∀ {R : Type u_1} {S : Type v} {A : Type w} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing A]
  [inst_3 : Algebra R S] [inst_4 : Algebra R A] [inst_5 : Algebra S A] [IsScalarTower R S A] [IsDomain S]
  [NoZeroDivisors A] {ι : Type u_2} {v : ι → A},
  IsTranscendenceBasis R v → (Algebra.IsAlgebraic S A ↔ ∀ (i : ι), IsAlgebraic S (v i))