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Nevanlinna/stronglyMeromorphicOn_eliminate.lean

@@ -376,8 +376,14 @@ theorem MeromorphicOn.decompose₃'
     intro z hz
     exact stronglyMeromorphicOn_ratlPolynomial₃ d z trivial
   have h₂h₁ : h₁h₁.meromorphicOn.divisor = d := by
-    sorry
+    apply stronglyMeromorphicOn_divisor_ratlPolynomial_U
+    rwa [h₁d]
+    --
+    rw [h₁d]
+    exact (StronglyMeromorphicOn.meromorphicOn h₁f).divisor.supportInU
   have h₃h₁ : ∀ (z : ℂ) (hz : z ∈ U), (h₁h₁ z hz).meromorphicAt.order ≠ ⊤ := by
+    intro z hz
+
     sorry
   let g' := f * h₁
   have h₁g' : MeromorphicOn g' U := h₁f.meromorphicOn.mul h₁h₁.meromorphicOn

Nevanlinna/stronglyMeromorphicOn_ratlPolynomial.lean

@@ -61,6 +61,13 @@ theorem stronglyMeromorphicOn_ratlPolynomial₃
     apply AnalyticOn.stronglyMeromorphicOn
     apply analyticOnNhd_const
 
+theorem stronglyMeromorphicOn_ratlPolynomial₃U
+  (d : ℂ → ℤ)
+  (U : Set ℂ) :
+  StronglyMeromorphicOn (∏ᶠ u, fun z ↦ (z - u) ^ d u) U := by
+  intro z hz
+  exact stronglyMeromorphicOn_ratlPolynomial₃ d z trivial
+
 
 theorem stronglyMeromorphicOn_divisor_ratlPolynomial₁
   {z : ℂ}
@@ -142,3 +149,21 @@ theorem stronglyMeromorphicOn_divisor_ratlPolynomial
   simp
   rw [stronglyMeromorphicOn_divisor_ratlPolynomial₁ d h₁d]
   simp
+
+theorem stronglyMeromorphicOn_divisor_ratlPolynomial_U
+  {U : Set ℂ}
+  (d : ℂ → ℤ)
+  (h₁d : Set.Finite d.support)
+  (h₂d : d.support ⊆ U) :
+  (stronglyMeromorphicOn_ratlPolynomial₃U d U).meromorphicOn.divisor = d := by
+
+  funext z
+  rw [MeromorphicOn.divisor]
+  simp
+  by_cases hz : z ∈ U
+  · simp [hz]
+    rw [stronglyMeromorphicOn_divisor_ratlPolynomial₁ d h₁d]
+    simp
+  · simp [hz]
+    rw [eq_comm, ← Function.nmem_support]
+    tauto