Proof Abelian_Group.op_id sum_abelian_group.
(** 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.

(** Asserts that every element has a right inverse. *)
Theorem sum_inv_r_ex
:  forall x : E, exists y : E, sum_is_inv_r x y.