
theorem
  for M be non empty MetrSpace,
      S be non empty Subset of M holds
    (M|S is totally_bounded complete)
    iff S is sequentially_compact
  proof
    let M be non empty MetrSpace,
        S be non empty Subset of M;
    hereby
      assume M|S is totally_bounded complete; then
      (M|S) is sequentially_compact by Th12;
      hence S is sequentially_compact by Th14;
    end;
    assume S is sequentially_compact; then
    (M|S) is sequentially_compact by Th14;
    hence M|S is totally_bounded complete by Th12;
  end;
