unfold not in |- *.
intros E0 A x y H' H'0 H'1; try assumption.
apply H'.
apply in_part_comp_l with y; auto with algebra.
Qed.

(* Specifications: *)
Lemma not_in_comp_r :
forall (E : Setoid) (A B : part_set E) (x : E),
~ in_part x A -> Equal A B -> ~ in_part x B.