theorem
  for S be non empty compact TopSpace,
      T be NormedLinearTopSpace st T is complete holds
   R_NormSpace_of_ContinuousFunctions(S,T) is complete
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  assume T is complete; then
  for seq be sequence of Z st seq is Cauchy_sequence_by_Norm holds
    seq is convergent by Th51;
  hence thesis by LOPBAN_1:def 15;
end;
