English
Let F be a (possibly infinite) family of compact convex sets in a finite-dimensional space. If every subfamily of F with cardinality ≤ d+1 has nonempty intersection, then the whole family has nonempty intersection.
Русский
Пусть F — семейство компактных выпуклых множеств в конечномерном пространстве. Если каждая подподсемейство F размерности ≤ d+1 имеет непустое пересечение, то пересечение всей F непусто.
LaTeX
$$$\\displaystyle\\text{Let } F = \\{F_i\\}_{i\\in I} \\text{ be a possibly infinite family of compact convex sets in } E. \\\\ \\text{If for every finite } J \\subseteq I, |J| \\le d+1, \\bigcap_{i\\in J} F_i \\neq \\emptyset, \\text{ then } \\bigcap_{i\\in I} F_i \\neq \\emptyset.$$$
Lean4
theorem left_mem_segment (x y : E) : x ∈ [x -[𝕜] y] :=
⟨1, 0, zero_le_one, le_refl 0, add_zero 1, by rw [zero_smul, one_smul, add_zero]⟩
