 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,
      G be Subset of Funcs(the carrier of M, the carrier of T),
      H be non empty Subset of
        (MetricSpaceNorm R_NormSpace_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 MetricSpaceNorm T
       st Hx = {f.x where f is Function of S,T :f in H }
     holds (MetricSpaceNorm T) | Cl(Hx) is compact
proof
  let M be non empty MetrSpace,S be non empty compact TopSpace,
      T be NormedLinearTopSpace;
  let G be Subset of Funcs(the carrier of M, the carrier of T),
      H be non empty Subset of
    MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T);
  assume A1: S = TopSpaceMetr(M) & T is complete;
  assume A2: G = H;
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  (MetricSpaceNorm Z) | H is totally_bounded iff
  Cl(H) is sequentially_compact by A1,Th11;
  hence thesis by A1,A2,Th16;
end;
