English
A collection of simp-lemmas describes how subset relations interact with intersections (sInter) in the closed-sets lattice.
Русский
Набор вспомогательных свойств (simp-лемм) описывает, как отношения включения взаимодействуют с пересечением (sInter) в решётке замкнутых множеств.
LaTeX
$$$\\text{simp}_{1,3}: \\text{Subset}(t, S\\;\\mathop{\\mathrm{sInter}}) \\iff \\forall t' \\in S, t \\subseteq t'$.$$
Lean4
/-- Intermediate value theorem for two functions: if `f` and `g` are two continuous functions
on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. -/
theorem intermediate_value_univ₂ [PreconnectedSpace X] {a b : X} {f g : X → α} (hf : Continuous f) (hg : Continuous g)
(ha : f a ≤ g a) (hb : g b ≤ f b) : ∃ x, f x = g x :=
by
obtain ⟨x, _, hfg, hgf⟩ : (univ ∩ {x | f x ≤ g x ∧ g x ≤ f x}).Nonempty :=
isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ _ _ (isClosed_le hf hg) (isClosed_le hg hf)
(fun _ _ => le_total _ _) ⟨a, trivial, ha⟩ ⟨b, trivial, hb⟩
exact ⟨x, le_antisymm hfg hgf⟩
