
theorem Th14:
  for S be non empty compact TopSpace,T be non empty MetrSpace
    st T is complete holds
  for H be non empty Subset of
      (MetricSpace_of_ContinuousFunctions(S,T)) holds
      Cl(H) is sequentially_compact iff
      (MetricSpace_of_ContinuousFunctions(S,T))
   | H is totally_bounded
  proof
    let S be non empty compact TopSpace,
        T be non empty MetrSpace;
    assume A1: T is complete;
    set Z = MetricSpace_of_ContinuousFunctions(S,T);
    let H be non empty Subset of
        MetricSpace_of_ContinuousFunctions(S,T);
    Z is complete by A1,Th13; then
    Z | Cl(H) is totally_bounded
      iff Cl(H) is sequentially_compact by TOPMETR4:17,Th3;
    hence thesis by Th4;
  end;
