 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 Th18:
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
      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
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);
  reconsider H = F as non empty Subset of
    (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T));
  assume A1: S = TopSpaceMetr(M) & T is complete;
  assume A2: G = F;
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  (MetricSpaceNorm Z) | H is totally_bounded iff
    Cl(F) is compact by A1,Th12;
  hence thesis by A1,A2,Th16;
end;
