Use new result Union_any_inter_equiv.

WIP: add is_Open_Union_any_closure.

Lebesgue/Set_theory/Set_system/Set_system_any.v

@@ -464,16 +464,20 @@ rewrite full_equiv, HPB3; f_equal; set_ext_auto.
 Qed.
 
 Lemma is_Basisp_inter :
-  Inter T -> is_Basisp T PB ->
-  forall B1 B2 x, PB B1 -> PB B2 -> inter B1 B2 x ->
-    exists B3, PB B3 /\ incl B3 (inter B1 B2) /\ B3 x.
+  Inter T -> is_Basisp T PB -> Union_any_inter PB.
 Proof.
-intros HT HPB B1 B2 x HB1 HB2 Hx.
+intros HT HPB; apply Union_any_inter_equiv; intros B1 B2 x HB1 HB2 Hx.
 apply is_Basisp_equiv in HPB; destruct HPB as [HPB1 HPB2].
 apply HPB2; try easy.
 apply HT; apply HPB1; easy.
 Qed.
 
+Lemma is_Open_Union_any_closure :
+  full (unionp_any PB) -> Union_any_inter PB ->
+  is_Open (Union_any_closure PB).
+Proof.
+Admitted.
+
 Lemma is_Basisp_Union_any_closure : is_Basisp (Union_any_closure PB) PB.
 Proof.
 split.