 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th44:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace
  for F,G,H being Point of R_NormSpace_of_ContinuousFunctions(S,T)
  for f,g,h be Function of S,T holds
  (f=F & g=G & h=H implies
  (H = F+G iff for x be Element of S holds h.x = f.x + g.x))
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  let F,G,H be Point of R_NormSpace_of_ContinuousFunctions(S,T);
  let f,g,h be Function of S,T;
  reconsider f1=F, g1=G, h1=H as VECTOR of
    R_VectorSpace_of_ContinuousFunctions(S,T);
  H=F+G iff h1=f1+g1;
  hence thesis by Th7;
end;
