
theorem Th2:
   for M,N be AbGroup
   for f,g be Homomorphism of M,N holds ADD(M,N).(f,g) is Homomorphism of M,N
   proof
     let M,N be AbGroup;
     let f,g be Homomorphism of M,N;
     reconsider f,g
       as Element of Funcs(the carrier of M, the carrier of N) by FUNCT_2:8;
     reconsider F = ADD(M,N).(f,g)
       as Element of Funcs(the carrier of M, the carrier of N);
     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;
        F.(x+y) = f.(x+y) + g.(x+y) by Th1
        .= f.x+ f.y + g.(x+y) by VECTSP_1:def 20
        .= f.x+ f.y + (g.x+ g.y) by VECTSP_1:def 20
        .= f.x+ (f.y + (g.x+ g.y)) by RLVECT_1:def 3
        .= f.x+ ((f.y + g.x) + g.y) by RLVECT_1:def 3
        .= (f.x+ (g.x + f.y)) + g.y by RLVECT_1:def 3
        .= ((f.x+ g.x) + f.y) + g.y by RLVECT_1:def 3
        .= (F.x + f.y) + g.y by Th1
        .= F.x + (f.y + g.y) by RLVECT_1:def 3 .= F.x + F.y by Th1;
        hence thesis;
      end; then
      F is additive;
      hence thesis;
    end;
