∀ {R : Type u_2} {S : Type u_1} {M : Type u_3} [inst : CommRing R] [inst_1 : Field S] [inst_2 : AddCommGroup M]
  [inst_3 : Algebra R S] [inst_4 : Module R M] [inst_5 : Module S M] [inst_6 : IsScalarTower R S M]
  (B : LinearMap.BilinForm S M) [inst_7 : IsDomain R] [inst_8 : Module.IsTorsionFree R S] (N : Submodule R M)
  (x y : ↥(B.dualSubmodule N)) (z : ↥N),
  (algebraMap R S) ({ toFun := B.dualSubmoduleParing (x + y), map_add' := ⋯, map_smul' := ⋯ } z) =
    (algebraMap R S)
      (({ toFun := B.dualSubmoduleParing x, map_add' := ⋯, map_smul' := ⋯ } +
          { toFun := B.dualSubmoduleParing y, map_add' := ⋯, map_smul' := ⋯ })
        z)