∀ {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] (c : Set (IntermediateField.Lifts F E K)),
  IsChain (fun x1 x2 => x1 ≤ x2) (insert ⊥ c) →
    ∀ (i j : ↑(insert ⊥ c)) (h : ((fun i => (↑i).carrier) i).toSubalgebra ≤ ((fun i => (↑i).carrier) j).toSubalgebra),
      (↑i).emb = (↑j).emb.comp (Subalgebra.inclusion h)