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 Th37:
   for R be comRing,M be LeftMod of R holds rho(M) is additive homogeneous
   proof
     let R be comRing,M be LeftMod of R;
     reconsider EM = AbGrLMod(AbGr(M),canHom(M)) as LeftMod of R;
     for x be Element of the carrier of M, a be Element of R holds
     (rho(M)).(a*x) = a*(rho(M)).x
     proof
       let x be Element of the carrier of M, a be Element of R;
       reconsider z = a*x as Element of the carrier of M;
       consider F be Endomorphism of AbGr(M) such that
A7:    F = (canHom(M)).a & (LModlmult(AbGr(M),canHom(M))).(a,x) = F.x by Def12;
       consider f be Endomorphism of R,M such that
A8:    f = (curry (the lmult of M)).a & (canHom(M)).a = AbGr(f) by Def27;
       reconsider x1 = x as Element of the carrier of AbGr(M);
       reconsider x2 = x1 as Element of the carrier of M;
       (AbGr(f)).x1 = f.x2 by Def26;
       hence thesis by LOPBAN_8:7,A8,A7;
     end;
     hence thesis;
   end;
