Update stronglyMeromorphicOn_eliminate.lean

Nevanlinna/stronglyMeromorphicOn_eliminate.lean

@@ -172,7 +172,40 @@ theorem MeromorphicOn.decompose₂
 
   have h₅g₀ : StronglyMeromorphicAt g₀ u := by
 
-    sorry
+    rw [stronglyMeromorphicAt_of_mul_analytic (g := ∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) (z₀ := u) (f := g₀)]
+    rw [← h₄g₀]
+    exact hf u u.2
+    --
+    have : (∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) = (fun z => ∏ p : P, (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) := by
+      funext w
+      simp
+    rw [this]
+    apply Finset.analyticAt_prod
+    intro p hp
+    apply AnalyticAt.zpow
+    apply AnalyticAt.sub
+    apply analyticAt_id
+    apply analyticAt_const
+    --
+    by_contra hCon
+    rw [sub_eq_zero] at hCon
+    have : p.1 = u := by
+      exact SetCoe.ext (_root_.id (Eq.symm hCon))
+    rw [← this] at hu
+    simp [hp] at hu
+    --
+    simp only [Finset.prod_apply]
+    rw [Finset.prod_ne_zero_iff]
+    intro p hp
+    apply zpow_ne_zero
+    by_contra hCon
+    rw [sub_eq_zero] at hCon
+    have : p.1 = u := by
+      exact SetCoe.ext (_root_.id (Eq.symm hCon))
+    rw [← this] at hu
+    simp [hp] at hu
+
+
   have h₆g₀ : (h₁g₀ u u.2).order ≠ ⊤ := by
     sorry