Proof fun cs
(H  : num_containers cs < num_things cs)
(H0 : containers_ok cs)
=> le_not_lt
(num_things cs)
(num_containers cs)
(num_things_num_containers cs H0)
H.

(** This lemma proves that if we have a collection of containers and the number of things in the containers is greater than the number of containers, then one of the containers cannot be empty or have only one item in it. *)
Lemma lemma_1
:  forall cs : containers,
num_things cs > num_containers cs ->
Exists (fun c : container => ~ container_ok c) cs.