 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 Th12:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace
   st T is complete holds
  for F be non empty Subset of
           R_NormSpace_of_ContinuousFunctions(S,T),
      H be non empty Subset of
           MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T) st
    H = F holds
    Cl(F) is 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 F be non empty Subset of Z, H be non empty Subset of MetricSpaceNorm Z;
  assume F = H; then
  A3:Cl(F) = Cl(H) by Th1;
  Cl(H) is sequentially_compact iff
    (MetricSpaceNorm R_NormSpace_of_ContinuousFunctions(S,T) )
    | H is totally_bounded by Th11,A1;
  hence thesis by TOPMETR4:18,A3;
end;
