have RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ := by
  refine fun s ⟨hs, hs1⟩ ↦ not_or.mpr ⟨?_, hs1⟩
  simpa [Complex.ext_iff] using fun _ _ ↦ by linarith [h_mem hs]
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹