
theorem Th12:
  for X being non empty TopSpace
  for F, G, H being VECTOR of (C_Algebra_of_ContinuousFunctions X)
  for f, g, h being Function of the carrier of X,COMPLEX
                    st f = F & g = G & h = H holds
  ( H = F * G iff for x being Element of the carrier of X holds
                          h . x = (f . x) * (g . x) )
proof
  let X be non empty TopSpace;
  let F, G, H be VECTOR of (C_Algebra_of_ContinuousFunctions X);
  let f, g, h be Function of the carrier of X,COMPLEX;
  assume
A1: f = F & g = G & h = H;
A2:C_Algebra_of_ContinuousFunctions X
           is ComplexSubAlgebra of CAlgebra the carrier of X by CC0SP1:2;
  reconsider f1 = F, g1 = G, h1 = H
        as VECTOR of (CAlgebra the carrier of X) by TARSKI:def 3;
  hereby
    assume
A3: H = F * G;
    let x be Element of the carrier of X;
    h1 = f1 * g1 by A2,A3,CC0SP1:3;
    hence h . x = (f . x) * (g . x) by A1,CFUNCDOM:2;
  end;
  assume for x being Element of X holds h . x = (f . x) * (g . x);
  then h1 = f1 * g1 by A1,CFUNCDOM:2;
  hence H = F * G by A2,CC0SP1:3;
end;
