let X, Y be RealNormSpace; :: thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators X,Y) st f = 0. (R_NormSpace_of_BoundedLinearOperators X,Y) holds
0 = ||.f.||
let f be Point of (R_NormSpace_of_BoundedLinearOperators X,Y); :: thesis: ( f = 0. (R_NormSpace_of_BoundedLinearOperators X,Y) implies 0 = ||.f.|| )
assume A1: f = 0. (R_NormSpace_of_BoundedLinearOperators X,Y) ; :: thesis: 0 = ||.f.||
thus ||.f.|| = 0 :: thesis: verum

proof

reconsider g = f as bounded LinearOperator of X,Y by Def10;
set z = the carrier of X --> (0. Y);
reconsider z = the carrier of X --> (0. Y) as Function of the carrier of X,the carrier of Y ;
consider r0 being set such that
A2: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A2;
A3: ( ( for s being real number st s in PreNorms g holds
s <= 0 ) implies sup (PreNorms g) <= 0 ) by SEQ_4:62;
A4: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th32;
A5: z = g by A1, Th37;

A6: now

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 such that
A7: r = ||.(g . t).|| and
||.t.|| <= 1 ;
||.(g . t).|| = ||.(0. Y).|| by A5, FUNCOP_1:13
.= 0 ;
hence ( 0 <= r & r <= 0 ) by A7; :: thesis: verum

end;

then 0 <= r0 by A2;
then sup (PreNorms g) = 0 by A6, A4, A2, A3, SEQ_4:def 4;
then (BoundedLinearOperatorsNorm X,Y) . f = 0 by Th36;
hence ||.f.|| = 0 by NORMSP_1:def 1; :: thesis: verum

end;