let X be non empty set ; :: thesis: for Y being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) holds 0 <= ||.f.||
let Y be RealNormSpace; :: thesis: for f being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) holds 0 <= ||.f.||
let f be Point of (R_NormSpace_of_BoundedFunctions (X,Y)); :: thesis: 0 <= ||.f.||
reconsider g = f as bounded Function of X, the carrier of Y by Def5;
consider r0 being object such that
A1: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A1;
A2: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th11;

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 ; :: thesis: verum

end;

then 0 <= r0 by A1;
then 0 <= upper_bound (PreNorms g) by A2, A1, SEQ_4:def 1;
hence 0 <= ||.f.|| by Th14; :: thesis: verum