
theorem Th26:
  for X be RealUnitarySpace, f be Point of DualSp X,
      g be Lipschitzian linear-Functional of X
    st g=f
   holds for t be VECTOR of X holds |.g.t.| <= ||.f.|| * ||.t.||
proof
  let X be RealUnitarySpace;
  let f be Point of DualSp X;
  let g be Lipschitzian linear-Functional of X such that
A1: g=f;
    let t be VECTOR of X;
    per cases;
    suppose
A3:  t = 0.X; then
A4:  ||.t.|| = 0 by BHSP_1:26;
     g.t = g.(0 * 0.X) by A3
        .= 0 * g.(0.X) by HAHNBAN:def 3
        .= 0;
     hence |.g.t.| <= ||.f.||*||.t.|| by A4,COMPLEX1:44;
    end;
    suppose
A5:  t <> 0.X;
     reconsider t1 = ( ||.t.||")*t as VECTOR of X;
A6:  ||.t.|| <> 0 by A5,BHSP_1:26; then
B61: 0 < ||.t.|| by BHSP_1:28;
A7:  |. ||.t.||".| = |. 1*||.t.||".|
                  .= |. 1/||.t.||.| by XCMPLX_0:def 9
                  .= 1/||.t.|| by B61,ABSVALUE:def 1
                  .= 1*||.t.||" by XCMPLX_0:def 9
                  .= ||.t.||";
A8:  |.g.t.|/||.t.|| = |.g.t.|*||.t.||" by XCMPLX_0:def 9
          .=|. ||.t.||"*g.t.| by A7,COMPLEX1:65
          .=|.g.t1.| by HAHNBAN:def 3;
     ||.t1.|| =|. ||.t.||".|*||.t.|| by BHSP_1:27
          .=1 by A6,A7,XCMPLX_0:def 7;
     then |.g.t.|/||.t.|| in {|.g.s.| where s is VECTOR of X : ||.s.||
        <= 1 } by A8;
     then |.g.t.|/||.t.|| <= upper_bound PreNorms g by SEQ_4:def 1;
     then
A9:  |.g.t.|/||.t.|| <= ||.f.|| by A1,Th24;
A10: |.g.t.|/||.t.||*||.t.||
           = |.g.t.|*||.t.||"*||.t.|| by XCMPLX_0:def 9
          .=|.g.t.|*(||.t.||"*||.t.||)
          .=|.g.t.|*1 by A6,XCMPLX_0:def 7
          .=|.g.t.|;
     0 <= ||.t.|| by BHSP_1:28;
     hence |.g.t.| <= ||.f.||*||.t.|| by A9,A10,XREAL_1:64;
  end;
end;
