--- a/cardinal.agda	Sat Aug 10 12:31:25 2019 +0900
+++ b/cardinal.agda	Sun Aug 11 08:10:13 2019 +0900
@@ -21,6 +21,13 @@
 open _∨_
 open Bool
 
+
+func : (f : Ordinal → Ordinal ) → ( dom cod : OD ) → OD 
+func f dom cod = record { def = λ z → {x y : Ordinal} → (z ≡ omax x y ) ∧  def dom x  ∧ def cod (f x ) }
+
+-- Func :  ( dom cod : OD ) → OD
+-- Func dom cod = record { def = λ x → x o< sup-o ( λ y → (f : Ordinal → Ordinal ) → y ≡ od→ord (func f dom cod) )  }
+
 ------------
 --
 -- Onto map
@@ -44,7 +51,7 @@
 cardinal :  (X  : OD ) → Cardinal X
 cardinal  X = record {
        cardinal = sup-o ( λ x → proj1 ( cardinal-p x) )
-     ; conto = onto
+     ; conto =  x∋minimul onto-set ∃-onto-set 
      ; cmax = cmax
    } where
     cardinal-p : (x  : Ordinal ) →  ( Ordinal  ∧ Dec (Onto (Ord x) X) )
@@ -52,31 +59,11 @@
     cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True }
     cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False }
     onto-set : OD 
-    onto-set = record { def = λ x →  {!!} } -- Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X }
-    onto : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X
-    onto = record {
-               xmap = xmap
-            ;  ymap = ymap
-            ;  ymap-on-X  = ymap-on-X
-            ;  onto-iso = onto-iso
-      } where
-       --
-       -- Ord cardinal itself has no onto map, but if we have x o< cardinal, there is one
-       --    od→ord X o< cardinal, so if we have def Y y or def X y, there is an Onto (Ord y) X
-       Y = (Ord (sup-o (λ x → proj1 (cardinal-p x))))
-       lemma1 : (y : Ordinal ) → def Y y  →  Onto (Ord y) X
-       lemma1 y y<Y with sup-o<  {λ x → proj1 ( cardinal-p x)} {y} 
-       ... | t = {!!}
-       lemma2 :  def Y (od→ord X)
-       lemma2 = {!!}
-       xmap : (x : Ordinal ) → def Y x → Ordinal 
-       xmap = {!!}
-       ymap : (y : Ordinal ) → def X y → Ordinal 
-       ymap = {!!}
-       ymap-on-X  : {y :  Ordinal  } → (lty : def X y ) → def Y (ymap y lty)  
-       ymap-on-X  = {!!}
-       onto-iso   : {y :  Ordinal  } → (lty : def X y ) → xmap  (ymap y lty) (ymap-on-X lty ) ≡ y
-       onto-iso = {!!}
+    onto-set = record { def = λ x →  Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X }
+    ∃-onto-set : ¬ ( onto-set == od∅ )
+    ∃-onto-set record { eq→ = eq→ ; eq← = eq← } = ¬x<0 {_} ( eq→ lemma ) where
+          lemma : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X 
+          lemma = {!!}
     cmax : (y : Ordinal) → sup-o (λ x → proj1 (cardinal-p x)) o< y → ¬ Onto (Ord y) X
     cmax y lt ontoy = o<> lt (o<-subst  {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))}
        (sup-o<  {λ x → proj1 ( cardinal-p x)}{y}  ) lemma refl ) where
@@ -85,20 +72,6 @@
           lemma | case1 x = refl
           lemma | case2 not = ⊥-elim ( not ontoy )
 
-func : (f : Ordinal  → Ordinal ) → OD 
-func  f = record { def = λ y → (x : Ordinal ) → y ≡ f x }
-
-Func : OD 
-Func  = record { def = λ x →  (f : Ordinal  → Ordinal ) → x ≡ od→ord (func f) }
-
-odmap : { x : OD  } → Func ∋ x → Ordinal  → OD 
-odmap  {f} lt x = record { def = λ y → def f y } 
-
-lemma1 :  { x : OD  } → Func ∋ x → {!!} -- ¬ ( (f : Ordinal  → Ordinal ) →  ¬ ( x ≡ od→ord (func f)  ))
-lemma1 = {!!}
-
-
------
 --  All cardinal is ℵ0,  since we are working on Countable Ordinal, 
 --  Power ω is larger than ℵ0, so it has no cardinal.