have h3 : ∀ z ∈ U, f z = (z - z₀) * g z :=
  λ z _ => by simpa only [smul_eq_mul, hfz₀, sub_zero] using (sub_smul_dslope f z₀ z).symm
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall c r, g z ≠ 0
h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0
z : ℂ
hz : z ∈ sphere c r
⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall c r, g z ≠ 0
h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0
z : ℂ
hz : z ∈ sphere c r
h3 : ∀ z ∈ U, f z = (z - z₀) * g z
⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z