inversion H;trivial.
inversion H0.
apply cons_is_set.
apply IHinsertion;trivial.
intro;apply H4;apply (insertion_inclusion A x l l');trivial.
Qed.

(** if (insertion x l l') and l' is a set then x is not in l *)
Lemma is_set_insertion_in : forall (A:Set)(l l':list A)(x:A),(insertion A x l l')->(is_set A l')->~(In x l).
induction l;simpl;try tauto;intros.