unfold included in |- *.
simpl in |- *; auto with algebra.
intros A x H'; exists x; split; [ try assumption | idtac ]; auto with algebra.
Qed.
Hint Resolve invimage_image image_invimage: algebra.

Lemma invimage_image_invimage :
forall A : part_set F,
Equal (invimage f (image f (invimage f A))) (invimage f A).