exact ne_of_mem_sphere hz hr.1.ne.symm
case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
  ball z₀ R ⊆
    {x |
      (fun x =>
          g x ≠ 0 ∧
            (x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
              DifferentiableAt ℂ g x)
        x}
r : ℝ
hr : r ∈ Set.Ioo 0 R
z✝ : ℂ
hz : z✝ ∈ sphere z₀ r
⊢ z✝ ≠ z₀
no goals