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.||

then upper_bound (PreNorms g) = 0 by A7, A3, A4, A5, SEQ_4:def 1;

hence 0 = ||.F.|| by A1, A2, Th13; :: thesis: verum

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 )

then
0 <= r0
by A4;( 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;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

then upper_bound (PreNorms g) = 0 by A7, A3, A4, A5, SEQ_4:def 1;

hence 0 = ||.F.|| by A1, A2, Th13; :: thesis: verum