(** 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.

Theorem prod_n1_x_inv_r : forall x : E, sum_is_inv_r x ({-1} # x).