let X be RealUnitarySpace; :: thesis:
let f be Point of (DualSp X); :: thesis:
let g be Lipschitzian linear-Functional of X; :: thesis:
assume A1: g = f ; :: thesis:
let t be VECTOR of X; :: thesis:

suppose A3: t = 0. X ; :: thesis:

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; :: thesis:

end;

suppose A5: t <> 0. X ; :: thesis:

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; :: thesis:

end;

end;