let X, Y be RealNormSpace; :: thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators X,Y) holds 0 <= ||.f.||
let f be Point of (R_NormSpace_of_BoundedLinearOperators X,Y); :: thesis: 0 <= ||.f.||
reconsider g = f as bounded LinearOperator of X,Y by Def10;

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 VECTOR of X such that
A3: ( r = ||.(g . t).|| & ||.t.|| <= 1 ) by A2;
thus 0 <= r by A3, NORMSP_1:8; :: thesis: verum

end;

A4: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th32;
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;
(BoundedLinearOperatorsNorm X,Y) . f = sup (PreNorms g) by Th36;
hence 0 <= ||.f.|| by A6, NORMSP_1:def 1; :: thesis: verum