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 Lm33:
   for f,g,h be Endomorphism of R,M st
   h = f*g holds AbGr(h) = AbGr(f) * AbGr(g)
   proof
     let f,g, h be Endomorphism of R,M;
     assume
A1:  h = f*g;
     for x being Element of the carrier of AbGr(M) holds
     (AbGr(h)).x = (AbGr(f) * (AbGr(g))).x
     proof
       let x be Element of the carrier of AbGr(M);
       reconsider x2 = x as Element of the carrier of M;
A2:    (AbGr(g)).x = g.x2 by Def26;
A3:    (AbGr(h)).x = h.x2 by Def26;
       reconsider x0 = x as Element of the carrier of M;
A4:    (AbGr(h)).x = f.(g.x0) by A1,A3,FUNCT_2:15;
       reconsider y0 = g.x0 as Element of the carrier of AbGr(M);
       reconsider y1 = y0 as Element of the carrier of M;
       (AbGr(f)).y0 = f.y1 by Def26;
       hence thesis by A4,A2,FUNCT_2:15;
     end;
     hence thesis;
   end;
