∀ {R : Type u_3} {M : Type u_1} {N : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : AddCommGroup N] [inst_4 : Module R N] (m : Submodule R M) (n : Submodule R N),
  TensorProduct.lift
        (m.liftQ
          (n.liftQ
              ((TensorProduct.mk R M N).flip.compr₂
                ((TensorProduct.map m.subtype LinearMap.id).range ⊔
                    (TensorProduct.map LinearMap.id n.subtype).range).mkQ)
              ⋯).flip
          ⋯) ∘ₗ
      ((TensorProduct.map m.subtype LinearMap.id).range ⊔ (TensorProduct.map LinearMap.id n.subtype).range).liftQ
        (TensorProduct.map m.mkQ n.mkQ) ⋯ =
    LinearMap.id