let X be non empty set ; :: thesis: for F being Point of (R_Normed_Algebra_of_BoundedFunctions X) holds 0 <= ||.F.||
let F be Point of (R_Normed_Algebra_of_BoundedFunctions X); :: thesis: 0 <= ||.F.||
F in BoundedFunctions X ;
then consider g being Function of X,REAL such that
A0: ( F = g & g | X is bounded ) ;

A1: now

let r be Real; :: thesis: ( r in PreNorms g implies 0 <= r )
assume r in PreNorms g ; :: thesis: 0 <= r
then ex t being Element of X st r = abs (g . t) ;
hence 0 <= r by COMPLEX1:132; :: thesis: verum

end;

A4: ( not PreNorms g is empty & PreNorms g is bounded_above ) by A0, ThB13;
consider r0 being set such that
A5: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A5;
0 <= r0 by A1, A5;
then 0 <= sup (PreNorms g) by A4, A5, SEQ_4:def 4;
hence 0 <= ||.F.|| by A0, ThB17; :: thesis: verum