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;

then upper_bound (PreNorms g) = 0 by A6, A2, A3, SEQ_4:def 1;

hence 0 = ||.f.|| by Th30; :: thesis: verum

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 )

then
0 <= r0
by A2;( 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;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

then upper_bound (PreNorms g) = 0 by A6, A2, A3, SEQ_4:def 1;

hence 0 = ||.f.|| by Th30; :: thesis: verum