Proof
functional_extensionality
(prod {-1}) {-}
(fun x
=> sum_inv_uniq x (- x) ({-1} # x)
(sum_neg_def x)
(prod_n1_x_inv x)).

(** Accepts one element, x, and proves that x -1 equals the additive negation of x. *)
Theorem prod_x_n1_neg
:  forall x : E, x # {-1} = - x.