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 Th35:
   for R be comRing, M be LeftMod of R holds
   LModlmult(AbGr(M),canHom(M)) = the lmult of M
   proof
     let R be comRing, M be LeftMod of R;
     set F = LModlmult(AbGr(M),canHom(M));
     for z be object st z in [:the carrier of R,the carrier of M:] holds
     F.z = (the lmult of M).z
     proof
       let z be object;
       assume
A1:    z in [:the carrier of R,the carrier of M:];
       consider x,y be object such that
A2:    x in the carrier of R & y in the carrier of M & z = [x,y]
       by A1,ZFMISC_1:def 2;
       reconsider y0 = y as Element of M by A2;
       reconsider y1 = y as Element of AbGr(M) by A2;
       reconsider x0 = x as Element of R by A2;
       consider h be Endomorphism of AbGr(M) such that
A3:    h = (canHom(M)).x0 & F.(x0,y0) = h.y0 by Def12;
       consider f be Endomorphism of R,M such that
A4:    f = (curry (the lmult of M)).x0 & (canHom(M)).x0 = AbGr(f) by Def27;
       reconsider y2 = y1 as Element of the carrier of M;
A5:    (AbGr(f)).y1 = f.y2 by Def26;
       F.(x0,y0) = (the lmult of M).(x0,y0) by A4,A5,A3,LOPBAN_8:7;
       hence thesis by A2;
     end;
     hence thesis;
   end;
