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.

Theorem Inclusion_is_transitive : Transitive (Ensemble U) (Included U).