Section Union1.
Variable E : Setoid.
Definition union : 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; try exact H'1; clear H' | intros H'1; clear H' ].
left; try assumption.
apply in_part_comp_l with x; auto with algebra.
right; try assumption.
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.
Parameter
in_part_union_or :
forall (A B : part_set E) (x : E),
in_part x A \/ in_part x B -> in_part x (union A B).
intros A B x; try assumption.
unfold union in |- *; intuition.
Qed.
unfold union in |- *; simpl in |- *; intuition.
Qed.

Lemma included2_union :
forall A B C : part_set E,
included A C -> included B C -> included (union A B) C.