
theorem Th34:
  for X, Y, Z be RealNormSpace
  for f being Point of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
  st f = 0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
  holds 0 = ||.f.||
  proof
    let X,Y,Z be RealNormSpace;
    let f being Point of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
    such that
    A1: f = 0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z);
      reconsider g=f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
      set z = (the carrier of [:X,Y:]) --> 0.Z;
      reconsider z as Function of the carrier of [:X,Y:],the carrier of Z;
      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;
      A5: z=g by A1,Th31;
      A6: now
        let r be Real;
        assume r in PreNorms(g); then
        consider t be VECTOR of X,s be VECTOR of Y such that
        A7: r = ||.g.(t,s).|| and
            ||.t.|| <= 1 & ||.s.|| <= 1;
        [t,s] is Point of [:X,Y:]; then
        g.(t,s) = 0.Z by FUNCOP_1:7,A5;
        hence 0 <= r & r <=0 by A7;
      end;
      then 0 <= r0 by A2;
      then upper_bound PreNorms(g) = 0 by A6,A2,A3,SEQ_4:def 1;
      hence thesis by Th30;
  end;
