unfold injective in |- *.
intros A B f H'; try assumption.
cut
(ex
(fun x : A =>
~ (forall y : A, ~ (~ Equal x y /\ Equal (Ap f x) (Ap f y))))).
intros H'0; try assumption.
elim H'0; intros x E; try exact E; clear H'0.
exists x; try assumption.
cut (ex (fun y : A => ~ ~ (~ Equal x y /\ Equal (Ap f x) (Ap f y)))).
intros H'0; try assumption.
elim H'0; intros y E0; try exact E0; clear H'0.
exists y; try assumption.
apply NNPP; auto with *.
apply
not_all_ex_not
with (P := fun y : A => ~ (~ Equal x y /\ Equal (Ap f x) (Ap f y)));
auto with *.
apply
not_all_ex_not
with
(P := fun x : A =>
forall y : A, ~ (~ Equal x y /\ Equal (Ap f x) (Ap f y)));
auto with *.
red in |- *.
intros H'0; try assumption.
apply H'.
intros x y H'1; try assumption.
specialize  H'0 with (n := x) (y := y); rename H'0 into H'3; try exact H'3.
apply NNPP; tauto.
Qed.

Lemma not_surjective_prop :
forall (A B : Setoid) (f : MAP A B),
~ surjective f -> exists y : B, ~ in_part y (image_map f).