∀ {K : Type u_3} {V : Type u} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {s : Set V}
  {a : V}, LinearIndepOn K id s → (a ∉ Submodule.span K s ↔ LinearIndepOn K id (insert a s) ∧ a ∉ s)