 reserve S, T for RealNormSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);
 reserve S,Z for RealNormSpace;
 reserve T for RealBanachSpace;
 reserve F for Subset of Funcs(the carrier of S,the carrier of T);

theorem
  for M be non empty MetrSpace,S be non empty compact TopSpace,
      T be NormedLinearTopSpace,
      F be non empty Subset of
         R_NormSpace_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
      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 MetricSpaceNorm T
       st Fx = {f.x where f is Function of S,T :f in F }
     holds (MetricSpaceNorm T) | Cl(Fx) is compact
proof
  let M be non empty MetrSpace,S be non empty compact TopSpace,
      T be NormedLinearTopSpace;
  let F be non empty Subset of
        R_NormSpace_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
  Cl(F) is compact iff G is equicontinuous &
  for x be Point of S, Fx be non empty Subset of MetricSpaceNorm T
       st Fx = {f.x where f is Function of S,T :f in F } holds
    (MetricSpaceNorm T) | Cl(Fx) is compact by Th18,A1;
  hence thesis by Th5,A1;
end;
