intros X H' H'0; elim H'0; intros maj E; try exact E; clear H'0.
red in |- *; simpl in |- *.
intros H'0 x y H'1; red in |- *; simpl in |- *.
intros H'2 H'3; exists maj; split.
elim E; auto with sets.
elim E; auto with sets.
Qed.

Theorem Consistent_downward_stable :
forall B : Ensemble U,
Included U B (Carrier_of U D) ->
Consistent U D B ->
forall A : Ensemble U, Included U A B -> Consistent U D A.