m be Nat;
  assume p <= n & p <= m; then
  dist ((S2 . n),(S2 . m)) < r by A7; then
  dist ((S1 . n),(S1 . m)) < r by TOPMETR:def 1;
  hence ||.((seq2 . n) - (seq2 . m)).|| < r by NORMSP_2:def 1;
end; then
A8: seq2 is convergent by A1,RSSPACE3:8; then
lim seq2 in Cl(F) by NFCONT_1:def 3,A6; then
reconsider L = lim seq2 as Point of N by TOPMETR:def 2,A2;
reconsider L0 = L as Point of MetricSpaceNorm Z;
for r being Real st 0 < r holds
  ex m being Nat st
 for n being Nat st m <= n holds
  ||.((seq2 . n) - lim seq2).|| < r by A8,NORMSP_1:def 7; then
A9:S1 is_convergent_in_metrspace_to L0 by NORMSP_2:4;
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,A9;
  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;
hence thesis;
end;
