theorem Th41:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace holds
  0.R_NormSpace_of_ContinuousFunctions(S,T)
    = 0.R_NormSpace_of_BoundedFunctions(the carrier of S,T)
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  thus 0.R_NormSpace_of_ContinuousFunctions(S,T)
= (the carrier of S)-->0.T by Th40
    .= 0.R_NormSpace_of_BoundedFunctions(the carrier of S,T) by RSSPACE4:15;
end;
