∀ {F : Type u} {E : Type v} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {A : IntermediateField F E}
  {L : Type w} [inst_3 : Field L] [inst_4 : Algebra F L] [inst_5 : Algebra L E] [inst_6 : IsScalarTower F L E],
  A.LinearDisjoint L → ∀ {ι : Type u_1} {a : ι → ↥A}, LinearIndependent F a → LinearIndependent L (⇑A.val ∘ a)