Section Quaternion.
Variable n : nat.
Hypothesis n_gt2 : n > 2.
Let m := (2 ^ n)%N.
Let q := (2 ^ n.-1)%N.
Let r := (2 ^ n.-2)%N.
Let GrpQ := 'Q_m \isog Grp (x : y : (x ^+ q, y ^+ 2 = x ^+ r, x ^ y = x^-1)).
Let defQ :  #|'Q_m| = m /\ GrpQ.

(* Specifications: *)
Lemma card_quaternion : #|'Q_m| = m. Proof. by case defQ. Qed.
Lemma Grp_quaternion : GrpQ. Proof. by case defQ. Qed.
End Quaternion.
Lemma eq_Mod8_D8 : 'Mod_8 = 'D_8. Proof. by []. Qed.
Section ExtremalStructure.