
theorem Th31:
  for X,Y,Z be RealNormSpace holds
  (the carrier of [:X,Y:]) --> 0.Z
    = 0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
  proof
    let X, Y, Z be RealNormSpace;
    ((the carrier of [:X,Y:]) --> 0.Z)
      = 0.R_VectorSpace_of_BoundedBilinearOperators(X,Y,Z) by Th26
     .= 0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z);
    hence thesis;
  end;
