intros x y z H' H'0; try assumption.
elim Strict_Rel_is_Strict_Included.
unfold contains.
intros H'1 H'2; try assumption.
apply H'1.
apply Strict_Rel_Transitive_with_Rel with (y := y); auto with sets.
Qed.

Lemma Strict_inclusion_is_transitive_with_inclusion_left :
forall x y z:Ensemble U,
Included U x y -> Strict_Included U y z -> Strict_Included U x z.