intro H'; elim H'; auto 10.
intro H'0; generalize (H'0 (Empty_set U)); intro h; lapply h;
[ intro H'1; lapply H'1;
[ intro h0; elim h0; intros bsup E; clear h H'1 h0; elim E | clear h ]
| clear h ]; auto 10.
intros H'1 H'2; exists bsup; apply Bottom_definition; auto 10.
elim H'1; auto 10.
intros y H'3; try assumption; auto 10.
apply H'2; auto 10.
apply Upper_Bound_definition; auto 10.
intros y0 H'4; elim H'4; auto 10.
Qed.
Hint Resolve Conditionally_complete_has_a_bottom.

Theorem Compatible_is_reflexive : Reflexive U (Compatible U D).
red in |- *; intro x; red in |- *.