obtain ⟨r, h2⟩ := hp.has_fpower_series_iterate_dslope_fslope p.order
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
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
case 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
r : ENNReal
h2 :
  HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
    (FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z