English
Change the index set of a sequence by a function f: ι' → ι; the transformed sequences preserve the similarity relation.
Русский
Изменение индексации последовательности по отображению f сохраняет отношение подобия между двумя семействами последовательностей.
LaTeX
$$$\\text{Similar}(v_1 \\circ f, v_2 \\circ f) \\iff \\text{Similar}(v_1, v_2).$$$
Lean4
/-- Shrinking lemma for coverings by open balls in a proper metric space. A point-finite open cover
of a proper metric space by nonempty open balls can be shrunk to a new cover by nonempty open balls
so that each of the new balls has strictly smaller radius than the old one. -/
theorem exists_iUnion_ball_eq_radius_pos_lt {r : ι → ℝ} (hr : ∀ i, 0 < r i)
(uf : ∀ x, {i | x ∈ ball (c i) (r i)}.Finite) (uU : ⋃ i, ball (c i) (r i) = univ) :
∃ r' : ι → ℝ, ⋃ i, ball (c i) (r' i) = univ ∧ ∀ i, r' i ∈ Ioo 0 (r i) :=
let ⟨r', hU, hv⟩ := exists_subset_iUnion_ball_radius_pos_lt hr isClosed_univ (fun x _ => uf x) uU.ge
⟨r', univ_subset_iff.1 hU, hv⟩
