interior_closedBall' — Mathlib

English

For any x ∈ E and r ∈ ℝ, the boundary of the closed ball centered at x with radius r equals the sphere of radius r about x.

Русский

Для любых x ∈ E и r ∈ ℝ граница замкнутого шара радиуса r вокруг x равна сфере радиуса r вокруг x.

LaTeX

$$$$ \forall x \in E, \forall r \in \mathbb{R}, \partial \overline{B}(x,r) = S(x,r). $$$$

Lean4

theorem interior_closedBall' (x : E) (r : ℝ) : interior (closedBall x r) = ball x r :=
  by
  rcases eq_or_ne r 0 with (rfl | hr)
  · rw [closedBall_zero, ball_zero, interior_singleton]
  · exact interior_closedBall x hr

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