let X be non empty set ; :: thesis: for F being Point of st F = 0. holds
0 =

let F be Point of ; :: thesis: ( F = 0. implies 0 = )
set z = X --> 0;
reconsider z = X --> 0c as Function of X,COMPLEX ;
F in ComplexBoundedFunctions X ;
then consider g being Function of X,COMPLEX such that
A1: g = F and
A2: g | X is bounded ;
A3: ( not PreNorms g is empty & PreNorms g is bounded_above ) by ;
consider r0 being object such that
A4: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A4;
A5: ( ( for s being Real st s in PreNorms g holds
s <= 0 ) implies upper_bound () <= 0 ) by SEQ_4:45;
assume A6: F = 0. ; :: thesis:
A7: 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
A8: r = |.(g . t).| ;
z = g by A6, A1, Th18;
then |.(g . t).| =
.= 0 ;
hence ( 0 <= r & r <= 0 ) by A8; :: thesis: verum
end;
then 0 <= r0 by A4;
then upper_bound () = 0 by ;
hence 0 = by A1, A2, Th13; :: thesis: verum