let X be non empty set ; :: thesis: for F being Point of (C_Normed_Algebra_of_BoundedFunctions X) st F = 0. (C_Normed_Algebra_of_BoundedFunctions X) holds
0 = ||.F.||
let F be Point of (C_Normed_Algebra_of_BoundedFunctions X); :: thesis: ( F = 0. (C_Normed_Algebra_of_BoundedFunctions X) implies 0 = ||.F.|| )
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 A2, Th11;
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 (PreNorms g) <= 0 ) by SEQ_4:45;
assume A6: F = 0. (C_Normed_Algebra_of_BoundedFunctions X) ; :: thesis: 0 = ||.F.||

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.|
.= 0 ;
hence ( 0 <= r & r <= 0 ) by A8; :: thesis: verum

end;

then 0 <= r0 by A4;
then upper_bound (PreNorms g) = 0 by A7, A3, A4, A5, SEQ_4:def 1;
hence 0 = ||.F.|| by A1, A2, Th13; :: thesis: verum