let X, Y, Z be RealNormSpace; :: thesis: for f being Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) holds 0 <= ||.f.||
let f be Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)); :: thesis: 0 <= ||.f.||
reconsider g = f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
consider r0 being object such that
A1: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A1;
A3: (BoundedBilinearOperatorsNorm (X,Y,Z)) . f = upper_bound (PreNorms g) by Th30;

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 VECTOR of X ex s being VECTOR of Y st
( r = ||.(g . (t,s)).|| & ||.t.|| <= 1 & ||.s.|| <= 1 ) ;
hence 0 <= r ; :: thesis: verum

end;

then 0 <= r0 by A1;
hence 0 <= ||.f.|| by A1, A3, SEQ_4:def 1; :: thesis: verum