Update stronglyMeromorphicOn_eliminate.lean

Nevanlinna/stronglyMeromorphicOn_eliminate.lean

@@ -10,14 +10,6 @@ import Nevanlinna.mathlibAddOn
 open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
 
-lemma untop_eq_untop'
-  {n : WithTop ℤ}
-  (hn : n ≠ ⊤) :
-  n.untop' 0 = n.untop hn := by
-  rw [WithTop.untop'_eq_iff]
-  simp
-
-
 theorem MeromorphicOn.decompose₁
   {f : ℂ → ℂ}
   {U : Set ℂ}
@@ -149,3 +141,19 @@ theorem MeromorphicOn.decompose₁
           simp
           apply zpow_ne_zero
           rwa [sub_ne_zero]
+
+
+theorem MeromorphicOn.decompose₂
+  {f : ℂ → ℂ}
+  {U : Set ℂ}
+  {P : Finset ℂ}
+  (hP : P.toSet ⊆ U)
+  (h₁f : MeromorphicOn f U)
+  (h₂f : ∀ hp : p ∈ P, StronglyMeromorphicAt f p)
+  (h₃f : ∀ hp : p ∈ P, (h₂f hp).meromorphicAt.order ≠ ⊤) :
+  ∃ g : ℂ → ℂ, (MeromorphicOn g U)
+    ∧ (∀ p ∈ P, AnalyticAt ℂ g p)
+    ∧ (∀ p ∈ P, g p ≠ 0)
+    ∧ (f = g * ∏ p ∈ P, fun z ↦ (z - p) ^ (h₁f.divisor p)) := by
+
+  sorry