∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Ring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommGroup E]
  [inst_3 : Module 𝕜 E] (K : Geometry.SimplicialComplex 𝕜 E) (faces : Set (Finset E)) (subset : faces ⊆ K.faces)
  (down_closed : IsLowerSet faces), (K.ofSubcomplex faces subset down_closed).faces = faces