reserve M,N for AbGroup;

theorem Th5:
   for M be AbGroup
   for f,g be Endomorphism of M holds (FuncComp(M)).(f,g) is Endomorphism of M
   proof
     let M be AbGroup;
     let f,g be Endomorphism of M;
  reconsider f, g as Element of Funcs(the carrier of M, the carrier of M)
  by FUNCT_2:8;
  reconsider F = (FuncComp(M)).(f,g)
  as Element of Funcs(the carrier of M, the carrier of M);
     F is additive
     proof
       for x,y being Element of the carrier of M holds F.(x+y) = F.x + F.y
       proof
         let x,y be Element of the carrier of M;
         reconsider z = x + y as Element of the carrier of M;
A1:      F.y = (f*g).y by Def4;
         F.(x+y) = (f*g).z by Def4
         .= f.(g.(x+y)) by FUNCT_2:15 .= f.(g.x+g.y) by VECTSP_1:def 20
         .= f.(g.x)+ f.(g.y) by VECTSP_1:def 20
         .= (f*g).x + f.(g.y) by FUNCT_2:15 .= (f*g).x + (f*g).y by FUNCT_2:15
         .= F.x + F.y by Def4,A1;
         hence thesis;
       end;
       hence thesis;
     end;
     hence thesis;
   end;
