
theorem Th30:
  for X, Y, Z be RealNormSpace for f be Lipschitzian BilinearOperator of X,Y,Z
  holds BoundedBilinearOperatorsNorm(X,Y,Z).f = upper_bound PreNorms(f)
  proof
    let X, Y, Z be RealNormSpace;
    let f be Lipschitzian BilinearOperator of X,Y,Z;
    reconsider f9 = f as set;
    f in BoundedBilinearOperators(X,Y,Z) by Def9;
    hence BoundedBilinearOperatorsNorm(X,Y,Z).f
      = upper_bound PreNorms(modetrans(f9,X,Y,Z)) by Def13
     .= upper_bound PreNorms(f);
  end;
