Icc_subset — Mathlib

English

If s is preconnected and a,b ∈ s, then the closed interval [a,b] lies inside s.

Русский

Если s предсоединено и a,b ∈ s, то отрезок [a,b] целиком contained в s.

LaTeX

$$$\forall a,b,\ a\in s \land b\in s \Rightarrow [a,b]\subseteq s$$$

Lean4

/-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/
theorem Icc_subset {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : Icc a b ⊆ s := by
  simpa only [image_id] using hs.intermediate_value ha hb continuousOn_id

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