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
A1: F = g and
A2: g | X is bounded ;
A3: ( not PreNorms g is empty & PreNorms g is bounded_above ) by A2, Th17;
consider r0 being object such that
A4: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A4;

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 Element of X st r = |.(g . t).| ;
hence 0 <= r by COMPLEX1:46; :: thesis: verum

end;

then 0 <= r0 by A4;
then 0 <= upper_bound (PreNorms g) by A3, A4, SEQ_4:def 1;
hence 0 <= ||.F.|| by A1, A2, Th20; :: thesis: verum