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;
consider r0 being set 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 Th32;
A3: (BoundedLinearOperatorsNorm X,Y) . f = upper_bound (PreNorms g) by Th36;

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 VECTOR of X st
( r = ||.(g . t).|| & ||.t.|| <= 1 ) ;
hence 0 <= r by NORMSP_1:8; :: thesis: verum

end;

then 0 <= r0 by A1;
then 0 <= upper_bound (PreNorms g) by A2, A1, SEQ_4:def 4;
hence 0 <= ||.f.|| by A3; :: thesis: verum