let X be non empty set ; :: thesis:
let a be Complex; :: thesis:
let F, G be Point of (C_Normed_Algebra_of_BoundedFunctions X); :: thesis:

A1: now :: thesis:

set z = X --> 0c;
reconsider z = X --> 0c as Function of X,COMPLEX ;
F in ComplexBoundedFunctions X ;
then consider g being Function of X,COMPLEX such that
A2: F = g and
A3: g | X is bounded ;
A4: ( not PreNorms g is empty & PreNorms g is bounded_above ) by A3, Th11;
consider r0 being object such that
A5: r0 in PreNorms g by XBOOLE_0:def 1;
reconsider r0 = r0 as Real by A5;
A6: ( ( for s being Real st s in PreNorms g holds
s <= 0 ) implies upper_bound (PreNorms g) <= 0 ) by SEQ_4:45;
assume F = 0. (C_Normed_Algebra_of_BoundedFunctions X) ; :: thesis:
then A7: z = g by A2, Th18;

A8: now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms g ; :: thesis:
then consider t being Element of X such that
A9: r = |.(g . t).| ;
|.(g . t).| = |.0.| by A7
.= 0 ;
hence ( 0 <= r & r <= 0 ) by A9; :: thesis:

end;

then 0 <= r0 by A5;
then upper_bound (PreNorms g) = 0 by A8, A4, A6, A5, SEQ_4:def 1;
hence ||.F.|| = 0 by A2, A3, Th13; :: thesis:

end;

A10: ||.(F + G).|| <= ||.F.|| + ||.G.||

proof

F + G in ComplexBoundedFunctions X ;
then consider h1 being Function of X,COMPLEX such that
A11: h1 = F + G and
A12: h1 | X is bounded ;
G in ComplexBoundedFunctions X ;
then consider g1 being Function of X,COMPLEX such that
A13: g1 = G and
A14: g1 | X is bounded ;
F in ComplexBoundedFunctions X ;
then consider f1 being Function of X,COMPLEX such that
A15: f1 = F and
A16: f1 | X is bounded ;

A17: now :: thesis:

let t be Element of X; :: thesis:
( |.(f1 . t).| <= ||.F.|| & |.(g1 . t).| <= ||.G.|| ) by A15, A16, A13, A14, Th19;
then A18: |.(f1 . t).| + |.(g1 . t).| <= ||.F.|| + ||.G.|| by XREAL_1:7;
( |.(h1 . t).| = |.((f1 . t) + (g1 . t)).| & |.((f1 . t) + (g1 . t)).| <= |.(f1 . t).| + |.(g1 . t).| ) by A15, A13, A11, Th22, COMPLEX1:56;
hence |.(h1 . t).| <= ||.F.|| + ||.G.|| by A18, XXREAL_0:2; :: thesis:

end;

A19: now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms h1 ; :: thesis:
then ex t being Element of X st r = |.(h1 . t).| ;
hence r <= ||.F.|| + ||.G.|| by A17; :: thesis:

end;

( ( for s being Real st s in PreNorms h1 holds
s <= ||.F.|| + ||.G.|| ) implies upper_bound (PreNorms h1) <= ||.F.|| + ||.G.|| ) by SEQ_4:45;
hence ||.(F + G).|| <= ||.F.|| + ||.G.|| by A11, A12, A19, Th13; :: thesis:

end;

A20: ||.(a * F).|| = |.a.| * ||.F.||

proof

F in ComplexBoundedFunctions X ;
then consider f1 being Function of X,COMPLEX such that
A21: f1 = F and
A22: f1 | X is bounded ;
a * F in ComplexBoundedFunctions X ;
then consider h1 being Function of X,COMPLEX such that
A23: h1 = a * F and
A24: h1 | X is bounded ;

A25: now :: thesis:

let t be Element of X; :: thesis:
|.(h1 . t).| = |.(a * (f1 . t)).| by A21, A23, Th23;
then |.(h1 . t).| = |.a.| * |.(f1 . t).| by COMPLEX1:65;
hence |.(h1 . t).| <= |.a.| * ||.F.|| by A21, A22, Th19, XREAL_1:64; :: thesis:

end;

A26: now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms h1 ; :: thesis:
then ex t being Element of X st r = |.(h1 . t).| ;
hence r <= |.a.| * ||.F.|| by A25; :: thesis:

end;

( ( for s being Real st s in PreNorms h1 holds
s <= |.a.| * ||.F.|| ) implies upper_bound (PreNorms h1) <= |.a.| * ||.F.|| ) by SEQ_4:45;
then A27: ||.(a * F).|| <= |.a.| * ||.F.|| by A23, A24, A26, Th13;

per cases ;

suppose A28: a <> 0 ; :: thesis:

A29: now :: thesis:

let t be Element of X; :: thesis:
|.(a ").| = |.(1 / a).| ;
then A30: |.(a ").| = 1 / |.a.| by COMPLEX1:80;
h1 . t = a * (f1 . t) by A21, A23, Th23;
then (a ") * (h1 . t) = ((a ") * a) * (f1 . t) ;
then A31: (a ") * (h1 . t) = 1 * (f1 . t) by A28, XCMPLX_0:def 7;
( |.((a ") * (h1 . t)).| = |.(a ").| * |.(h1 . t).| & 0 <= |.(a ").| ) by COMPLEX1:65;
hence |.(f1 . t).| <= (|.a.| ") * ||.(a * F).|| by A23, A24, A31, A30, Th19, XREAL_1:64; :: thesis:

end;

A32: now :: thesis:

let r be Real; :: thesis:
assume r in PreNorms f1 ; :: thesis:
then ex t being Element of X st r = |.(f1 . t).| ;
hence r <= (|.a.| ") * ||.(a * F).|| by A29; :: thesis:

end;

( ( for s being Real st s in PreNorms f1 holds
s <= (|.a.| ") * ||.(a * F).|| ) implies upper_bound (PreNorms f1) <= (|.a.| ") * ||.(a * F).|| ) by SEQ_4:45;
then ||.F.|| <= (|.a.| ") * ||.(a * F).|| by A21, A22, A32, Th13;
then |.a.| * ||.F.|| <= |.a.| * ((|.a.| ") * ||.(a * F).||) by XREAL_1:64;
then |.a.| * ||.F.|| <= (|.a.| * (|.a.| ")) * ||.(a * F).|| ;
then |.a.| * ||.F.|| <= 1 * ||.(a * F).|| by A28, XCMPLX_0:def 7;
hence ||.(a * F).|| = |.a.| * ||.F.|| by A27, XXREAL_0:1; :: thesis:

end;

suppose A33: a = 0 ; :: thesis:

reconsider fz = F as VECTOR of (C_Algebra_of_BoundedFunctions X) ;
a * fz = (a + a) * fz by A33
.= (a * fz) + (a * fz) by CLVECT_1:def 3 ;
then 0. (C_Algebra_of_BoundedFunctions X) = (- (a * fz)) + ((a * fz) + (a * fz)) by VECTSP_1:16;
then 0. (C_Algebra_of_BoundedFunctions X) = ((- (a * fz)) + (a * fz)) + (a * fz) by RLVECT_1:def 3;
then 0. (C_Algebra_of_BoundedFunctions X) = (0. (C_Algebra_of_BoundedFunctions X)) + (a * fz) by VECTSP_1:16;
then A34: a * F = 0. (C_Normed_Algebra_of_BoundedFunctions X) by RLVECT_1:4;
|.a.| * ||.F.|| = 0 * ||.F.|| by A33;
hence ||.(a * F).|| = |.a.| * ||.F.|| by A34, Th21; :: thesis:

end;

now :: thesis:

set z = X --> 0c;
reconsider z = X --> 0c as Function of X,COMPLEX ;
F in ComplexBoundedFunctions X ;
then consider g being Function of X,COMPLEX such that
A35: F = g and
A36: g | X is bounded ;
assume A37: ||.F.|| = 0 ; :: thesis:

now :: thesis:

let t be Element of X; :: thesis:
|.(g . t).| = 0 by A35, A36, A37, Th19;
hence g . t = 0
.= z . t ;
:: thesis:

end;

then g = z by FUNCT_2:63;
hence F = 0. (C_Normed_Algebra_of_BoundedFunctions X) by A35, Th18; :: thesis:

end;

hence ( ( ||.F.|| = 0 implies F = 0. (C_Normed_Algebra_of_BoundedFunctions X) ) & ( F = 0. (C_Normed_Algebra_of_BoundedFunctions X) implies ||.F.|| = 0 ) & ||.(a * F).|| = |.a.| * ||.F.|| & ||.(F + G).|| <= ||.F.|| + ||.G.|| ) by A1, A20, A10; :: thesis: