Qed.
Definition inj_submonoid : Hom (H:MONOID) G.
apply (Build_monoid_hom (E:=H) (F:=G) (monoid_sgroup_hom:=inj_subsgroup H)).
red in |- *.
auto with algebra.
Defined.

Lemma inj_subsmonoid_injective : injective inj_submonoid.
red in |- *.