--- a/cardinal.agda	Mon Aug 26 12:27:20 2019 +0900
+++ b/cardinal.agda	Tue Aug 27 14:13:27 2019 +0900
@@ -61,13 +61,60 @@
 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 = {!!}
+open _==_
+
+exg-pair : { x y : OD } → (x , y ) == ( y , x )
+exg-pair {x} {y} = record { eq→ = left ; eq← = right } where
+    left : {z : Ordinal} → def (x , y) z → def (y , x) z 
+    left (case1 t) = case2 t
+    left (case2 t) = case1 t
+    right : {z : Ordinal} → def (y , x) z → def (x , y) z 
+    right (case1 t) = case2 t
+    right (case2 t) = case1 t
+
+==-trans : { x y z : OD } →  x == y →  y == z →  x ==  z
+==-trans x=y y=z  = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← =  λ {m} t → eq← x=y (eq← y=z t) }
+
+==-sym : { x y  : OD } →  x == y →  y == x 
+==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← =  λ {m} t → eq→ x=y t }
+
+ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y
+ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq )
+
+prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
+prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where
+    lemma0 : {x y z : OD } → ( x , x ) == ( z , y ) → x ≡ y
+    lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y) 
+    lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl) 
+    lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a )
+    lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a )
+    lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b
+    lemma0 {x} {y} eq | tri> ¬a ¬b c  with eq← eq {od→ord y} (case2 refl) 
+    lemma0 {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c )
+    lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c )
+    lemma2 : {x y z : OD } → ( x , x ) == ( z , y ) → z ≡ y
+    lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq )  where
+        lemma3 : ( x , x ) == ( y , z )
+        lemma3 = ==-trans eq exg-pair
+    lemma1 : {x y : OD } → ( x , x ) == ( y , y ) → x ≡ y
+    lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl)
+    lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s)
+    lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s)
+    lemma4 : {x y z : OD } → ( x , y ) == ( x , z ) → y ≡ z
+    lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl)
+    lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z
+    ... | refl with lemma2 (==-sym eq )
+    ... | refl = refl
+    lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z
+    lemmax : x ≡ x'
+    lemmax with eq→ eq {od→ord (x , x)} (case1 refl) 
+    lemmax | case1 s = lemma1 (ord→== s )  -- (x,x)≡(x',x')
+    lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y'
+    ... | refl = lemma1 (ord→== s )
+    lemmay : y ≡ y'
+    lemmay with lemmax
+    ... | refl with lemma4 eq -- with (x,y)≡(x,y')
+    ... | eq1 = lemma4 (ord→== (cong (λ  k → od→ord k )  eq1 ))
 
 postulate
    def-eq :  { P Q p q : OD } →  P ≡ Q → p ≡ q → (pt : P ∋ p ) → (qt : Q ∋ q ) → pt ≅ qt