
theorem Th46:
  for a be Complex
  for X be non empty TopSpace
  for F,G be Point of C_Normed_Space_of_C_0_Functions(X) holds
  (||.F.|| = 0 iff F = 0.C_Normed_Space_of_C_0_Functions(X) ) &
  ||.a*F.|| = |.a.| * ||.F.|| & ||.F+G.|| <= ||.F.|| + ||.G.||
proof
  let a be Complex;
  let X be non empty TopSpace;
  let F,G be Point of C_Normed_Space_of_C_0_Functions(X);
A1:||.F.|| = 0 iff F = 0.C_Normed_Space_of_C_0_Functions(X)
  proof
    reconsider FB=F as
         Point of C_Normed_Algebra_of_BoundedFunctions the carrier of X
           by Th42;
A2: ||.FB.||=||.F.|| by FUNCT_1:49;
A3: 0.C_Normed_Algebra_of_BoundedFunctions the carrier of X
                      =X-->0 by CC0SP1:18;
A4: 0.C_Normed_Space_of_C_0_Functions(X) =X-->0 by Th45;
    ||.FB.|| = 0 iff
          FB = 0.C_Normed_Algebra_of_BoundedFunctions the carrier of X
                                                    by CC0SP1:25;
    hence thesis by A2,A3,A4;
  end;
A5:||.a*F.|| = |.a.| * ||.F.||
  proof
    reconsider FB=F as
         Point of C_Normed_Algebra_of_BoundedFunctions the carrier of X
           by Th42;
A6: ||.FB.||=||.F.|| by FUNCT_1:49;
A7:   a*FB=a*F by Lm14;
      reconsider aFB=a*FB as Point of C_Normed_Algebra_of_BoundedFunctions
                                                 the carrier of X;
    reconsider aF=a*F as Point of C_Normed_Space_of_C_0_Functions(X);
A8: ||.aFB.||=||.aF.|| by A7,FUNCT_1:49;
    ||.a*FB.|| = |.a.| * ||.FB.|| by CC0SP1:25;
    hence thesis by A6,A8;
  end;
  ||.F+G.|| <= ||.F.|| + ||.G.||
  proof
A9:   F in ComplexBoundedFunctions the carrier of X &
        G in ComplexBoundedFunctions the carrier of X by Th42;
      reconsider FB=F,GB=G as
        Point of C_Normed_Algebra_of_BoundedFunctions the carrier of X
                                             by A9;
A10:||.FB.||=||.F.|| & ||.GB.||=||.G.|| by FUNCT_1:49;
      FB+GB=F+G by Lm13; then
A11:  ||.FB+GB.||=||.F+G.|| by FUNCT_1:49;
      reconsider aFB=FB+GB as Point of C_Normed_Algebra_of_BoundedFunctions
                                                 the carrier of X;
    reconsider aF=F,aG=G as Point of C_Normed_Space_of_C_0_Functions(X);
    ||.FB+GB.|| <= ||.FB.|| + ||.GB.|| by CC0SP1:25;
    hence thesis by A11,A10;
  end;
  hence thesis by A1,A5;
end;
