∀ {R : Type u_1} {A : Type w} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A]
  [inst_3 : FaithfulSMul R A] [inst_4 : IsDomain A] {s t : Set A},
  (AlgebraicIndependent.matroid R A).IsBasis s t =
    (AlgebraicIndepOn R id s ∧ s ⊆ t ∧ ∀ a ∈ t, IsAlgebraic (↥(Algebra.adjoin R s)) a)