∀ {ι : Type u_1} {R : Type u_2} [inst : CommRing R] {M : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  {s : ι → M} (this : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ≠ 0 ∧ a • s i ∈ Submodule.span R (s '' s_1)),
  LinearIndepOn R s this.choose ∧ ∀ i ∉ this.choose, ∃ a, a ≠ 0 ∧ a • s i ∈ Submodule.span R (s '' this.choose)