intros.
inversion H.
set (B' := seq_set (Map_embed x)) in *.
assert (has_n_elements n B').
red in |- *.
exists (seq_set_seq (Map_embed x)).
split.
unfold B' in |- *.
simpl in |- *.
red in |- *; simpl in |- *.
split; intro; auto.
clear H3.
unfold subtype_image_equal in |- *.
simpl in |- *.
destruct x0.
simpl in subtype_prf.
rename subtype_elt into a.
inversion_clear subtype_prf.
exists x0.
simpl in |- *.
auto.
red in |- *.
intros.
simpl in |- *.
unfold subtype_image_equal in |- *.
simpl in |- *.
intro; (apply (H2 i j); auto with algebra).
assert (exists a : A, in_part a B' /\ ~ in_part a C).
apply has_extra_element with m n; auto with algebra.
assert (included B' B).
red in |- *.
intros a Ha.
simpl in Ha.
inversion_clear Ha.
apply in_part_comp_l with (subtype_elt (x x0)); auto with algebra.
inversion_clear H4.
rename x0 into a.
exists a.
inversion_clear H6.
split; auto.
Qed.

(* Specifications: *)
Lemma not_at_least_then_at_most :
forall (A : Setoid) (n : Nat),
~ has_at_least_n_elements (S n) A -> has_at_most_n_elements n A.