
theorem Th5:
  for X being non empty TopSpace
  for F,G,H being VECTOR of R_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 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 R_Algebra_of_ContinuousFunctions(X);
  let f,g,h be RealMap of X;
  assume
A1: f=F & g=G & h=H;
A2:R_Algebra_of_ContinuousFunctions(X) is
     Subalgebra of RAlgebra the carrier of X by C0SP1:6;
   reconsider f1=F, g1=G, h1=H as VECTOR of
   RAlgebra 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,C0SP1:8;
     hence h.x = f.x * g.x by A1,FUNCSDOM:2;
   end;
   assume for x be Element of X holds h.x = f.x * g.x;
   then
   h1 = f1 * g1 by A1,FUNCSDOM:2;
   hence H = F * G by A2,C0SP1:8;
end;
