English
The insert of an element into a set preserves the intersection property exactly when the element is not bottom and it remains non-disjoint with all members of the original set.
Русский
Добавление элемента в множество сохраняет свойство пересечения точно при условии, что элемент не нижний и не_DISJOINT с любым элементом множества.
LaTeX
$$$$\\text{insert } a\\ s \\text{ is intersecting } \\iff s \\text{ is intersecting and } a \\neq \\bot \\text{ and } (\\forall b\\in s, \\lnot Disjoint(a,b)).$$$$
Lean4
protected theorem insert (hs : s.Intersecting) (ha : a ≠ ⊥) (h : ∀ b ∈ s, ¬Disjoint a b) : (insert a s).Intersecting :=
by
rintro b (rfl | hb) c (rfl | hc)
· rwa [disjoint_self]
· exact h _ hc
· exact fun H => h _ hb H.symm
· exact hs hb hc
