
theorem Th33:
  for X,Y,Z be RealNormSpace
  for f being Point of 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);
    reconsider g=f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
    consider r0 be object such that
    A1: r0 in PreNorms(g) by XBOOLE_0:def 1;
    reconsider r0 as Real by A1;
    A3: BoundedBilinearOperatorsNorm(X,Y,Z).f
      = upper_bound PreNorms(g) by Th30;
    now
      let r be Real;
      assume r in PreNorms(g); then
      ex t be VECTOR of X,s be VECTOR of Y
      st r = ||.g.(t,s).|| & ||.t.|| <= 1 & ||.s.|| <= 1;
      hence 0 <= r;
    end;
    then 0 <= r0 by A1;
    hence thesis by A1,A3,SEQ_4:def 1;
  end;
