have : (2 : ℝ) ^ ε * (2 : ℝ) ^ (-ε) = (2 : ℝ) ^ (ε - ε) := by
  rw [← rpow_add (by norm_num), add_neg_self, sub_self]
case calc_3
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
x ε : ℝ
hx : 1 + 2 * Real.log 2 * ε ≤ x
hε : ε ∈ Ioo 0 1
⊢ 2 ^ ε * (1 - 2 ^ (-ε)) / ε = (2 ^ ε - 1) / ε
case calc_3
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
x ε : ℝ
hx : 1 + 2 * Real.log 2 * ε ≤ x
hε : ε ∈ Ioo 0 1
this : 2 ^ ε * 2 ^ (-ε) = 2 ^ (ε - ε)
⊢ 2 ^ ε * (1 - 2 ^ (-ε)) / ε = (2 ^ ε - 1) / ε