Section Def.
Variable M : MONOID.
Variable S : SET.
Definition operation := Hom M (Endo_SET S).
Variable op : operation.
intros x y s; try assumption.
apply
Trans
with
(Ap
(sgroup_law (Endo_SET S) (Ap (sgroup_map (monoid_sgroup_hom op)) x)
(Ap (sgroup_map (monoid_sgroup_hom op)) y)) s);
auto with algebra.
cut (Equal (op (sgroup_law _ x y)) (sgroup_law (Endo_SET S) (op x) (op y))).
auto with algebra.
apply (sgroup_hom_prf op).
Qed.

Lemma operation_unit : forall s : S, Equal (op (monoid_unit M) s) s.