
theorem Th15:
  for X being compact non empty TopSpace
  for F,G,H being Point of R_Normed_Algebra_of_ContinuousFunctions(X)
  for f,g,h be RealMap of X
    holds
  (f=F & g=G & h=H implies
  (H = F+G iff for x be Element of X holds h.x = f.x + g.x))
proof
  let X be compact non empty TopSpace;
  let F,G,H be Point of R_Normed_Algebra_of_ContinuousFunctions(X);
  let f,g,h be RealMap of X;
  reconsider f1=F, g1=G, h1=H as VECTOR of R_Algebra_of_ContinuousFunctions(X);
  H=F+G iff h1=f1+g1;
  hence thesis by Th3;
end;
