English
If s is convex and x,y ∈ s, then the midpoint of x and y lies in s.
Русский
Если s выпукло и x,y ∈ s, то середина x и y belongs к s.
LaTeX
$$$$ \\text{midpoint}_{\\mathbb{K}}(x,y) \\in s $$$$
Lean4
theorem midpoint_mem [Ring 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [Invertible (2 : 𝕜)]
{s : Set E} {x y : E} (h : Convex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) : midpoint 𝕜 x y ∈ s :=
h.segment_subset hx hy <| midpoint_mem_segment x y