English
If a is the least upper bound of s and b is the least upper bound of t in a SemilatticeSup, then a ⊔ b is the least upper bound of s ∪ t.
Русский
Если a — наименьшая верхняя граница s, а b — наименьшая верхняя граница t в полулепладе, то a ⊔ b — наименьшая верхняя граница s ∪ t.
LaTeX
$$$[SemilatticeSup γ] \; {a b : γ} {s t : Set γ} (hs : IsLUB s a) (ht : IsLUB t b) : IsLUB (s \cup t) (\sup a b)$$$
Lean4
theorem inter_Ici_of_mem [LinearOrder γ] {s : Set γ} {a b : γ} (ha : IsLUB s a) (hb : b ∈ s) : IsLUB (s ∩ Ici b) a :=
⟨fun _ hx => ha.1 hx.1, fun c hc =>
have hbc : b ≤ c := hc ⟨hb, le_rfl⟩
ha.2 fun x hx => ((le_total x b).elim fun hxb => hxb.trans hbc) fun hbx => hc ⟨hx, hbx⟩⟩
