(** Represents -1 and proves that it exists. *)
Definition E_n1_strong
  :  { x : E | sum_is_inv 1 x }
  := constructive_definite_description (sum_is_inv 1) (sum_inv_uniq_ex 1).(** Represents -1. *)
Definition E_n1 : E := proj1_sig E_n1_strong.(**
  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 : ring_scope.

Theorem prod_n1_x_inv : forall x : E, sum_is_inv x ({-1} # x).