intros.
apply subset_has_at_least_n_elements with B; auto with algebra.
apply has_n_elements_then_has_at_least_n_elements; auto with algebra.
Qed.

Lemma has_extra_element_strong :
forall (A : Setoid) (B C : part_set A) (m n : Nat),
has_at_least_n_elements n B ->
has_n_elements m C -> m < n -> exists a : A, in_part a B /\ ~ in_part a C.