Section Inter1.
Variable E : Setoid.
Definition inter : part_set E -> part_set E -> part_set E.
intros A B.
apply (Build_Predicate (Pred_fun:=fun x : E => in_part x A /\ in_part x B)).
red in |- *.
intros x y H' H'0; try assumption.
elim H'; intros H'1 H'2; try exact H'1; clear H'.
split; [ try assumption | idtac ].
apply in_part_comp_l with x; auto with algebra.
apply in_part_comp_l with x; auto with algebra.
Defined.
unfold included in |- *; simpl in |- *; intuition.
Qed.
unfold included in |- *; simpl in |- *; intuition.
Qed.
simpl in |- *; intuition.
Qed.
simpl in |- *; intuition.
Qed.
simpl in |- *. intuition.
Qed.
simpl in |- *; intuition.
Qed.
simpl in |- *; intuition.
Qed.

Lemma included2_inter :
forall A B C : part_set E,
included A C -> included B C -> included (inter A B) C.