theorem Th30:
  for X, Y be RealNormSpace for f be Lipschitzian LinearOperator of X,Y
  holds BoundedLinearOperatorsNorm(X,Y).f = upper_bound PreNorms(f)
proof
  let X, Y be RealNormSpace;
  let f be Lipschitzian LinearOperator of X,Y;
  reconsider f9=f as set;
  f in BoundedLinearOperators(X,Y) by Def9;
  hence BoundedLinearOperatorsNorm(X,Y).f =
  upper_bound PreNorms(modetrans(f9,X,Y)) by Def13
    .= upper_bound PreNorms(f) by Th29;
end;
