∀ {α : Type u_1} [inst : SemilatticeSup α] {s t : Finset α} {a : α} [inst_1 : DecidableLE α] [inst_2 : OrderTop α]
  [inst_3 : DecidableEq α], a ∈ lowerClosure ↑s → a ∉ lowerClosure ↑t → (s ∪ t).truncatedSup a = s.truncatedSup a