simpl in |- *.
unfold eq_part in |- *.
intros A x; split; [ idtac | intros H'; try assumption ].
simpl in |- *.
intros H'; try assumption.
elim H'; intros x0 E0; elim E0; intros H'0 H'1; try exact H'0; clear E0 H'.
apply in_part_comp_l with (Ap f x0); auto with algebra.
auto with algebra.
Qed.

Lemma image_invimage_image :
forall A : part_set E, Equal (image f (invimage f (image f A))) (image f A).