{ι : Type u_1} →
  {R : Type u_2} →
    {M : Type u_3} →
      {N : Type u_4} →
        [inst : CommRing R] →
          [inst_1 : AddCommGroup M] →
            [inst_2 : Module R M] →
              [inst_3 : AddCommGroup N] →
                [inst_4 : Module R N] →
                  {P : RootPairing ι R M N} →
                    (support : Finset ι) →
                      LinearIndepOn R ⇑P.root ↑support →
                        LinearIndepOn R ⇑P.coroot ↑support →
                          (∀ (i : ι),
                              P.root i ∈ AddSubmonoid.closure (⇑P.root '' ↑support) ∨
                                -P.root i ∈ AddSubmonoid.closure (⇑P.root '' ↑support)) →
                            (∀ (i : ι),
                                P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' ↑support) ∨
                                  -P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' ↑support)) →
                              P.Base