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

theorem Th8:
  for S be non empty TopSpace, T be non empty LinearTopSpace
  for f,h be VECTOR of R_VectorSpace_of_ContinuousFunctions(S,T)
  for a be Real holds h = a*f iff for x be Element of S holds h.x = a * f.x
proof
  let S be non empty TopSpace, T be non empty LinearTopSpace;
  let f,h be VECTOR of R_VectorSpace_of_ContinuousFunctions(S,T);
  set G = { f where f is Function of the carrier of S,
      the carrier of T :f is continuous };
A1: f in G & h in G; then
  ex x be Function of the carrier of S,
    the carrier of T st f=x & x is continuous; then
  reconsider f9=f as Function of S,T;
  ex x be Function of the carrier of S,
    the carrier of T st h=x & x is continuous by A1; then
  reconsider h9=h as Function of S,T;
  let a be Real;
A2: R_VectorSpace_of_ContinuousFunctions(S,T)
    is Subspace of RealVectSpace(the carrier of S,T) by Th5,RSSPACE:11;
  then reconsider f1=f as VECTOR of RealVectSpace(the carrier of S,T)
    by RLSUB_1:10;
  reconsider h1=h as VECTOR of RealVectSpace(the carrier of S,T)
    by A2,RLSUB_1:10;
A3: now
    assume
A4: h = a*f;
    let x be Element of S;
    h1=a*f1 by A2,A4,RLSUB_1:14;
    hence h9.x=a*f9.x by LOPBAN_1:2;
  end;
  now
    assume for x be Element of S holds h9.x=a*f9.x;
    then h1=a*f1 by LOPBAN_1:2;
    hence h =a*f by A2,RLSUB_1:14;
  end;
  hence thesis by A3;
end;
