unfold comp_map_map in |- *; simpl in |- *.
unfold Map_eq in |- *; simpl in |- *; auto with algebra.
Qed.
Hint Resolve comp_map_assoc: algebra.
Comments "We define now the identity map:".
Definition Id : forall A : Setoid, MAP A A.
intros A; try assumption.
apply (Build_Map (A:=A) (B:=A) (Ap:=fun x : A => x)).
red in |- *.
auto with algebra.
Defined.
Comments "Identity map is a unit element for composition:".

Lemma Id_unit_r :
forall (A B : Setoid) (f : MAP A B), Equal (comp_map_map f (Id A)) f.