(**)
intros U F F' F'' F''' A B A' B'. cut(Intersection _ A B=Complement _(Union _(
Complement _ A)(Complement _ B))). intro H. rewrite H. apply F''. apply(
sigalg_P0 _ F F' F'' F'''). apply F''. auto. apply F''. auto.
(**)
apply Extensionality_Ensembles. split. red. unfold Complement;unfold In;unfold
not. intros x x' x''. elim(Intersection_inv _ _ _ _ x'). elim(Union_inv _ _ _ _
x''). auto. auto. split. apply NNPP. auto with sets. apply NNPP. auto with sets
. Qed.

Lemma sigalg_P3:
forall(U:Type)(F:Ensemble(Ensemble U))(F':sigalg_0 _ F)(F'':sigalg_1 _ F)(F''':
sigalg_2 _ F),
F(Full_set U).