
theorem Th16:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for f,g,h be VECTOR of R_VectorSpace_of_MultilinearOperators(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_MultilinearOperators(X,Y);
    reconsider f9=f,g9=g,h9=h as MultilinearOperator of X,Y by Def6;
    A1: R_VectorSpace_of_MultilinearOperators(X,Y) is Subspace of
    RealVectSpace(the carrier of product X,Y) by RSSPACE:11; then
    reconsider f1=f, h1=h, g1=g as VECTOR of
      RealVectSpace(the carrier of product X,Y) by RLSUB_1:10;
    A2: now
      assume
      A3: h = f+g;
      let x be Element of product X;
      h1 = f1+g1 by A1,A3,RLSUB_1:13;
      hence h9.x=f9.x+g9.x by LOPBAN_1:1;
    end;
    now
      assume for x be Element of product X holds h9.x = f9.x + g9.x; then
      h1 = f1 + g1 by LOPBAN_1:1;
      hence h = f + g by A1,RLSUB_1:13;
    end;
    hence thesis by A2;
  end;
