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

Lemma image_invimage :
forall A : part_set E, included A (invimage f (image f A)).