Section Lemmas_on_partial_orders.
Variable U : Type.
Variable B : Ensemble U.
Variable D : PO U.
elim D; simpl in |- *; auto.
intros C R H' H'0; elim H'0; auto 10.
Qed.
Hint Resolve Rel_of_reflexive.
elim D; simpl in |- *; auto.
intros C R H' H'0; elim H'0; auto 10.
Qed.
Hint Resolve Rel_of_antisymmetric.
elim D; simpl in |- *; auto.
intros C R H' H'0; elim H'0; auto 10.
Qed.
Hint Resolve Rel_of_transitive.
intros x y H' H'0; red in |- *; auto 20.
intros x0 H'1; elim H'1; auto 20.
Qed.
Hint Resolve Couple_included_in_carrier.

Theorem Lub_is_in_Carrier :
forall (bsup : U) (X : Ensemble U),
Included U X (Carrier_of U D) ->
Lub U D X bsup -> In U (Carrier_of U D) bsup.