
theorem Th19:
  for M be non empty MetrSpace,S be non empty compact TopSpace,
      T be non empty MetrSpace,
      F be non empty Subset of MetricSpace_of_ContinuousFunctions(S,T),
    G be Subset of Funcs(the carrier of M, the carrier of T)
  st S = TopSpaceMetr(M) & T is complete & G = F holds
    MetricSpace_of_ContinuousFunctions(S,T) |  Cl(F) is compact
    iff
     G is equicontinuous
    &
     for x be Point of S,
         Fx be non empty Subset of T
      st Fx = { f.x where f is Function of S,T : f in F } holds
        T | Cl(Fx) is compact
  proof
    let M be non empty MetrSpace,S be non empty compact TopSpace,
        T be non empty MetrSpace;
    let F be non empty Subset of MetricSpace_of_ContinuousFunctions(S,T),
        G be Subset of Funcs(the carrier of M, the carrier of T);
    assume A1: S = TopSpaceMetr(M) & T is complete;
    assume A2: G = F;
    set Z = MetricSpace_of_ContinuousFunctions(S,T);
A3: Cl(F) is sequentially_compact iff
      Z | F is totally_bounded by Th14,A1;
    Cl(F) is sequentially_compact iff Z | Cl(F) is compact by TOPMETR4:14;
    hence thesis by A1,A2,Th17,A3;
  end;
