Update stronglyMeromorphicAt.lean

Nevanlinna/stronglyMeromorphicAt.lean

@@ -233,12 +233,41 @@ theorem makeStronglyMeromorphic_id
       simp [B]
       exact Filter.EventuallyEq.eq_of_nhds h₁f
     · obtain ⟨n, g, h₁g, h₂g, h₃g⟩ := h₁f
-
+      rw [Filter.EventuallyEq.eq_of_nhds h₃g]
+      have : h₀f.meromorphicAt.order = n := by
+        rw [MeromorphicAt.order_eq_int_iff (StronglyMeromorphicAt.meromorphicAt h₀f) n]
+        use g
+        constructor
+        · assumption
+        · constructor
+          · assumption
+          · exact eventually_nhdsWithin_of_eventually_nhds h₃g
       by_cases h₃f : h₀f.meromorphicAt.order = 0
       · simp [h₃f]
-        sorry
+        have hn : n = (0 : ℤ) := by
+          rw [h₃f] at this
+          exact WithTop.coe_eq_zero.mp (id (Eq.symm this))
+        simp_rw [hn]
+        simp
+        let t₀ : h₀f.meromorphicAt.order = (0 : ℤ) := by
+          exact h₃f
+        let A := (h₀f.meromorphicAt.order_eq_int_iff 0).1 t₀
+        have : g =ᶠ[𝓝 z₀] (Classical.choose A) := by
+          obtain ⟨h₀, h₁, h₂⟩  := Classical.choose_spec A
+          apply localIdentity h₁g h₀
+          rw [hn] at h₃g
+          simp at h₃g
+          simp at h₂
+          have h₄g : f =ᶠ[𝓝[≠] z₀] g := by
+            apply Filter.EventuallyEq.filter_mono h₃g
+            exact nhdsWithin_le_nhds
+          exact Filter.EventuallyEq.trans (Filter.EventuallyEq.symm h₄g) h₂
+        exact Filter.EventuallyEq.eq_of_nhds this
       · simp [h₃f]
-
-        sorry
+        left
+        apply zero_zpow n
+        by_contra hn
+        rw [hn] at this
+        tauto
 
   · exact m₁ (StronglyMeromorphicAt.meromorphicAt hf) z hz