
theorem Th33:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for f being Point of R_NormSpace_of_BoundedMultilinearOperators(X,Y)
  holds 0 <= ||.f.||
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace;
    let f be Point of R_NormSpace_of_BoundedMultilinearOperators(X,Y);
    reconsider g = f as Lipschitzian MultilinearOperator of X,Y by Def9;
    consider r0 be object such that
    A1: r0 in PreNorms(g) by XBOOLE_0:def 1;
    reconsider r0 as Real by A1;
    A2: PreNorms(g) is non empty bounded_above by Th27;
    A3: BoundedMultilinearOperatorsNorm(X,Y).f = upper_bound PreNorms(g)
      by Th30;
    now
      let r be Real;
      assume r in PreNorms(g); then
      ex t be VECTOR of product X
      st r = ||.g.t .||
       & for i be Element of dom X holds ||.t.i.|| <= 1;
      hence 0 <= r;
    end; then
    0 <= r0 by A1;
    hence thesis by A1,A2,A3,SEQ_4:def 1;
  end;
