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
   for R be comRing, M be strict LeftMod of R holds
   AbGrLMod(AbGr(M),canHom(M)) = M
   proof
     let R be comRing, M be strict LeftMod of R;
     AbGrLMod(AbGr(M),canHom(M)) is Submodule of M
     proof
      set N = AbGrLMod(AbGr(M),canHom(M));
      reconsider N as LeftMod of R;
A1:   0.N = 0.M;
A2:   the addF of N = (the addF of M) || (the carrier of N)
      proof
        the addF of N
        = (the addF of M) | dom(the addF of M) by RELAT_1:69
        .= (the addF of M) ||(the carrier of N) by FUNCT_2:def 1;
        hence thesis;
      end;
      the lmult of N = (the lmult of M)|[:the carrier of R,the carrier of N:]
      proof
        dom(the lmult of M) = [:the carrier of R,the carrier of M:]
          by FUNCT_2:def 1; then
        (the lmult of M)|[:the carrier of R,the carrier of N:]
         = the lmult of M by RELAT_1:69
        .= the lmult of N by Th35;
        hence thesis;
       end;
       hence thesis by A1,A2,VECTSP_4:def 2;
     end;
     hence thesis by VECTSP_4:31;
   end;
