let X be RealUnitarySpace; :: thesis: for f being Point of (DualSp X) holds 0 <= ||.f.||
let f be Point of (DualSp X); :: thesis: 0 <= ||.f.||
reconsider g = f as Lipschitzian linear-Functional of X by Def10;
consider r0 being object such that
A1: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A1;
A3: (BoundedLinearFunctionalsNorm X) . f = upper_bound (PreNorms g) by Th24;

now :: thesis: for r being Real st r in PreNorms g holds
0 <= r

let r be Real; :: thesis: ( r in PreNorms g implies 0 <= r )
assume r in PreNorms g ; :: thesis: 0 <= r
then ex t being VECTOR of X st
( r = |.(g . t).| & ||.t.|| <= 1 ) ;
hence 0 <= r by COMPLEX1:46; :: thesis: verum

end;

then 0 <= r0 by A1;
hence 0 <= ||.f.|| by A1, SEQ_4:def 1, A3; :: thesis: verum