 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 Th11:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace
    st T is complete holds
  for H be non empty Subset of
      (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T))
  holds
      Cl(H) is sequentially_compact iff
      (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T) )
   | H is totally_bounded
proof
  let S be non empty compact TopSpace,
          T be NormedLinearTopSpace;
  assume A1: T is complete;
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  let H be non empty Subset of
      MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T);
  reconsider F = H as non empty Subset of Z;
  set N = (MetricSpaceNorm Z) | H;
  Z is complete by A1,C0SP3:52; then
  (MetricSpaceNorm Z) | Cl(H) is totally_bounded
     iff Cl(H) is sequentially_compact by TOPMETR4:17,Th7;
  hence thesis by Th8;
end;
