--- a/HOD.agda	Tue Jul 16 09:57:01 2019 +0900
+++ b/HOD.agda	Wed Jul 17 10:52:31 2019 +0900
@@ -57,8 +57,8 @@
   -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
   od→ord : {n : Level} → OD {n} → Ordinal {n}
   ord→od : {n : Level} → Ordinal {n} → OD {n} 
-  c<→o<  : {n : Level} {x y : OD {n} }      → def y ( od→ord x ) → od→ord x o< od→ord y
-  oiso   : {n : Level} {x : OD {n}}     → ord→od ( od→ord x ) ≡ x
+  c<→o<  : {n : Level} {x y : OD {n} }   → def y ( od→ord x ) → od→ord x o< od→ord y
+  oiso   : {n : Level} {x : OD {n}}      → ord→od ( od→ord x ) ≡ x
   diso   : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
   -- we should prove this in agda, but simply put here
   ==→o≡ : {n : Level} →  { x y : OD {suc n} } → (x == y) → x ≡ y
@@ -319,10 +319,9 @@
          union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
               ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
          union← :  (X z : OD) (X∋z : Union X ∋ z) →  ¬  ( (u : OD ) → ¬ ((X ∋  u) ∧ (u ∋ z )))
-         union← X z UX∋z =  TransFiniteExists' _ lemma UX∋z where
+         union← X z UX∋z =  TransFiniteExists _ lemma UX∋z where
               lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
               lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } 
-              -- ( λ {y} xx →  
 
          ψiso :  {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y   → ψ y
          ψiso {ψ} t refl = t
@@ -377,22 +376,27 @@
               lemma :  od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x)))
               lemma = sup-o<   
 
+         -- double-neg-eilm : {n  : Level } {A : Set n} → ¬ ¬ A → A
+         -- double-neg-eilm {n} {A} notnot = ⊥-elim (notnot (λ A → {!!} ))
          -- 
          -- Every set in OD is a subset of Ordinals
          -- 
          -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y )
-         power→ :  ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x
-         power→ A t P∋t {x} t∋x = TransFiniteExists _ -- (λ y → (t ==  (A ∩ ord→od y)))
-                 lemma4 lemma5  where
+
+         -- we have oly double negation form because of the replacement axiom
+         --
+         power→ :  ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
+         power→ A t P∋t {x} t∋x = TransFiniteExists _ lemma5 lemma4 where
               a = od→ord A
               lemma2 : ¬ ( (y : OD) → ¬ (t ==  (A ∩ y)))
               lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t
-              lemma3 : (y : OD) → t == ( A ∩ y ) → A ∋ x
-              lemma3 y eq = proj1 (eq→ eq t∋x)
+              lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
+              lemma3 y eq not = not (proj1 (eq→ eq t∋x))
               lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y)))
               lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 ))
-              lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → def A (od→ord x)
-              lemma5 {y} eq = lemma3 (ord→od y) eq
+              lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) →  ¬ ¬  (def A (od→ord x))
+              lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
+
          power← :  (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
          power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where 
               a = od→ord A