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.

Lemma card_quaternion : #|'Q_m| = m. Proof. by case defQ. Qed.
Lemma Grp_quaternion : GrpQ. Proof. by case defQ. Qed.