
theorem Th30:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for f be Lipschitzian MultilinearOperator of X,Y
  holds BoundedMultilinearOperatorsNorm(X,Y).f = upper_bound PreNorms(f)
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace;
    let f be Lipschitzian MultilinearOperator of X,Y;
    reconsider f9 = f as set;
    f in BoundedMultilinearOperators(X,Y) by Def9;
    hence BoundedMultilinearOperatorsNorm(X,Y).f
     = upper_bound PreNorms(modetrans(f9,X,Y)) by Def13
    .= upper_bound PreNorms(f);
  end;
