intros x H'; try assumption.
cut (~ in_part x X).
intros H'0; try assumption.
absurd (in_part x X); auto with algebra.
simpl in |- *.
unfold X_def in |- *.
apply not_inpart_comp_r with X; auto with algebra.
unfold not in |- *.
intros H'0; try assumption.
absurd (in_part x X); auto with algebra.
apply not_inpart_comp_r with (Ap f x); auto with algebra.
Qed.
End Part_set_greater.

Theorem part_set_is_strictly_greater_than_set :
forall (E : Setoid) (f : MAP E (part_set E)), ~ surjective f.