let X be non empty set ; :: thesis: for Y being ComplexNormSpace
for f being Point of () st f = 0. () holds
0 =

let 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 bounded Function of X, the carrier of Y by Def5;
set z = X --> (0. Y);
reconsider z = X --> (0. Y) as Function 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 Th12;
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 Element of X such that
A7: r = ||.(g . t).|| ;
||.(g . t).|| = ||.(0. Y).|| by
.= 0 ;
hence ( 0 <= r & r <= 0 ) by A7; :: thesis: verum
end;
then 0 <= r0 by A2;
then upper_bound () = 0 by ;
then (ComplexBoundedFunctionsNorm (X,Y)) . f = 0 by Th15;
hence ||.f.|| = 0 ; :: thesis: verum
end;