
theorem Th25:
  for X being non empty set,
      a being Complex,
      F, G being Point of C_Normed_Algebra_of_BoundedFunctions(X) holds
  ((||.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.||)
proof
  let X be non empty set, a be Complex,
      F, G be Point of C_Normed_Algebra_of_BoundedFunctions(X);
A1:now
   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 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); then
A7:   z = g by A2,Th18;
A8:now
    let r be Real;
    assume r in PreNorms g;
    then consider t be Element of X such that
A9: r=|.(g.t).|;
    |.(g.t).| = |.(0).| by A7
            .= 0;
    hence 0 <= r & r <=0 by A9;
   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;
  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
     let t be Element of X;
     |. 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;
    end;
A19:now
     let r be Real;
     assume r in PreNorms h1;
     then ex t being Element of X st r = |.(h1.t).|;
     hence r <= ||.F.|| + ||.G.|| by A17;
    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;
  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
     let t be Element of X;
     |.(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;
    end;
A26:now
     let r be Real;
     assume r in PreNorms h1;
     then ex t being Element of X st r = |.(h1.t).|;
     hence r <= |.a.| * ||.F.|| by A25;
    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;
A29:  now
       let t be Element of X;
       |.(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;
      end;
A32:  now
       let r be Real;
       assume r in PreNorms f1;
       then ex t being Element of X st r = |.(f1.t).|;
       hence r <= ((|.a.|) " ) * ||.(a * F).|| by A29;
      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;
    end;
    suppose
A33:  a = 0;
      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;
    end;
    end;
    now
     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;
    now
     let t be Element of X;
     |.(g.t).| = 0 by A35,A36,A37,Th19;
     hence g.t = 0
              .= z.t;
    end;
    then g = z by FUNCT_2:63;
    hence F = 0.C_Normed_Algebra_of_BoundedFunctions(X) by A35,Th18;
  end;
  hence thesis by A1,A20,A10;
end;
