
theorem
  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
   ( for x be Point of M holds G is_equicontinuous_at x ) &
   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 G = F; then
    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 by Th19,A1;
    hence thesis by Lm2,A1;
  end;
