--- a/HOD.agda	Mon Jul 15 19:10:08 2019 +0900
+++ b/HOD.agda	Tue Jul 16 09:57:01 2019 +0900
@@ -69,8 +69,8 @@
   sup-o  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
   sup-o< : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → ∀ {x : Ordinal {n}} →  ψ x  o<  sup-o ψ 
   -- contra-position of mimimulity of supermum required in Power Set Axiom
-  sup-x  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
-  sup-lb : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → {z : Ordinal {n}}  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
+  -- sup-x  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
+  -- sup-lb : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → {z : Ordinal {n}}  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
   -- sup-lb : {n : Level } → ( ψ : Ordinal {n} →  Ordinal {n}) → ( ∀ {x : Ordinal {n}} →  ψx  o<  z ) →  z o< osuc ( sup-o ψ ) 
   minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} 
   -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox  ( minimum of x )
@@ -315,15 +315,14 @@
          ⊆→o< {x} {y}  lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl )
          ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
 
-         union-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → OD {suc n}
-         union-u {X} {z} U>z = ord→od ( osuc ( od→ord z ) )
          union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
          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 _ UX∋z
-             ( λ {y} xx not → not (ord→od y) (record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } ))
+         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
@@ -354,29 +353,12 @@
          --- ZFSubset A x =  record { def = λ y → def A y ∧  def x y }                   subset of A
          --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )       Power X is a sup of all subset of A
          --
-         --  if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t 
-         --    then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x
-         --    In case of later, ZFSubset A ∋ t and t ∋ x implies A ∋ x by transitivity of Ordinals
          --
          ∩-≡ :  { a b : OD {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
          ∩-≡ {a} {b} inc = record {
             eq→ = λ {x} x<a → record { proj2 = x<a ;
                  proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {a} {_} x<a refl (sym diso) )) refl diso  } ;
             eq← = λ {x} x<a∩b → proj2 x<a∩b }
-         -- ord-power→ : (a : Ordinal ) ( t : OD) → Def (Ord a) ∋ t → {x : OD} → t ∋ x → Ord a ∋ x
-         -- ord-power→ a t P∋t {x} t∋x with osuc-≡<  (sup-lb  P∋t )
-         -- ... | case1 eq = proj1 (def-subst t∋x (sym (subst₂ (λ j k → j ≡ k ) oiso oiso ( cong (λ k → ord→od k) (sym eq) ))) refl )  
-         -- ... | case2 lt = lemma3 where
-         --      sp =  sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x)))
-         --      minsup :  OD
-         --      minsup =  ZFSubset (Ord a) ( ord→od ( sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x))))) 
-         --      Ltx :   {n : Level} → {x t : OD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x
-         --      Ltx {n} {x} {t} lt = c<→o< lt
-         --      -- lemma1 hold because a subset of ordinals is ordinal
-         --      lemma1 : od→ord t o< od→ord minsup → minsup ∋ Ord (od→ord t)
-         --      lemma1 lt = {!!}
-         --      lemma3 : od→ord x o< a
-         --      lemma3 = otrans (proj1 (lemma1 lt)) (c<→o< {suc n} {x} {Ord (od→ord t)} (Ltx t∋x) )
          -- 
          -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t
          -- Power A is a sup of ZFSubset A t, so Power A ∋ t
@@ -436,8 +418,6 @@
                   lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) 
                   lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A  )))
 
-         ∅-iso :  {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) 
-         ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq  
          regularity :  (x : OD) (not : ¬ (x == od∅)) →
             (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
          proj1 (regularity x not ) = x∋minimul x not
@@ -454,7 +434,6 @@
          eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d  
          eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d  
 
-
          infinity∅ : infinite  ∋ od∅ {suc n}
          infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} iφ refl lemma where
               lemma : o∅ ≡ od→ord od∅