
theorem Th25:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for f,h be VECTOR of R_VectorSpace_of_BoundedMultilinearOperators(X,Y)
  for a be Real
  holds
    h = a * f
  iff
    for x be VECTOR of product X holds
    h.x = a * f.x
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace;
    let f,h be VECTOR of R_VectorSpace_of_BoundedMultilinearOperators(X,Y);
    let a be Real;
    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 as VECTOR of
      R_VectorSpace_of_MultilinearOperators(X,Y) by RLSUB_1:10;
    hereby
      assume
      A2: h = a * f;
      let x be Element of product X;
      h1 = a * f1 by A1,A2,RLSUB_1:14;
      hence h.x = a * f.x by Th17;
    end;
    assume for x be Element of product X holds h.x = a * f.x; then
    h1 = a * f1 by Th17;
    hence thesis by A1,RLSUB_1:14;
  end;
