--- a/ODC.agda	Sat May 09 09:40:18 2020 +0900
+++ b/ODC.agda	Sat May 09 16:41:40 2020 +0900
@@ -94,47 +94,3 @@
          choice : (X : OD  ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A 
          choice X {A} X∋A not = x∋minimal A not
 
-         --- With assuption of OD is ordered,  p ∨ ( ¬ p ) <=> axiom of choice
-         ---
-         record choiced  ( X : OD) : Set (suc n) where
-          field
-             a-choice : OD
-             is-in : X ∋ a-choice
-         
-         choice-func' :  (X : OD ) → (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X
-         choice-func'  X p∨¬p not = have_to_find where
-                 ψ : ( ox : Ordinal ) → Set (suc n)
-                 ψ ox = (( x : Ordinal ) → x o< ox  → ( ¬ def X x )) ∨ choiced X
-                 lemma-ord : ( ox : Ordinal  ) → ψ ox
-                 lemma-ord  ox = TransFinite {ψ} induction ox where
-                    ∋-p : (A x : OD ) → Dec ( A ∋ x ) 
-                    ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) -- LEM
-                    ∋-p A x | case1 (lift t)  = yes t
-                    ∋-p A x | case2 t  = no (λ x → t (lift x ))
-                    ∀-imply-or :  {A : Ordinal  → Set n } {B : Set (suc n) }
-                        → ((x : Ordinal ) → A x ∨ B) →  ((x : Ordinal ) → A x) ∨ B
-                    ∀-imply-or  {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) -- LEM
-                    ∀-imply-or  {A} {B} ∀AB | case1 (lift t) = case1 t
-                    ∀-imply-or  {A} {B} ∀AB | case2 x  = case2 (lemma (λ not → x (lift not ))) where
-                         lemma : ¬ ((x : Ordinal ) → A x) →  B
-                         lemma not with p∨¬p B
-                         lemma not | case1 b = b
-                         lemma not | case2 ¬b = ⊥-elim  (not (λ x → dont-orb (∀AB x) ¬b ))
-                    induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x
-                    induction x prev with ∋-p X ( ord→od x) 
-                    ... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } )
-                    ... | no ¬p = lemma where
-                         lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X
-                         lemma1 y with ∋-p X (ord→od y)
-                         lemma1 y | yes y<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } )
-                         lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) )
-                         lemma :  ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X
-                         lemma = ∀-imply-or lemma1
-                 have_to_find : choiced X
-                 have_to_find = dont-or  (lemma-ord (od→ord X )) ¬¬X∋x where
-                     ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → def X x → ⊥)
-                     ¬¬X∋x nn = not record {
-                            eq→ = λ {x} lt → ⊥-elim  (nn x (def→o< lt) lt) 
-                          ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt )
-                        }
-