
theorem
  for X being RealNormSpace-Sequence,Y be RealNormSpace
  for f,g,h be Point of R_NormSpace_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 Point of R_NormSpace_of_BoundedMultilinearOperators(X,Y);
    reconsider f9=f,g9=g,h9=h as Lipschitzian MultilinearOperator of X,Y
      by Def9;
    hereby
      assume h = f-g; then
      h+g = f-(g-g) by RLVECT_1:29; then
      A1: h+g = f-0.R_NormSpace_of_BoundedMultilinearOperators(X,Y)
          by RLVECT_1:15;
      now
        let x be VECTOR of product X;
        f9.x = h9.x + g9.x by A1,Th35; then
        f9.x - g9.x = h9.x + (g9.x - g9.x) by RLVECT_1:def 3; then
        f9.x - g9.x = h9.x + 0.Y by RLVECT_1:15;
        hence f9.x - g9.x = h9.x;
      end;
      hence for x be VECTOR of product X holds h.x = f.x - g.x;
    end;
    assume
    A2: for x be VECTOR of product X holds h.x = f.x - g.x;
    now
      let x be VECTOR of product X;
      h9.x = f9.x - g9.x by A2; then
      h9.x + g9.x = f9.x - (g9.x - g9.x) by RLVECT_1:29;
      then h9.x + g9.x = f9.x - 0.Y by RLVECT_1:15;
      hence h9.x + g9.x = f9.x;
    end; then
    f = h+g by Th35; then
    f-g = h+(g-g) by RLVECT_1:def 3; then
    f-g = h+0.R_NormSpace_of_BoundedMultilinearOperators(X,Y) by RLVECT_1:15;
    hence thesis;
  end;
