cases' HC' with n hn
case neg
P : ℤ[X]
t : ℤ
ht : t ∈ periodicPts fun x => Polynomial.eval x P
C : Cycle ℤ := Cycle.map (fun x => Polynomial.eval x P - x) (periodicOrbit (fun x => Polynomial.eval x P) t)
HC : ∀ {n : ℕ}, (fun x => Polynomial.eval x P)^[n + 1] t - (fun x => Polynomial.eval x P)^[n] t ∈ C
Hdvd : Cycle.Chain (fun x x_1 => x ∣ x_1) C
Habs :
  ∀ (m n : ℕ),
    ((fun x => Polynomial.eval x P)^[m + 1] t - (fun x => Polynomial.eval x P)^[m] t).natAbs =
      ((fun x => Polynomial.eval x P)^[n + 1] t - (fun x => Polynomial.eval x P)^[n] t).natAbs
HC' :
  ∃ n,
    Polynomial.eval ((fun x => Polynomial.eval x P)^[n] t) P - (fun x => Polynomial.eval x P)^[n] t ≠
      Polynomial.eval ((fun x => Polynomial.eval x P)^[n + 1] t) P - (fun x => Polynomial.eval x P)^[n + 1] t
⊢ IsPeriodicPt (fun x => Polynomial.eval x P) 2 t
case neg.intro
P : ℤ[X]
t : ℤ
ht : t ∈ periodicPts fun x => Polynomial.eval x P
C : Cycle ℤ := Cycle.map (fun x => Polynomial.eval x P - x) (periodicOrbit (fun x => Polynomial.eval x P) t)
HC : ∀ {n : ℕ}, (fun x => Polynomial.eval x P)^[n + 1] t - (fun x => Polynomial.eval x P)^[n] t ∈ C
Hdvd : Cycle.Chain (fun x x_1 => x ∣ x_1) C
Habs :
  ∀ (m n : ℕ),
    ((fun x => Polynomial.eval x P)^[m + 1] t - (fun x => Polynomial.eval x P)^[m] t).natAbs =
      ((fun x => Polynomial.eval x P)^[n + 1] t - (fun x => Polynomial.eval x P)^[n] t).natAbs
n : ℕ
hn :
  Polynomial.eval ((fun x => Polynomial.eval x P)^[n] t) P - (fun x => Polynomial.eval x P)^[n] t ≠
    Polynomial.eval ((fun x => Polynomial.eval x P)^[n + 1] t) P - (fun x => Polynomial.eval x P)^[n + 1] t
⊢ IsPeriodicPt (fun x => Polynomial.eval x P) 2 t