 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,T be RealNormSpace
  holds 0. R_VectorSpace_of_ContinuousFunctions(S,T)
    = (the carrier of S) --> 0.T
proof
  let S,T be RealNormSpace;
A1: 0.RealVectSpace(the carrier of S,T) = (the carrier of S) -->0.T;
  R_VectorSpace_of_ContinuousFunctions(S,T)
    is Subspace of RealVectSpace(the carrier of S,T) by Th11,RSSPACE:11;
  hence thesis by A1,RLSUB_1:11;
end;
