--- a/cardinal.agda	Mon Aug 26 02:50:16 2019 +0900
+++ b/cardinal.agda	Mon Aug 26 12:27:20 2019 +0900
@@ -61,13 +61,49 @@
 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
 eq-prod refl refl = refl
 
+-- prod-eq : { x x' y y' : OD } → < x , y > ≡ < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
+-- prod-eq {x} {x'} {y} {y'} eq = {!!} where
+--     lemma : < x , y > ≡ < x , y' > → y ≡ y'
+--     lemma eq1 with trio< (od→ord x) (od→ord y) 
+--     lemma eq1 | tri< a ¬b ¬c = {!!}
+--     lemma eq1 | tri≈ ¬a b ¬c = {!!}
+--     lemma eq1 | tri> ¬a ¬b c = {!!}
+
 postulate
    def-eq :  { P Q p q : OD } →  P ≡ Q → p ≡ q → (pt : P ∋ p ) → (qt : Q ∋ q ) → pt ≅ qt
 
+∈-to-ord : {p : Ordinal } → ( ZFProduct ∋  ord→od p ) → ord-pair p
+∈-to-ord {p} lt = def-subst {ZFProduct} {(od→ord (ord→od p))} {_} {_} lt refl diso
+
+ord-to-∈ : {p : Ordinal } → ord-pair p → ZFProduct ∋  ord→od p 
+ord-to-∈ {p} lt = def-subst {_} {_} {ZFProduct} {(od→ord (ord→od p))} lt refl (sym diso)
+
+lemma333 : { A a  : OD } → { x : A ∋ a } → def-subst {A} {od→ord a} (def-subst {A} {od→ord a} x refl refl ) refl refl ≡ x
+lemma333 = refl
+
+lemma334 : { A B  : OD } → {a b : Ordinal} → { x : A ∋ ord→od a } → { y : B ∋ ord→od b } → (f1 : A ≡ B) → (f2 : a ≡ b)
+   → def-subst {B} {od→ord (ord→od b)} (def-subst {A} { od→ord (ord→od a)} x f1 (cong (λ k → od→ord (ord→od k)) f2 )) refl refl ≅ x
+lemma334 {A} {A} {a} {a} {x} {y} refl refl with def-eq {A} {A} {ord→od a} {ord→od a} refl refl x y
+... | HE.refl = HE.refl
+
+lemma335 : { A B C : OD } → {a b c : Ordinal} → { x : A ∋ ord→od a } → { y : C ∋ ord→od c } → (f1 : A ≡ B) → (f2 : a ≡ b) → (g1 : B ≡ C) → (g2 : b ≡ c)
+   → def-subst {B} {od→ord (ord→od b)} (def-subst {A} { od→ord (ord→od a)} x f1 (cong (λ k → od→ord (ord→od k)) f2 )) g1 (cong (λ k → od→ord (ord→od k)) g2 )
+      ≅ def-subst {A} { od→ord (ord→od a)} {C } { od→ord (ord→od c)} x (trans f1 g1)
+          (trans (cong (λ k → od→ord (ord→od k)) f2 ) (cong (λ k → od→ord (ord→od k)) g2 ))
+lemma335 {A} {A} {A} {a} {a} {a} {x} {y} refl refl refl refl with def-eq {A} {A} {ord→od a} {ord→od a} refl refl x y
+... | HE.refl = HE.refl
+
+∈-to-ord-oiso : { p : Ordinal } → { x : ord-pair p } → ∈-to-ord (ord-to-∈ x) ≡ x
+∈-to-ord-oiso {p} {x} = {!!} where
+   lemma : def-subst {_} {_} {ZFProduct} {{!!}} (def-subst {_} {_} {ZFProduct} {{!!}} x refl (sym diso)) refl diso ≡ x
+   lemma = {!!}
+
 lemma34 : { p q : Ordinal } → (x :  ord-pair p ) → (y :  ord-pair q ) → p ≡ q → x ≅ y
-lemma34 {p} {q} x y eq = {!!} where
+lemma34 {p} {q} x y refl = subst₂ (λ j k → j ≅ k) ∈-to-ord-oiso ∈-to-ord-oiso (HE.cong (λ k → ∈-to-ord k) lemma1 ) where
    lemma : (pt : ZFProduct ∋ ord→od p ) → (qt : ZFProduct ∋ ord→od q ) → p ≡ q → pt ≅ qt
-   lemma pt qt eq = def-eq {ZFProduct} {ZFProduct} refl {!!} {!!} {!!}
+   lemma pt qt eq = def-eq {ZFProduct} {ZFProduct} refl (cong (λ k → ord→od k) eq) pt qt
+   lemma1 : (ord-to-∈ x) ≅ (ord-to-∈ y )
+   lemma1 = lemma (ord-to-∈ x) (ord-to-∈ y ) refl
 
 π1-cong :  { p q : OD } →  p ≡ q → (pt : ZFProduct ∋ p ) → (qt : ZFProduct ∋ q ) → π1 pt ≅ π1 qt
 π1-cong {p} {q} refl s t = HE.cong (λ k → pi1 k ) (def-eq {ZFProduct} {ZFProduct} refl refl s t )