
theorem Th7:
  for X being non empty set,
      F, G, H being VECTOR of C_Algebra_of_BoundedFunctions(X),
      f, g, h being Function of X,COMPLEX st f=F & g=G & h=H holds
  ( H = F*G iff for x be Element of X holds h.x = (f.x)*(g.x))
proof
  let X be non empty set,
      F, G, H be VECTOR of C_Algebra_of_BoundedFunctions(X),
      f, g, h be Function of X,COMPLEX;
   assume
A1:f=F & g=G & h=H;
A2:C_Algebra_of_BoundedFunctions X is ComplexSubAlgebra of CAlgebra X by Th2;
   reconsider f1=F, g1=G, h1=H as VECTOR of CAlgebra X by TARSKI:def 3;
   hereby assume
A3: H = F*G;
    let x be Element of X;
    h1 = f1*g1 by A2,A3,Th3;
    hence h.x = f.x * g.x by A1,CFUNCDOM:2;
   end;
   assume for x be Element of X holds h.x = f.x * g.x; then
   h1 = f1 * g1 by A1,CFUNCDOM:2;
   hence H = F * G by A2,Th3;
end;
