reserve M,N for AbGroup;
 reserve R for Ring;
 reserve r for Element of R;
reserve M,N for LeftMod of R;
reserve f,g,h for Element of Funcs(the carrier of M, the carrier of N);
reserve a,b for Element of the carrier of R;
reserve R for comRing;
reserve M,M1,N,N1 for LeftMod of R;

theorem Th18:
   for R,M,N
   for f,g be Homomorphism of R,M,N holds
   ADD(M,N).(f,g) is Homomorphism of R,M,N
   proof
     let R,M,N;
     let f,g be Homomorphism of R,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);
A1:  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 Th15
       .= f.x+ f.y + g.(x+y) by Def10,VECTSP_1:def 20
       .= f.x+ f.y + (g.x+ g.y) by Def10,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 Th15
       .= F.x + (f.y + g.y) by RLVECT_1:def 3
       .= F.x + F.y by Th15;
       hence thesis;
     end;
     for a be Element of R, x being Element of the carrier of M holds
     F.(a*x) = a*F.x
     proof
       let a be Element of R, x be Element of the carrier of M;
       reconsider z = a*x as Element of the carrier of M;
       F.z = f.(a*x) + g.(a*x) by Th15
       .= a*f.x + g.(a*x) by Def10,MOD_2:def 2
       .= a*f.x + a*g.x by Def10,MOD_2:def 2
       .= a*(f.x + g.x) by VECTSP_1:def 14
       .= a*F.x by Th15;
       hence thesis;
     end; then
     F is homogeneous;
     hence thesis by A1, Def10,VECTSP_1:def 20;
   end;
