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 Th19:
   for M,N,M1 be LeftMod of R,
   f be Homomorphism of R,M,N, u be Homomorphism of R,M1,M holds
   f*u is Homomorphism of R,M1,N
   proof
     let M,N,M1 be LeftMod of R,
     f be Homomorphism of R,M,N,
     u be Homomorphism of R,M1,M;
A1:  for x1,y1 be Element of the carrier of M1, a be Element of R holds
     (f*u).(x1+y1) = (f*u).x1 + (f*u).y1 & a*(f*u).x1 = a*((f*u).x1)
     proof
       let x1,y1 be Element of the carrier of M1, a be Element of R;
reconsider ux1 = u.x1, uy1 = u.y1 as Element of the carrier of M;
       (f*u).(x1+y1) = f.(u.(x1+y1)) by FUNCT_2:15
       .= f.(ux1+uy1) by Def10,VECTSP_1:def 20
       .= f.(ux1)+ f.(uy1) by Def10,VECTSP_1:def 20
       .= (f*u).x1 + f.(u.y1) by FUNCT_2:15
       .= (f*u).x1 + (f*u).y1 by FUNCT_2:15;
       hence thesis;
     end;
     for x1 be Element of the carrier of M1, a be Element of R holds
     (f*u).(a*x1) = a*((f*u).x1)
     proof
       let x1 be Element of the carrier of M1, a be Element of R;
       reconsider ux1 = u.x1 as Element of the carrier of M;
       (f*u).(a*x1) = f.(u.(a*x1)) by FUNCT_2:15
       .= f.(a*ux1) by Def10,MOD_2:def 2
       .= a*f.(ux1) by Def10,MOD_2:def 2
       .= a*(f*u).x1 by FUNCT_2:15;
       hence thesis;
     end; then
     f*u is homogeneous;
     hence thesis by A1,Def10,VECTSP_1:def 20;
   end;
