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