 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 holds
  (R_VectorSpace_of_ContinuousFunctions(S,T) is
    Subspace of R_VectorSpace_of_BoundedFunctions(the carrier of S,T))

proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
A1:the carrier of R_VectorSpace_of_ContinuousFunctions(S,T)
    c= the carrier of R_VectorSpace_of_BoundedFunctions(the carrier of S,T)
  by Th34;
A2:R_VectorSpace_of_ContinuousFunctions(S,T) is
    Subspace of RealVectSpace(the carrier of S,T) by Th5,RSSPACE:11;
  R_VectorSpace_of_BoundedFunctions(the carrier of S,T) is
    Subspace of RealVectSpace(the carrier of S,T) by RSSPACE4:7;
  hence thesis by A1,A2,RLSUB_1:28;
end;
