case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R R₁ R₂ : ℝ
f : ℂ → E
z z₀ : ℂ
hd : DifferentiableOn ℂ f (ball z R₁)
h_maps : MapsTo f (ball z R₁) (ball (f z) R₂)
hz : z ∈ ball z R₁
⊢ dist (f z) (f z) ≤ R₂ / R₁ * dist z z

case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R R₁ R₂ : ℝ
f : ℂ → E
c z z₀ : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)
hz : z ∈ ball c R₁
hne : z ≠ c
⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R R₁ R₂ : ℝ
f : ℂ → E
c z z₀ : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)
hz : z ∈ ball c R₁
⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c
rcases eq_or_ne z c with (rfl | hne)
no goals
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R R₁ R₂ : ℝ
f : ℂ → E
c z z₀ : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)
hz : z ∈ ball c R₁
hne : z ≠ c
⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c
simpa only [dslope_of_ne _ hne, slope_def_module, norm_smul, norm_inv, ← div_eq_inv_mul, ←
  dist_eq_norm, div_le_iff (dist_pos.2 hne)] using norm_dslope_le_div_of_mapsTo_ball hd h_maps hz
no goals
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R R₁ R₂ : ℝ
f : ℂ → E
z z₀ : ℂ
hd : DifferentiableOn ℂ f (ball z R₁)
h_maps : MapsTo f (ball z R₁) (ball (f z) R₂)
hz : z ∈ ball z R₁
⊢ dist (f z) (f z) ≤ R₂ / R₁ * dist z z
simp only [dist_self, mul_zero, le_rfl]