Section Theorems.
(** Represents an arbitrary group.  Note: we use Variable rather than Parameter to ensure that the following theorems are generalized w.r.t g. *)
Variable g : Group.
(** Represents the set of group elements. *)
Let E := E g.
(** Represents the monoid structure formed by op over E. *)
Definition op_monoid := Monoid.monoid E 0 {+} op_is_assoc op_id_l op_id_r.
(** Accepts one group element, x, and asserts that x is the left identity element. *)
Definition op_is_id_l := Monoid.op_is_id_l op_monoid.
(** Accepts one group element, x, and asserts that x is the right identity element. *)
Definition op_is_id_r := Monoid.op_is_id_r op_monoid.
(** Accepts one group element, x, and asserts that x is the identity element. *)
Definition op_is_id := Monoid.op_is_id op_monoid.

(** Proves that 0 is the identity element. *)
Theorem op_id
:  op_is_id 0.