 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 Th9:
  for Z be RealNormSpace,
      F be non empty Subset of Z,
      H be non empty Subset of MetricSpaceNorm Z
   st Z is complete & H = F
      & (MetricSpaceNorm Z) | H is totally_bounded
holds
  Cl(H) is sequentially_compact
  &
  (MetricSpaceNorm Z) | Cl(H) is compact
  &
  Cl(F) is compact
proof
  let Z be RealNormSpace,
      F be non empty Subset of Z,
      H be non empty Subset of MetricSpaceNorm Z;
  set K = Cl(H);
  set M = MetricSpaceNorm Z;
  assume A1: Z is complete & H = F & M | H is totally_bounded; then
  A2: Cl(H) = Cl(F) by Th1;
  (MetricSpaceNorm Z) | K is complete by Th7,A1; then
  K is sequentially_compact by TOPMETR4:17,A1,Th8;
  hence thesis by TOPMETR4:14,TOPMETR4:18,A2;
end;
