∀ {K : Type v} {L : Type w} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] {p : Polynomial K}
  {F : IntermediateField K L},
  (Polynomial.map (algebraMap K L) p).Splits → (∀ x ∈ p.rootSet L, x ∈ F) → (Polynomial.map (algebraMap K ↥F) p).Splits