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 Th38:
   for R be comRing,M be LeftMod of R holds rho(M) is one-to-one onto
   proof
     let R be comRing,M be LeftMod of R;
     reconsider EM = AbGrLMod(AbGr(M),canHom(M)) as LeftMod of R;
     set f = rho(M);
     for y being object st y in the carrier of EM
     ex x being object st x in the carrier of M & y = f.x
     proof
       let y be object;
       assume y in the carrier of EM; then
       consider x be object such that
A6:    x = y & x in the carrier of M;
       reconsider x0 = x as Element of the carrier of M by A6;
       f.x0 = x0;
       hence thesis by A6;
     end;
     hence thesis by FUNCT_2:10;
   end;
