
theorem
  for a being Complex
  for X being non empty compact TopSpace
  for f, g being Function of the carrier of X,COMPLEX
  for F, G being Point of C_Normed_Algebra_of_ContinuousFunctions X st
              f = F & g = G holds
  (G = a*F iff for x being Element of X holds g.x = a*(f.x))
proof
  let a be Complex;
  let X be non empty compact TopSpace;
  let f, g be Function of the carrier of X,COMPLEX;
  let F, G be Point of C_Normed_Algebra_of_ContinuousFunctions X;
  reconsider f1 = F, g1 = G as VECTOR of C_Algebra_of_ContinuousFunctions X;
  (G = a*F iff g1 = a*f1);
  hence thesis by Th11;
end;
