
theorem Th22:
  for X being non empty set,
      f, g, h being Function of X,COMPLEX,
      F, G, H being Point of C_Normed_Algebra_of_BoundedFunctions(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 set,
      f, g, h be Function of X,COMPLEX,
      F, G, H be Point of C_Normed_Algebra_of_BoundedFunctions(X);
  reconsider f1 = F, g1 = G, h1 = H
                       as VECTOR of C_Algebra_of_BoundedFunctions(X);
A1: H = F + G iff h1 = f1 + g1;
  assume f = F & g = G & h = H;
  hence thesis by A1,Th5;
 end;
