intros H' x C H H'0; elim H'0.
elim (Bottom_is_compact U D);
[ intros bot E; elim E; intros H'4 H'5; elim H'4; clear E | idtac ];
auto with sets.
exists bot; split; auto with sets; auto with sets.
apply Lub_definition; auto with sets.
intros y H'1; elim H'1; auto with sets.
intros A H'1 H'2 x0 H'3.
cut (forall x : U, Union U A (Singleton U x) = Add U A x); auto with sets.
intro H'4; rewrite <- (H'4 x0).
clear H'4; intro H'4.
elim H'2; [ intros bsup E; elim E; intros H'6 H'7; clear E H'2 | clear H'2 ];
auto with sets.
red in H'4.
lapply Lub_of_Union;
[ intro H'2; lapply (H'2 A (Singleton U x0));
[ intro H'9; lapply (H'9 x);
[ intro H'11; lapply H'11;
[ intro H'12; elim (H'12 bsup x0);
[ intros bsup0 E; elim E; intros H'17 H'18; try exact H'17;
clear E H'11
| clear H'11
| clear H'11 ]
| clear H'11 ]
| idtac ]
| idtac ]
| idtac ]; auto with sets.
exists bsup0; split; [ assumption | auto with sets ].
lapply Lub_of_compacts_is_compact;
[ intro H'5; elim (H'5 bsup x0 x);
[ intros bsup1 E; elim E; intros H'15 H'16; try exact H'16; clear E
| idtac
| idtac ]
| idtac ]; auto with sets.
lapply (Lub_is_unique U D);
[ intro H'11; rewrite (H'11 (Couple U bsup x0) bsup0 bsup1) | idtac ];
auto with sets.
red in |- *; simpl in |- *.
intros x1 H'8; elim H'8; auto with sets.
elim H'7; auto with sets.
apply Singleton_has_lub.
lapply (H'4 x0); [ intro H'8; elim H'8 | idtac ]; auto with sets.
apply Upper_Bound_definition; auto with sets.
intros y H'5; try assumption.
lapply (H'4 y); [ intro H'10; elim H'10 | idtac ]; auto with sets.
red in |- *; simpl in |- *.
intros x1 H'5; try assumption.
lapply (H'4 x1); [ intro H'9; elim H'9 | idtac ]; auto with sets.
Qed.

(* Specifications: *)
Theorem Approximants_directed_set :
Conditionally_complete U D ->
forall x : U, In U (Carrier_of U D) x -> Directed U D (Approximants U D x).
Theorem Inclusion_of_Approximants :
Domain U D ->
forall x y : U,
In U (Carrier_of U D) x ->
In U (Carrier_of U D) y ->
Included U (Approximants U D x) (Approximants U D y) -> Rel_of U D x y.