∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (S : Subfield L)
  (algebra_map_mem : ∀ (x : K), (algebraMap K L) x ∈ S), ↑(S.toIntermediateField algebra_map_mem) = ↑S