intros.
rewrite <- H1.
red in |- *.
red in H.
inversion_clear H.
rename x into Bseq.
exists (Map_to_equal_subsets H0 Bseq).
red in |- *; intros; intro; red in H2; (apply (H2 i j); auto with algebra).
simpl in |- *.
red in |- *.
apply Trans with (subtype_elt (Bseq i)); auto with algebra.
apply Trans with (subtype_elt (Bseq j)); auto with algebra.
apply Trans with (subtype_elt (Map_to_equal_subsets H0 Bseq i));
auto with algebra.
apply Trans with (subtype_elt (Map_to_equal_subsets H0 Bseq j));
auto with algebra.
Qed.

Lemma has_n_elements_then_has_at_least_n_elements :
forall (A : Setoid) (n : Nat),
has_n_elements n A -> has_at_least_n_elements n A.