Lemma for_exists_ab : forall u v m n : Z, v <= u -> u * u = m * m + n * n -> v * v = n * n - m * m -> 1 < u -> 1 < v -> Zodd v -> Zodd u -> rel_prime u v -> exists two : Z * Z, let (a, b) := two in (u - v = 4 * (a * a) /\ u + v = 2 * (b * b) \/ u - v = 2 * (b * b) /\ u + v = 4 * (a * a)) /\ 0 < a /\ 0 < b.