∀ {F : Type u} (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {K : Type w} [inst_3 : Field K]
  [inst_4 : Algebra F K] [inst_5 : Algebra E K] [IsScalarTower F E K] (S : IntermediateField F K)
  [Algebra.IsAlgebraic F ↥S] [IsPurelyInseparable F E],
  Field.sepDegree E ↥(IntermediateField.adjoin E ↑S) = Field.sepDegree F ↥S