let X, Y be ComplexNormSpace; :: thesis: for f being Point of st f = 0. holds
0 =

let f be Point of ; :: thesis: ( f = 0. implies 0 = )
assume A1: f = 0. ; :: thesis:
thus ||.f.|| = 0 :: thesis: verum
proof
reconsider g = f as Lipschitzian LinearOperator of X,Y by Def7;
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 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 () <= 0 ) by SEQ_4:45;
A4: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th26;
A5: z = g by ;
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 such that
A7: r = ||.(g . t).|| and
||.t.|| <= 1 ;
||.(g . t).|| = ||.(0. Y).|| by
.= 0 by NORMSP_0:def 6 ;
hence ( 0 <= r & r <= 0 ) by A7; :: thesis: verum
end;
then 0 <= r0 by A2;
then upper_bound () = 0 by ;
then (BoundedLinearOperatorsNorm (X,Y)) . f = 0 by Th29;
hence ||.f.|| = 0 ; :: thesis: verum
end;