--- a/BAlgbra.agda	Sun Jul 05 11:40:55 2020 +0900
+++ b/BAlgbra.agda	Sun Jul 05 12:32:09 2020 +0900
@@ -24,16 +24,16 @@
 open _∨_
 open Bool
 
-_∩_ : ( A B : OD  ) → OD
+_∩_ : ( A B : HOD  ) → HOD
 A ∩ B = record { def = λ x → def A x ∧ def B x } 
 
-_∪_ : ( A B : OD  ) → OD
+_∪_ : ( A B : HOD  ) → HOD
 A ∪ B = record { def = λ x → def A x ∨ def B x } 
 
-_＼_ : ( A B : OD  ) → OD
+_＼_ : ( A B : HOD  ) → HOD
 A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }
 
-∪-Union : { A B : OD } → Union (A , B) ≡ ( A ∪ B )
+∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B )
 ∪-Union {A} {B} = ==→o≡ ( record { eq→ =  lemma1 ; eq← = lemma2 } )  where
     lemma1 :  {x : Ordinal} → def (Union (A , B)) x → def (A ∪ B) x
     lemma1 {x} lt = lemma3 lt where
@@ -49,7 +49,7 @@
     lemma2 {x} (case2 B∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
        (record { proj1 = case2 refl ; proj2 = subst (λ k → def B k) (sym diso) B∋x}))
 
-∩-Select : { A B : OD } →  Select A (  λ x → ( A ∋ x ) ∧ ( B ∋ x )  ) ≡ ( A ∩ B )
+∩-Select : { A B : HOD } →  Select A (  λ x → ( A ∋ x ) ∧ ( B ∋ x )  ) ≡ ( A ∩ B )
 ∩-Select {A} {B} = ==→o≡ ( record { eq→ =  lemma1 ; eq← = lemma2 } ) where
     lemma1 : {x : Ordinal} → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → def (A ∩ B) x
     lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → def B k ) diso (proj2 (proj2 lt)) }
@@ -57,7 +57,7 @@
     lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 =
         record { proj1 = subst (λ k → def A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → def B k ) (sym diso) (proj2 lt) } }
 
-dist-ord : {p q r : OD } → p ∩ ( q ∪ r ) ≡   ( p ∩ q ) ∪ ( p ∩ r )
+dist-ord : {p q r : HOD } → p ∩ ( q ∪ r ) ≡   ( p ∩ q ) ∪ ( p ∩ r )
 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
     lemma1 :  {x : Ordinal} → def (p ∩ (q ∪ r)) x → def ((p ∩ q) ∪ (p ∩ r)) x
     lemma1 {x} lt with proj2 lt
@@ -67,7 +67,7 @@
     lemma2 {x} (case1 p∩q) = record { proj1 = proj1 p∩q ; proj2 = case1 (proj2 p∩q ) } 
     lemma2 {x} (case2 p∩r) = record { proj1 = proj1 p∩r ; proj2 = case2 (proj2 p∩r ) } 
 
-dist-ord2 : {p q r : OD } → p ∪ ( q ∩ r ) ≡   ( p ∪ q ) ∩ ( p ∪ r )
+dist-ord2 : {p q r : HOD } → p ∪ ( q ∩ r ) ≡   ( p ∪ q ) ∩ ( p ∪ r )
 dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
     lemma1 : {x : Ordinal} → def (p ∪ (q ∩ r)) x → def ((p ∪ q) ∩ (p ∪ r)) x
     lemma1 {x} (case1 cp) = record { proj1 = case1 cp ; proj2 = case1 cp }