import Mathlib.Analysis.Analytic.Meromorphic
import Nevanlinna.analyticAt
import Nevanlinna.divisor


open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral


theorem MeromorphicAt.eventually_eq_zero_or_eventually_ne_zero
  {f : ℂ → ℂ}
  {z₀ : ℂ}
  (hf : MeromorphicAt f z₀) :
  (∀ᶠ (z : ℂ) in nhds z₀, f z = 0) ∨ ∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z ≠ 0 := by
  sorry