English
Let F be a (possibly infinite) set of sets that are convex and compact. If for every finite subcollection G ⊆ F with |G| ≤ d+1 we have ⋂ G ≠ ∅, then ⋂ F ≠ ∅.
Русский
Пусть F — множество множеств, каждое из которых выпукло и компакто. Если для любого конечного подмножества G ⊆ F с |G| ≤ d+1 пересечение ⋂ G непусто, то ⋂ F непусто.
LaTeX
$$$\\displaystyle\\text{Let } F \\subseteq \\mathcal{P}(E) \\text{ be a set of convex compact sets. If for every finite } G \\subseteq F, |G| \\le d+1, we have } \\bigcap_{X \\in G} X \\neq \\emptyset, \\text{ then } \\bigcap_{X \\in F} X \\neq \\emptyset.$$$
Lean4
theorem segment_symm (x y : E) : [x -[𝕜] y] = [y -[𝕜] x] :=
Set.ext fun _ =>
⟨fun ⟨a, b, ha, hb, hab, H⟩ => ⟨b, a, hb, ha, (add_comm _ _).trans hab, (add_comm _ _).trans H⟩,
fun ⟨a, b, ha, hb, hab, H⟩ => ⟨b, a, hb, ha, (add_comm _ _).trans hab, (add_comm _ _).trans H⟩⟩
