Definition both_odd (x y : Z) := Zodd x /\ Zodd y.Definition distinct_parity (a b : Z) :=
   (Zeven a) /\ (Zodd b) \/ (Zodd a) /\ (Zeven b).

Lemma Zodd_def1 : forall z : Z, (Zodd z) -> exists k : Z, z = 2 * k + 1.