
theorem
  for X being compact non empty TopSpace
  for F,G,H being Point of R_Normed_Algebra_of_ContinuousFunctions(X)
  for f,g,h being RealMap of X holds
  (f=F & g=G & h=H implies
  (H = F*G iff for x be Element of X holds h.x = f.x * g.x))
proof
  let X be compact non empty TopSpace;
  let F,G,H be Point of R_Normed_Algebra_of_ContinuousFunctions(X);
  let f,g,h be RealMap of X;
  reconsider f1=F, g1=G, h1=H as VECTOR of R_Algebra_of_ContinuousFunctions(X);
  H=F*G iff h1=f1*g1;
  hence thesis by Th5;
end;
