
theorem Th17:
  for X, Y, Z be RealLinearSpace
  for f,h be VECTOR of R_VectorSpace_of_BilinearOperators(X,Y,Z)
  for a be Real
  holds
    h = a*f
  iff
    for x be VECTOR of X, y be VECTOR of Y
    holds h.(x,y) = a * f.(x,y)
  proof
    let X, Y, Z be RealLinearSpace;
    let f,h be VECTOR of R_VectorSpace_of_BilinearOperators(X,Y,Z);
    reconsider f9=f,h9=h as BilinearOperator of X,Y,Z by Def6;
    let a be Real;
    A1: R_VectorSpace_of_BilinearOperators(X,Y,Z) is Subspace of
      RealVectSpace(the carrier of [:X,Y:],Z) by RSSPACE:11; then
    reconsider f1=f as VECTOR of
      RealVectSpace(the carrier of [:X,Y:],Z) by RLSUB_1:10;
    reconsider h1=h as VECTOR of
      RealVectSpace(the carrier of [:X,Y:],Z) by A1,RLSUB_1:10;
    A2: now
      assume
      A3: h = a*f;
      let x be Element of X,y be Element of Y;
      A4: h1=a*f1 by A1,A3,RLSUB_1:14;
      [x,y] is Element of [:X,Y:];
      hence h9.(x,y)=a*f9.(x,y) by A4,LOPBAN_1:2;
    end;
    now
      assume
      A5: for x be Element of X,y be Element of Y
           holds h9.(x,y) = a * f9.(x,y);
      for z be Element of [:X,Y:] holds h9.z=a*f9.z
      proof
        let z be Element of [:X,Y:];
        consider x be Point of X,y be Point of Y such that
        A6: z=[x,y] by PRVECT_3:9;
        thus h9.z = h9.(x,y) by A6
          .= a*f9.(x,y) by A5
          .= a*f9.z by A6;
      end; then
      h1 = a*f1 by LOPBAN_1:2;
      hence h = a*f by A1,RLSUB_1:14;
    end;
    hence thesis by A2;
  end;
