simpl in |- *.
unfold eq_part, minus_part, diff, single in |- *; simpl in |- *.
intro A.
case A; unfold pred_compatible in |- *; simpl in |- *.
intros a pa x neg_a_x x0.
generalize (pa x0 x).
intuition auto with algebra.
Qed.
Hint Resolve minus_not_in: algebra.

Lemma minus_trans_not_in :
forall (A : part_set E) (x y : E),
~ in_part y A -> ~ in_part y (minus_part A x).