∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] {ι : Type u_4}
  [Nonempty ι] {K : ι → Subalgebra R A}, Directed (fun x1 x2 => x1 ≤ x2) K → ↑(iSup K) = ⋃ i, ↑(K i)