reserve V for non empty RealLinearSpace;
reserve S for Real_Sequence;
reserve k,n,m,m1 for Nat;
reserve g,h,r,x for Real;

theorem Th32:
  for X be RealNormSpace, 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 RealNormSpace;
  let f be Point of DualSp X;
  let g be Lipschitzian linear-Functional of X such that
A1: g=f;
  now
    let t be VECTOR of X;
    per cases;
    suppose
A3:  t = 0.X; then
A4:  ||.t.|| = 0;
     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,NORMSP_0:def 5;
A7:  |. ||.t.||".| =|. 1*||.t.||".| .=|. 1/||.t.||.| by XCMPLX_0:def 9
          .=1/||.t.|| by 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 NORMSP_1:def 1
          .=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,Th30;
     |.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.|;
     hence |.g.t.| <= ||.f.||*||.t.|| by A9,XREAL_1:64;
    end;
  end;
  hence thesis;
end;
