
theorem Th34:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for f being Point of R_NormSpace_of_BoundedMultilinearOperators(X,Y)
  st f = 0.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) such that
    A1: f = 0.R_NormSpace_of_BoundedMultilinearOperators(X,Y);
    reconsider g = f as Lipschitzian MultilinearOperator of X,Y by Def9;
    set z = (the carrier of product X) --> 0.Y;
    reconsider z as Function of the carrier of product X,the carrier of Y;
    consider r0 be object such that
    A2: r0 in PreNorms(g) by XBOOLE_0:def 1;
    reconsider r0 as Real by A2;
    A3: (for s be Real st s in PreNorms(g) holds s <= 0)
      implies upper_bound PreNorms(g) <= 0 by SEQ_4:45;
    A4: PreNorms(g) is non empty bounded_above by Th27;
    A5: z = g by A1,Th31;
    A6: now
      let r be Real;
      assume r in PreNorms(g); then
      consider t be VECTOR of product X such that
      A7: r = ||. g.t .|| and
      for i be Element of dom X holds ||.t.i.|| <= 1;
      ||.g.t.|| = ||.0.Y.|| by A5,FUNCOP_1:7
      .= 0;
      hence 0 <= r & r <=0 by A7;
    end; then
    0 <= r0 by A2; then
    upper_bound PreNorms(g) = 0 by A6,A4,A2,A3,SEQ_4:def 1;
    hence thesis by Th30;
  end;
