
theorem Th17:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for f,h be VECTOR of R_VectorSpace_of_MultilinearOperators(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_MultilinearOperators(X,Y);
    reconsider f9=f,h9=h as MultilinearOperator of X,Y by Def6;
    let a be Real;
    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 as VECTOR of
      RealVectSpace(the carrier of product X,Y) by RLSUB_1:10;
    A2: now
      assume
      A3: h = a * f;
      let x be Element of product X;
      h1 = a * f1 by A1,A3,RLSUB_1:14;
      hence h9.x = a * f9.x by LOPBAN_1:2;
    end;
    now
      assume for x be Element of product X holds h9.x = a * f9.x; then
      h1 = a * f1 by LOPBAN_1:2;
      hence h = a * f by A1,RLSUB_1:14;
    end;
    hence thesis by A2;
  end;
