let X be non empty set ; :: thesis: for F being Point of (C_Normed_Algebra_of_BoundedFunctions X) holds 0 <= ||.F.||

let F be Point of (C_Normed_Algebra_of_BoundedFunctions X); :: thesis: 0 <= ||.F.||

F in ComplexBoundedFunctions X ;

then consider g being Function of X,COMPLEX such that

A1: F = g and

A2: g | X is bounded ;

consider r0 being object such that

A3: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r1 = r0 as Real by A3;

( not PreNorms g is empty & PreNorms g is bounded_above ) by Th11, A2;

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

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

let F be Point of (C_Normed_Algebra_of_BoundedFunctions X); :: thesis: 0 <= ||.F.||

F in ComplexBoundedFunctions X ;

then consider g being Function of X,COMPLEX such that

A1: F = g and

A2: g | X is bounded ;

consider r0 being object such that

A3: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r1 = r0 as Real by A3;

now :: thesis: for r being Real st r in PreNorms g holds

0 <= r

then A4:
0 <= r1
by A3;0 <= r

let r be Real; :: thesis: ( r in PreNorms g implies 0 <= r )

assume r in PreNorms g ; :: thesis: 0 <= r

then ex t being Element of X st r = |.(g . t).| ;

hence 0 <= r ; :: thesis: verum

end;assume r in PreNorms g ; :: thesis: 0 <= r

then ex t being Element of X st r = |.(g . t).| ;

hence 0 <= r ; :: thesis: verum

( not PreNorms g is empty & PreNorms g is bounded_above ) by Th11, A2;

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

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