simp only [sub_self, taylorWithinEval_self, MulZeroClass.mul_zero, zero_div, zero_smul,
  eq_self_iff_true, exists_prop, and_true_iff, MulZeroClass.zero_mul]
case inl
f g g' : ℝ → ℝ
x₀ a b : ℝ
n : ℕ
hab : a < b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx : x₀ ∈ Icc a b
⊢ ∃ x',
    ∃ (_ : x' ∈ Ioo a b),
      x' ≠ x₀ ∧
        (f x₀ - taylorWithinEval f n (Icc a b) x₀ x₀) * g' x' =
          ((x₀ - x') ^ n / ↑n ! * (g x₀ - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x'
case inl
f g g' : ℝ → ℝ
x₀ a b : ℝ
n : ℕ
hab : a < b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx : x₀ ∈ Icc a b
⊢ ∃ x' ∈ Ioo a b, x' ≠ x₀