intro A.
case A.
simpl in |- *; unfold pred_compatible, eq_part, in_part in |- *;
simpl in |- *.
intuition eauto.
Qed.
Hint Resolve add_in: algebra.

Lemma add_part_in_el_diff :
forall (A : part_set E) (x y : E),
in_part y (add_part A x) -> ~ Equal y x -> in_part y A.