
theorem
  for X being non empty compact TopSpace
  for f, g, h being Function of the carrier of X,COMPLEX
  for F, G, H being Point of C_Normed_Algebra_of_ContinuousFunctions X st
              f = F & g = G & h = H holds
  ( H= F*G iff for x being Element of X holds h.x = (f.x)*(g.x))
proof
  let X be non empty compact TopSpace;
  let f, g, h be Function of the carrier of X,COMPLEX;
  let F, G, H be Point of C_Normed_Algebra_of_ContinuousFunctions X;
  reconsider f1 = F, g1 = G, h1 = H
                        as VECTOR of C_Algebra_of_ContinuousFunctions X;
  H = F*G iff h1 = f1*g1;
  hence thesis by Th12;
end;
