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;

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

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

then
0 <= r0
by A1;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;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

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