simp only [taylorWithinEval_self] at h
case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
  (g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y =
    (taylorWithinEval f n (Icc x x₀) x₀ x - taylorWithinEval f n (Icc x x₀) x x) * g' y
⊢ f x - taylorWithinEval f n (Icc x x₀) x₀ x =
    ((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y
case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
  (g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y =
    (taylorWithinEval f n (Icc x x₀) x₀ x - f x) * g' y
⊢ f x - taylorWithinEval f n (Icc x x₀) x₀ x =
    ((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y