
theorem Th25:
  for X, Y, Z be RealNormSpace
  for f,h be VECTOR of R_VectorSpace_of_BoundedBilinearOperators(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 RealNormSpace;
    let f,h be VECTOR of R_VectorSpace_of_BoundedBilinearOperators(X,Y,Z);
    let a be Real;
    A1: R_VectorSpace_of_BoundedBilinearOperators(X,Y,Z)
        is Subspace of R_VectorSpace_of_BilinearOperators(X,Y,Z)
        by RSSPACE:11; then
    reconsider f1=f as VECTOR of R_VectorSpace_of_BilinearOperators(X,Y,Z)
      by RLSUB_1:10;
    reconsider h1=h as VECTOR of R_VectorSpace_of_BilinearOperators(X,Y,Z)
      by A1,RLSUB_1:10;
    hereby
      assume
      A2: h = a*f;
      let x be Element of X,y be Element of Y;
      h1 = a*f1 by A1,A2,RLSUB_1:14;
      hence h.(x,y) = a*f.(x,y) by Th17;
    end;
    assume for x be Element of X,y be Element of Y
      holds h.(x,y)=a*f.(x,y); then
    h1 = a*f1 by Th17;
    hence thesis by A1,RLSUB_1:14;
  end;
