Section The_power_set_partial_order.
Variable U : Type.
Inductive Power_set (A:Ensemble U) : Ensemble (Ensemble U) :=
Definition_of_Power_set :
forall X:Ensemble U, Included U X A -> In (Ensemble U) (Power_set A) X.
Hint Resolve Definition_of_Power_set : core.
intro X; red.
intros x H'; elim H'.
Qed.
Hint Resolve Empty_set_minimal : core.
intro X.
apply Inhabited_intro with (Empty_set U); auto with sets.
Qed.
Hint Resolve Power_set_Inhabited : core.
auto 6 with sets.
Qed.
Hint Resolve Inclusion_is_an_order : core.
elim Inclusion_is_an_order; auto with sets.
Qed.
Hint Resolve Inclusion_is_transitive : core.
Definition Power_set_PO : Ensemble U -> PO (Ensemble U).
intro A; try assumption.
apply Definition_of_PO with (Power_set A) (Included U); auto with sets.
Defined.
Hint Unfold Power_set_PO : core.
auto with sets.
Qed.
Hint Resolve Strict_Rel_Transitive Strict_Rel_is_Strict_Included : core.

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