English
If there exists c with ||c|| > 1, then for any nonzero x there exists d with d ≠ 0 such that ||d x|| < ε, ε/||c|| ≤ ||d x||, and ||d||^{-1} ≤ ε^{-1} ||c|| ||x||.
Русский
Если существует c с ||c|| > 1, то для любого ненулевого x существует d ≠ 0 such that ||d x|| < ε, ε/||c|| ≤ ||d x|| и ||d||^{-1} ≤ ε^{-1} ||c|| ||x||.
LaTeX
$$$\exists d \in 𝕜, d \neq 0 \wedge \|d \cdot x\| < \varepsilon \wedge (\varepsilon / \|c\| ≤ \|d \cdot x\|) \wedge (\|d\|^{-1} ≤ \varepsilon^{-1} \cdot \|c\| \, \|x\|).$$$
Lean4
@[simp]
theorem arsinh_neg (x : ℝ) : arsinh (-x) = -arsinh x :=
by
rw [← exp_eq_exp, exp_arsinh, exp_neg, exp_arsinh]
apply eq_inv_of_mul_eq_one_left
rw [neg_sq, neg_add_eq_sub, add_comm x, mul_comm, ← sq_sub_sq, sq_sqrt, add_sub_cancel_right]
exact add_nonneg zero_le_one (sq_nonneg _)
