simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
  f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
    ((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
      iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
    iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!
case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
  f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
    (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
      iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
    iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!