
theorem
  for M be non empty MetrSpace,S be non empty compact TopSpace,
      T be non empty MetrSpace,
      G be Subset of Funcs(the carrier of M, the carrier of T),
      H be non empty Subset of
        (MetricSpace_of_ContinuousFunctions(S,T))
  st S = TopSpaceMetr(M) & T is complete & G = H
    holds
      Cl(H) is sequentially_compact
    iff
      G is equicontinuous
  &
  for x be Point of S, Hx be non empty Subset of T
       st Hx = { f.x where f is Function of S,T : f in H }
     holds T | Cl(Hx) is compact
  proof
    let M be non empty MetrSpace,S be non empty compact TopSpace,
        T be non empty MetrSpace;
    let G be Subset of Funcs(the carrier of M, the carrier of T),
        H be non empty Subset of MetricSpace_of_ContinuousFunctions(S,T);
    assume A1: S = TopSpaceMetr(M) & T is complete;
    assume A2: G = H;
    set Z = MetricSpace_of_ContinuousFunctions(S,T);
    Z | H is totally_bounded iff
    Cl(H) is sequentially_compact by A1,Th14;
    hence thesis by A1,A2,Th17;
  end;
