simpl in |- *; auto with algebra.
Qed.
End Restrictions1.
Hint Resolve restrict_prop: algebra.
Section Inverse_image1.
Variable E F : Setoid.
Section Inverse_image1_1.
Variable f : MAP E F.
Definition invimage : part_set F -> part_set E.
intros A.
apply (Build_Predicate (Pred_fun:=fun x : E => in_part (f x) A)).
red in |- *.
intros x y H' H'0; try assumption.
apply in_part_comp_l with (Ap f x); auto with algebra.
Defined.
End Inverse_image1_1.
Variable f : MAP E F.

Lemma invimage_in :
forall (A : part_set F) (x : E), in_part x (invimage f A) -> in_part (f x) A.