Proof Ring.prod_inv_0 ring.
(** Represents -1 and proves that it exists. *)
Definition E_n1_strong
:  { x : E | sum_is_inv 1 x }
:= Ring.E_n1_strong ring.
(** Represents -1. *)
Definition E_n1 : E := Ring.E_n1 ring.
(** Defines a symbolic representation for -1  Note: here we represent the inverse of 1 rather than the negation of 1. Letter we prove that the negation equals the inverse.  Note: brackets are needed to ensure Coq parses the symbol as a single token instead of a prefixed function call. *)
Notation "{-1}" := E_n1 : commutative_ring_scope.
(** Asserts that -1 is the additive inverse of 1. *)
Definition E_n1_def
:  sum_is_inv 1 {-1}
:= Ring.E_n1_def ring.

(** Asserts that -1 is the left inverse of 1. *)
Theorem E_n1_inv_l
:  sum_is_inv_l 1 {-1}.