refine crucial hU h20 hz (h18 ε) hgz e2 (λ w hw hwz => ?_)
case h.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r✝ : ℝ
f g✝ g : ℂ → ℂ
hU : IsOpen U
hg : DifferentiableOn ℂ g U
hi : InjOn g U
hz₀ : z₀ ∈ U
hgz₀ : g z₀ = 0
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt g p z₀
h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0
h17 : p ≠ 0
h✝ : deriv g z₀ = 0
h6 : 2 ≤ FormalMultilinearSeries.order p
r : ℝ
h7 : r > 0
h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p)
h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0
h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0
h20 : closedBall z₀ r ⊆ U
h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0
h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U
h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U
h24 : FormalMultilinearSeries.order p ≠ 0
this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0
ε : ℂ
h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0
hε : ε ∈ {0}ᶜ
z : ℂ
hz : z ∈ ball z₀ r
hgz : g z + ε = 0
e1 : z ≠ z₀
e2 : deriv (fun z => g z + ε) z ≠ 0
⊢ (cindex z₀ r fun z => g z + ε) = 1
case h.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r✝ : ℝ
f g✝ g : ℂ → ℂ
hU : IsOpen U
hg : DifferentiableOn ℂ g U
hi : InjOn g U
hz₀ : z₀ ∈ U
hgz₀ : g z₀ = 0
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt g p z₀
h25 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0
h17 : p ≠ 0
h✝ : deriv g z₀ = 0
h6 : 2 ≤ FormalMultilinearSeries.order p
r : ℝ
h7 : r > 0
h8 : cindex z₀ r g = ↑(FormalMultilinearSeries.order p)
h14 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → deriv g z ≠ 0
h21 : ∀ z ∈ closedBall z₀ r, z ≠ z₀ → g z ≠ 0
h20 : closedBall z₀ r ⊆ U
h22 : ∀ z ∈ sphere z₀ r, g z ≠ 0
h18 : ∀ (ε : ℂ), DifferentiableOn ℂ (fun z => g z + ε) U
h19 : TendstoLocallyUniformlyOn (fun ε z => g z + ε) g (𝓝[≠] 0) U
h24 : FormalMultilinearSeries.order p ≠ 0
this : ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + x = 0
ε : ℂ
h : ε ∈ {0}ᶜ → ∃ z ∈ ball z₀ r, g z + ε = 0
hε : ε ∈ {0}ᶜ
z : ℂ
hz : z ∈ ball z₀ r
hgz : g z + ε = 0
e1 : z ≠ z₀
e2 : deriv (fun z => g z + ε) z ≠ 0
w : ℂ
hw : w ∈ closedBall z₀ r
hwz : w ≠ z
⊢ g w + ε ≠ 0