 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
  for S,T be RealNormSpace
    for f,g,h be VECTOR of R_VectorSpace_of_ContinuousFunctions(S,T)
  holds h = f+g iff for x be Element of S holds h.x = f.x + g.x
proof
  let S,T be RealNormSpace;
  let f,g,h be VECTOR of R_VectorSpace_of_ContinuousFunctions(S,T);
set G = { f where f is Function of the carrier of S,
      the carrier of T :f is_continuous_on the carrier of S };
A1: f in G & g in G & h in G; then
ex x be Function of the carrier of S,
   the carrier of T st f=x & x is_continuous_on the carrier of S; then
reconsider f9=f as Function of S,T;
ex x be Function of the carrier of S,
   the carrier of T st g=x & x is_continuous_on the carrier of S by A1; then
reconsider g9=g as Function of S,T;
ex x be Function of the carrier of S,
   the carrier of T st h=x & x is_continuous_on the carrier of S by A1;
  then reconsider h9=h as Function of S,T;
A2: R_VectorSpace_of_ContinuousFunctions(S,T)
    is Subspace of RealVectSpace(the carrier of S,T) by Th11,RSSPACE:11;
  then reconsider f1=f as VECTOR of RealVectSpace(the carrier of S,T)
    by RLSUB_1:10;
  reconsider h1=h as VECTOR of RealVectSpace(the carrier of S,T)
    by A2,RLSUB_1:10;
  reconsider g1=g as VECTOR of RealVectSpace(the carrier of S,T)
    by A2,RLSUB_1:10;
A3: now
    assume
A4: h = f+g;
    let x be Element of S;
    h1=f1+g1 by A2,A4,RLSUB_1:13;
    hence h9.x=f9.x+g9.x by LOPBAN_1:1;
  end;
  now
    assume for x be Element of S holds h9.x=f9.x+g9.x;
    then h1=f1+g1 by LOPBAN_1:1;
    hence h =f+g by A2,RLSUB_1:13;
  end;
  hence thesis by A3;
end;
