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 Th20:
   for R,M,N
   for a be Element of the carrier of R, g be Homomorphism of R,M,N holds
   LMULT(M,N).[a,g] is Homomorphism of R,M,N
   proof
     let R,M,N;
     let a be Element of the carrier of R, g be Homomorphism of R,M,N;
  reconsider g as Element of Funcs(the carrier of M, the carrier of N)
  by FUNCT_2:8;
  reconsider F = LMULT(M,N).[a,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.z = a*(g.(x+y)) by Def16
       .= a*(g.x +g.y) by Def10,VECTSP_1:def 20
       .= a*g.x + a*g.y by VECTSP_1:def 14 .= F.x + a*g.y by Def16
       .= F.x + F.y by Def16;
       hence thesis;
     end;
     for b be Element of R, x being Element of the carrier of M holds
     F.(b*x) = b*F.x
     proof
       let b be Element of R, x being Element of the carrier of M;
       reconsider z = b*x as Element of the carrier of M;
       F.z = a*(g.(b*x))by Def16
       .= a*(b*g.x) by Def10,MOD_2:def 2
       .= (a*b)*g.x by VECTSP_1:def 16
       .= b*(a*g.x) by VECTSP_1:def 16 .= b*F.x by Def16;
       hence thesis;
     end;then
     F is homogeneous;
     hence thesis by A1,Def10,VECTSP_1:def 20;
   end;
