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

theorem Th9:
  for S be non empty TopSpace, T be non empty LinearTopSpace
    holds 0. R_VectorSpace_of_ContinuousFunctions(S,T)
      = (the carrier of S) --> 0.T
proof
  let S be non empty TopSpace, T be non empty LinearTopSpace;
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 Th5,RSSPACE:11;
  hence thesis by A1,RLSUB_1:11;
end;
