Update analyticOn_zeroSet.lean

Nevanlinna/analyticOn_zeroSet.lean

@@ -380,14 +380,33 @@ theorem AnalyticOnCompact.eliminateZeros
   obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
   use g
   use A
+
+  have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by
+    intro z
+    rw [h₃g z]
+    congr
+    funext a
+    congr
+    dsimp [n]
+    simp [a.2]
+
   constructor
   · exact h₁g
   · constructor
-    · sorry
-    · intro z
-      rw [h₃g z]
-      congr
-      funext a
-      congr
-      dsimp [n]
-      simp [a.2]
+    · intro z h₁z
+      by_cases h₂z : ⟨z, h₁z⟩  ∈ ↑A.toSet
+      · exact h₂g ⟨z, h₁z⟩ h₂z
+      · have : f z ≠ 0 := by
+          by_contra C
+          have : ⟨z, h₁z⟩ ∈ ↑A₁ := by
+            dsimp [A₁, ι]
+            simp
+            exact C
+          have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
+            dsimp [A]
+            simp
+            exact this
+          tauto
+        rw [inter z] at this
+        exact right_ne_zero_of_smul this
+    · exact inter