
theorem Th24:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for f,g,h be VECTOR of R_VectorSpace_of_BoundedMultilinearOperators(X,Y)
  holds
    h = f+g
  iff
    for x be VECTOR of product X
    holds h.x = f.x + g.x
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace;
    let f,g,h be VECTOR of R_VectorSpace_of_BoundedMultilinearOperators(X,Y);
    A1: R_VectorSpace_of_BoundedMultilinearOperators(X,Y) is Subspace of
        R_VectorSpace_of_MultilinearOperators(X,Y) by RSSPACE:11; then
    reconsider f1 = f, h1=h, g1=g as VECTOR of
      R_VectorSpace_of_MultilinearOperators(X,Y) by RLSUB_1:10;
    hereby
      assume
      A2: h = f+g;
      let x be Element of product X;
      h1 = f1+g1 by A1,A2,RLSUB_1:13;
      hence h.x = f.x+g.x by Th16;
    end;
    assume for x be Element of product X holds h.x = f.x+g.x; then
    h1 = f1+g1 by Th16;
    hence thesis by A1,RLSUB_1:13;
  end;
