let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for f being Point of (C_NormSpace_of_BoundedFunctions X,Y) holds 0 <= ||.f.||
let Y be ComplexNormSpace; :: thesis: for f being Point of (C_NormSpace_of_BoundedFunctions X,Y) holds 0 <= ||.f.||
let f be Point of (C_NormSpace_of_BoundedFunctions X,Y); :: thesis: 0 <= ||.f.||
reconsider g = f as bounded Function of X,the carrier of Y by Def5;

A1: now

let r be Real; :: thesis: ( r in PreNorms g implies 0 <= r )
assume A2: r in PreNorms g ; :: thesis: 0 <= r
consider t being Element of X such that
A3: r = ||.(g . t).|| by A2;
thus 0 <= r by A3, CLVECT_1:106; :: thesis: verum

end;

A4: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th14;
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 A6: 0 <= sup (PreNorms g) by A4, A5, SEQ_4:def 4;
(ComplexBoundedFunctionsNorm X,Y) . f = sup (PreNorms g) by Th18;
hence 0 <= ||.f.|| by A6, CLVECT_1:def 10; :: thesis: verum