{R : Type u} →
  {S : Type v} →
    [inst : CommSemiring R] →
      [inst_1 : CommSemiring S] →
        [inst_2 : Algebra R S] →
          {A B : Subalgebra R S} →
            A.LinearDisjoint B → A ⊔ B = ⊤ → {ι : Type u_1} → Module.Basis ι R ↥A → Module.Basis ι (↥B) S