
theorem Th5:
  for Z be non empty MetrSpace,
      H be non empty Subset of Z
   st Z is complete & Z | H is totally_bounded
      holds
  Cl(H) is sequentially_compact & Z | Cl(H) is compact
  proof
    let Z be non empty MetrSpace,
        H be non empty Subset of Z;
    set K = Cl(H);
    assume A1: Z is complete & Z | H is totally_bounded; then
    Z | K is complete by Th3; then
    K is sequentially_compact by TOPMETR4:17,A1,Th4;
    hence thesis by TOPMETR4:14;
  end;
