let X, Y, Z be RealNormSpace; :: thesis: for f being Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) st f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) holds
0 = ||.f.||
let f be Point of (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)); :: thesis: ( f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) implies 0 = ||.f.|| )
assume A1: f = 0. (R_NormSpace_of_BoundedBilinearOperators (X,Y,Z)) ; :: thesis: 0 = ||.f.||
reconsider g = f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
set z = the carrier of [:X,Y:] --> (0. Z);
reconsider z = the carrier of [:X,Y:] --> (0. Z) as Function of the carrier of [:X,Y:], the carrier of Z ;
consider r0 being object such that
A2: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A2;
A3: ( ( for s being Real st s in PreNorms g holds
s <= 0 ) implies upper_bound (PreNorms g) <= 0 ) by SEQ_4:45;
A5: z = g by A1, Th31;

A6: now :: thesis: for r being Real st r in PreNorms g holds
( 0 <= r & r <= 0 )

let r be Real; :: thesis: ( r in PreNorms g implies ( 0 <= r & r <= 0 ) )
assume r in PreNorms g ; :: thesis: ( 0 <= r & r <= 0 )
then consider t being VECTOR of X, s being VECTOR of Y such that
A7: r = ||.(g . (t,s)).|| and
( ||.t.|| <= 1 & ||.s.|| <= 1 ) ;
[t,s] is Point of [:X,Y:] ;
then g . (t,s) = 0. Z by FUNCOP_1:7, A5;
hence ( 0 <= r & r <= 0 ) by A7; :: thesis: verum

end;

then 0 <= r0 by A2;
then upper_bound (PreNorms g) = 0 by A6, A2, A3, SEQ_4:def 1;
hence 0 = ||.f.|| by Th30; :: thesis: verum