∀ {F : Type u_3} {E : Type u_1} {K : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Field K]
  [inst_3 : Algebra F E] [inst_4 : Algebra F K] (motive : (∃ m, Maximal (fun x => x ∈ {ϕ | ϕ.IsExtendible}) m) → Prop)
  (x : ∃ m, Maximal (fun x => x ∈ {ϕ | ϕ.IsExtendible}) m),
  (∀ (ϕ : IntermediateField.Lifts F E K) (hϕ : Maximal (fun x => x ∈ {ϕ | ϕ.IsExtendible}) ϕ), motive ⋯) → motive x