 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem
 for S be non empty compact TopSpace,
     T be NormedLinearTopSpace,
      f be Point of R_NormSpace_of_ContinuousFunctions(S,T),
      g be Function of S,T
   st f=g holds
  for t be Point of S holds ||.g.t.|| <= ||.f.||
  proof
    let S be non empty compact TopSpace,
        T be NormedLinearTopSpace;
    let f be Point of R_NormSpace_of_ContinuousFunctions(S,T),
        g be Function of S,T;
    assume A1: f=g; then
    g in BoundedFunctions (the carrier of S,T) by Th34; then
    reconsider h =g as bounded Function of the carrier of S,T
      by RSSPACE4:def 5;
    reconsider k =h as Point of
      R_NormSpace_of_BoundedFunctions(the carrier of S,T) by Th34,A1;
    ||.k.|| = ||.f.|| by A1,FUNCT_1:49;
    hence thesis by RSSPACE4:16;
  end;
