 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 a be Real
  for S be non empty compact TopSpace,T be NormedLinearTopSpace
  for F,G being Point of R_NormSpace_of_ContinuousFunctions(S,T)
  for f,g being Function of S,T holds
  (f=F & g=G implies ( G = a*F iff for x be Element of S holds g.x = a*f.x ))
proof
  let a be Real;
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  let F,G be Point of R_NormSpace_of_ContinuousFunctions(S,T);
  let f,g be Function of S,T;
  reconsider f1=F, g1=G as VECTOR of R_VectorSpace_of_ContinuousFunctions(S,T);
  G=a*F iff g1=a*f1;
  hence thesis by Th8;
end;
