
theorem Th22:
  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 Th10;
end;
