Proof Abelian_Group.op_inv_0_uniq sum_abelian_group.
(** Represents strongly-specified negation. *)
Definition sum_neg_strong
:  forall x : E, { y | sum_is_inv x y }
:= Abelian_Group.op_neg_strong sum_abelian_group.
(** Represents negation. *)
Definition sum_neg
:  E -> E
:= Abelian_Group.op_neg sum_abelian_group.
Notation "{-}" := (sum_neg) : ring_scope.
Notation "- x" := (sum_neg x) : ring_scope.
(** Asserts that the negation returns the inverse of its argument. *)
Definition sum_neg_def
:  forall x : E, sum_is_inv x (- x)
:= Abelian_Group.op_neg_def sum_abelian_group.
(** Proves that negation is one-to-one *)
Definition sum_neg_inj
:  is_injective E E {-}
:= Abelian_Group.op_neg_inj sum_abelian_group.

(** Proves the cancellation property for negation. *)
Theorem sum_cancel_neg
:  forall x : E, - (- x) = x.