
theorem Th32:
  for X,Y,Z be RealNormSpace
  for f being Point of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
  for g be Lipschitzian BilinearOperator of X,Y,Z st g=f
  holds
    for t be VECTOR of X, s be VECTOR of Y
    holds ||.g.(t,s).|| <= ||.f.|| * ||.t.|| * ||.s.||
  proof
    let X, Y, Z be RealNormSpace;
    let f being Point of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z);
    let g be Lipschitzian BilinearOperator of X,Y,Z such that
    A1: g = f;
    now
      let t be VECTOR of X,s be VECTOR of Y;
      now
        per cases;
        case
          A3: t = 0.X or s = 0.Y; then
          A4: ||.t.|| = 0 or ||.s.|| = 0;
          t = 0 * t or s = 0*s by A3; then
          g.(t,s) = 0 * g.(t,s) by LOPBAN_8:12
           .= 0.Z by RLVECT_1:10;
          hence ||.g.(t,s).|| <= ||.f.|| * ||.t.|| * ||.s.|| by A4;
        end;
        case
          A5: t <> 0.X & s <> 0.Y;
          reconsider t1 = ( ||.t.||") * t as VECTOR of X;
          reconsider s1 = ( ||.s.||") * s as VECTOR of Y;
          A6: ||.t.|| <> 0 & ||.s.|| <> 0 by A5,NORMSP_0:def 5; then
          A7: ||.t.|| * ||.s.|| <> 0 by XCMPLX_1:6;
          A8: |. ||.t.||" .| = |. 1 * ||.t.||".|
           .= |. 1 / ||.t.|| .| by XCMPLX_0:def 9
           .= 1 / ||.t.|| by ABSVALUE:def 1
           .= 1 * ||.t.||" by XCMPLX_0:def 9
           .= ||.t.||";
          A9: |. ||.s.||".| = |. 1 * ||.s.||".|
           .= |. 1 / ||.s.||.| by XCMPLX_0:def 9
           .= 1 / ||.s.|| by ABSVALUE:def 1
           .= 1 * ||.s.||" by XCMPLX_0:def 9
           .= ||.s.||";
          A10: |. ( ||.t.|| * ||.s.|| )".|
            = |. ||.t.||" * ||.s.||" .| by XCMPLX_1:204
           .= ||.t.||" * ||.s.||" by A8,A9,COMPLEX1:65
           .= ( ||.t.|| * ||.s.|| )" by XCMPLX_1:204;
          A11: ||.t1.|| = |. ||.t.||".| * ||.t.|| by NORMSP_1:def 1
           .= 1 by A6,A8,XCMPLX_0:def 7;
          A12: ||.s1.|| = |. ||.s.||".| * ||.s.|| by NORMSP_1:def 1
          .= 1 by A6,A9,XCMPLX_0:def 7;
          ||.g.(t,s).|| / ( ||.t.|| * ||.s.|| )
           = ||.g.(t,s).|| * ( ||.t.|| * ||.s.|| )" by XCMPLX_0:def 9
          .= ||. ( ||.t.|| * ||.s.|| )" * g.(t,s) .|| by A10,NORMSP_1:def 1
          .= ||. ( ||.t.||" * ||.s.||" ) * g.(t,s) .|| by XCMPLX_1:204
          .= ||. ||.t.||" * ( ||.s.||" * g.(t,s)) .|| by RLVECT_1:def 7
          .= ||. ||.t.||" * ( g.(t,s1)) .|| by LOPBAN_8:12
          .= ||. g.(t1,s1) .|| by LOPBAN_8:12;
          then
          ||.g.(t,s).|| / ( ||.t.||*||.s.||)
            in {||.g.(t,s).|| where t is VECTOR of X, s is VECTOR of Y
                  : ||.t.|| <= 1 & ||.s.|| <= 1 } by A11,A12;
          then
          ||.g.(t,s).|| / (||.t.|| * ||.s.||) <= upper_bound PreNorms(g)
            by SEQ_4:def 1;
          then
          A13: ||.g.(t,s).|| / (||.t.|| * ||.s.||) <= ||.f.|| by A1,Th30;
          ||.g.(t,s).|| / (||.t.|| * ||.s.||) * (||.t.|| * ||.s.||)
           = ||.g.(t,s).|| * (||.t.|| * ||.s.||)" * (||.t.|| * ||.s.||)
              by XCMPLX_0:def 9
          .= ||.g.(t,s).|| * ((||.t.|| * ||.s.||)" * (||.t.|| * ||.s.||))
          .= ||.g.(t,s).|| * 1 by A7,XCMPLX_0:def 7
          .= ||.g.(t,s).||;
          hence ||.g.(t,s).|| <= ||.f.|| * ( ||.t.|| * ||.s.|| )
            by A13,XREAL_1:64;
        end;
      end;
      hence ||.g.(t,s).|| <= ||.f.||*||.t.||*||.s.||;
    end;
    hence thesis;
  end;
