--- a/src/Topology.agda	Sat Dec 31 11:18:04 2022 +0900
+++ b/src/Topology.agda	Sat Dec 31 17:56:01 2022 +0900
@@ -141,7 +141,7 @@
        OS    = POS TP TQ
     ;  OS⊆PL = tp10
     ;  o∪ = tp13
-    ;  o∩ = ?
+    ;  o∩ = tp14
   } where
         --  B : (OS P ∋ x →  proj⁻¹ x ) ∨ (OS Q ∋ y  →  proj⁻¹ y )
         --  U ⊂ ZFP P Q  ∧ ( U ∋ ∀ x → B ∋ ∃ b → b ∋ x ∧  b ⊂ U )
@@ -162,7 +162,15 @@
         tp10 : POS TP TQ ⊆ Power (ZFP P Q)
         tp10 {x} record { b = b ; pb = pb ; bx = bx } z xz = tp11 (pb _ bx) xz
         tp13 : {U : HOD} → U ⊆ POS TP TQ → POS TP TQ ∋ Union U
-        tp13 {U} U⊆O  = record { b = ? ; pb = ? ; bx = ? } 
+        tp13 {U} U⊆O  = tp20 U U⊆O where
+             ind : {x : HOD} → ({y : HOD} → x ∋ y → y ⊆ POS TP TQ → POS TP TQ ∋ Union y) → x ⊆ POS TP TQ → POS TP TQ ∋ Union x
+             ind {x} prev x⊆O = record { b = ? ; pb = ? ; bx = ? }
+             tp20 : (U : HOD ) →  U ⊆ POS TP TQ  → POS TP TQ ∋ Union U 
+             tp20 U U⊆O = ε-induction0 { λ U →  U ⊆ POS TP TQ  → POS TP TQ ∋ Union U } ind U U⊆O
+        tp14 :  {p q : HOD} → POS TP TQ ∋ p → POS TP TQ ∋ q → POS TP TQ ∋ (p ∩ q)
+        tp14 {p} {q} op oq  = record { b = & tp15 ; pb = ? ; bx = ? } where
+             tp15 : HOD
+             tp15 = ?
 
 -- existence of Ultra Filter 
 
@@ -216,8 +224,3 @@
          LQQ : LQ ⊆ Power Q
          LQQ = ?
 
-
-
-
-
-