intros a x H'; try assumption.
apply (submodule_prop (R:=R) (M:=M) (s:=N)); auto with algebra.
Qed.
Definition inj_submodule : Hom (N:MODULE R) M.
apply
(BUILD_HOM_MODULE (R:=R) (Mod:=N:MODULE R) (Mod':=M)
(ff:=fun x : N => subtype_elt x)); auto with algebra.
Defined.

Lemma inj_submodule_injective : injective inj_submodule.
red in |- *.