Proper local homeomorphisms are covering maps

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_isCoveringMap_of_isProperMap_of_isLocalHomeomorph
    {E X : Type*} [TopologicalSpace E] [TopologicalSpace X] [T2Space E]
    {g : E → X} (hproper : IsProperMap g) (hlocal : IsLocalHomeomorph g) :
    IsCoveringMap g := by sorry