Two roots in one lemniscate component force a critical point

import Definitions.Def_ErdosProblems_Erdos1041_Counterexample_Defs
import Mathlib
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.SpecialFunctions.Complex.Log
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.EMetricSpace.BoundedVariation

/-! External source: ani, erdosproblems.com forum thread 1041, 7 Sept 2026.
Formalisation of Lemma 2.1 from `ani_degree7_counterexample.tex`. -/

/-
Integration: move the S2 declaration out of Defs before importing this module.
The supplied Defs is preserved under input/; see DEFS_DELTAS.md.
Repaired and compiled against Mathlib v4.29.1 (commit 5e932f97).
-/

noncomputable section

open Set Filter Topology

open Erdos1041.Counterexample

theorem Erdos1041.Counterexample.s2_two_zeros_gives_critical_point
    (p : Polynomial ℂ) (hp : 0 < p.natDegree) (z : ℂ) (hz : z ∈ Omega p)
    (w₁ w₂ : ℂ) (hw₁ : w₁ ∈ connectedComponentIn (Omega p) z)
    (hw₂ : w₂ ∈ connectedComponentIn (Omega p) z) (hne : w₁ ≠ w₂)
    (hr₁ : p.IsRoot w₁) (hr₂ : p.IsRoot w₂) :
    ∃ cc ∈ connectedComponentIn (Omega p) z, (Polynomial.derivative p).IsRoot cc := by sorry