∀ {F : Type u_1} {E : Type u_2} {K : Type u_3} [inst : Field F] [inst_1 : Field E] [inst_2 : Field K]
  [inst_3 : Algebra F E] [inst_4 : Algebra F K] [alg : Algebra.IsAlgebraic F E],
  (∀ (S : Finset E), ∃ σ, ↑S ⊆ ↑σ.carrier) → Nonempty (E →ₐ[F] K)