(**
  Accepts two elements, x and y, and
  asserts that y is x's left inverse.
*)
Definition sum_is_inv_l
  := Abelian_Group.op_is_inv_l sum_abelian_group.(**
  Accepts two elements, x and y, and
  asserts that y is x's right inverse.
*)
Definition sum_is_inv_r
  := Abelian_Group.op_is_inv_r sum_abelian_group.(**
  Accepts two elements, x and y, and
  asserts that y is x's inverse.
*)
Definition sum_is_inv
  := Abelian_Group.op_is_inv sum_abelian_group.(**
  Accepts one argument, x, and asserts that
  x has a left inverse.
*)
Definition sum_has_inv_l := Abelian_Group.has_inv_l sum_abelian_group.(**
  Accepts one argument, x, and asserts that
  x has a right inverse.
*)
Definition sum_has_inv_r := Abelian_Group.has_inv_r sum_abelian_group.(**
  Accepts one argument, x, and asserts that
  x has an inverse.
*)
Definition sum_has_inv := Abelian_Group.has_inv sum_abelian_group.

Theorem sum_inv_sym : forall x y : E, sum_is_inv x y <-> sum_is_inv y x.