∀ {ι : Type u_2} {R : Type u_3} {M : Type u_4} {N : Type u_1} [inst : Finite ι] [inst_1 : AddCommGroup M]
  [inst_2 : AddCommGroup N] [inst_3 : Field R] [CharZero R] [inst_5 : Module R M] [inst_6 : Module R N]
  (P : RootPairing ι R M N) [P.IsRootSystem] [P.IsCrystallographic] [P.IsReduced] (s : Set ι),
  LinearIndepOn R (⇑P.root) s →
    (∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s)) →
      ∀ (i : ι),
        P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' ↑⋯.toFinset) ∨
          -P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' ↑⋯.toFinset)