∀ {K : Type u_1} {V : Type u_2} [inst : Field K] [inst_1 : AddCommGroup V] [inst_2 : Module K V]
  (s : Projectivization.Subspace K V) {v : V} (hv : v ≠ 0),
  v ∈ Projectivization.Subspace.submodule s ↔ Projectivization.mk K v hv ∈ s