 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
  for Z be RealNormSpace,
      F be non empty Subset of Z,
      H be non empty Subset of MetricSpaceNorm Z,
      T being Subset of TopSpaceNorm Z
   st Z is complete & H = F & H = T
holds
( (MetricSpaceNorm Z) | H is totally_bounded
     iff Cl(H) is sequentially_compact) &
( (MetricSpaceNorm Z) | H is totally_bounded
  iff
 (MetricSpaceNorm Z) | Cl(H) is compact) &
( (MetricSpaceNorm Z) | H is totally_bounded
  iff
 Cl(F) is compact) &
( (MetricSpaceNorm Z) | H is totally_bounded
  iff
 Cl(T) is compact)
proof
  let Z be RealNormSpace,
      F be non empty Subset of Z,
      H be non empty Subset of MetricSpaceNorm Z,
      T be Subset of TopSpaceNorm Z;
  assume A1: Z is complete;
  assume A2: F = H & H = T; then
A3: Cl(F) = Cl(H) by Th1;
  consider HH be Subset of TopSpaceMetr MetricSpaceNorm Z such that
A4:HH = H & Cl(H) = Cl(HH) by Def1;
  (MetricSpaceNorm Z) | Cl(H) is complete by Th7,A1;
  hence
A5: (MetricSpaceNorm Z) | H is totally_bounded
     iff Cl(H) is sequentially_compact by TOPMETR4:17,Th8;
  thus ((MetricSpaceNorm Z) | H is totally_bounded
  iff (MetricSpaceNorm Z) | Cl(H) is compact) by A5,TOPMETR4:14;
  thus (MetricSpaceNorm Z) | H is totally_bounded
  iff
   Cl(F) is compact by A5,TOPMETR4:18,A3;
  hence (MetricSpaceNorm Z) | H is totally_bounded iff
    Cl(T) is compact by A3,A2,A4,TOPMETR4:19;
end;
