simpl in |- *.
unfold eq_part, add_part, minus_part, union, diff, single in |- *;
simpl in |- *.
intro.
case A; simpl in |- *.
intros a pa.
intros.
intuition.
apply pa with x; auto with algebra.
apply NNPP; intuition.
Qed.
Hint Resolve minus_add: algebra.

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