(**
  Represents the non-commutative ring formed
  by addition and multiplication over E.
*)
Definition ring := Ring.ring E 0 1 {+} {#} distinct_0_1 sum_is_assoc sum_is_comm sum_id_l sum_inv_l_ex prod_is_assoc prod_id_l prod_id_r prod_sum_distrib_l prod_sum_distrib_r.(**
  Represents the abelian group formed by
  addition over E.
*)
Definition sum_abelian_group := Ring.sum_abelian_group ring.(**
  Represents the group formed by addition
  over E.
*)
Definition sum_group := Ring.sum_group ring.(**
  Represents the monoid formed by addition
  over E.
*)
Definition sum_monoid := Ring.sum_monoid ring.(**
  Represents the monoid formed by
  multiplication over E.
*)
Definition prod_monoid := Ring.prod_monoid ring.

Theorem distinct_1_0 : E_1 (c := r) <> E_0 (c := r).