intros x y H'; red in |- *; simpl in |- *.
intros H'0 x0 y0 H'1; try assumption.
red in H'1.
lapply (H'1 x0); [ intro H'3 | auto with sets ].
lapply (H'1 y0); [ intro H'4 | auto with sets ].
elim H'4.
elim H'3; auto with sets.
elim H'3; auto with sets.
Qed.

Theorem Coherent_implies_Conditionally_Complete :
Coherent U D -> Conditionally_complete U D.