∀ {M : Type u_1} [inst : AddCommGroup M] {a b : AddSubgroup M},
  { toFun := fun S => { toAddSubmonoid := S.toAddSubmonoid, smul_mem' := ⋯ }, invFun := Submodule.toAddSubgroup,
          left_inv := ⋯, right_inv := ⋯ }
        a ≤
      { toFun := fun S => { toAddSubmonoid := S.toAddSubmonoid, smul_mem' := ⋯ }, invFun := Submodule.toAddSubgroup,
          left_inv := ⋯, right_inv := ⋯ }
        b ↔
    { toFun := fun S => { toAddSubmonoid := S.toAddSubmonoid, smul_mem' := ⋯ }, invFun := Submodule.toAddSubgroup,
          left_inv := ⋯, right_inv := ⋯ }
        a ≤
      { toFun := fun S => { toAddSubmonoid := S.toAddSubmonoid, smul_mem' := ⋯ }, invFun := Submodule.toAddSubgroup,
          left_inv := ⋯, right_inv := ⋯ }
        b