(* Step 5 *)

(* The full condition over p and q *)
Definition cond_pq (p q : Z) := cond_pqb p q /\ (distinct_parity p q).(* The (new) set of positive rational coordinates of the unit circle *)
Definition in_ucp_set (x y : R) :=
  exists p : Z, exists q : Z,
  (x = (fst (ucp p q)) /\ y = (snd (ucp p q)) \/
   x = (snd (ucp p q)) /\ y = (fst (ucp p q))) /\ (cond_pq p q).

Lemma nrat_pos_coo1 : forall x y : R, (in_ucp_set x y) -> (in_ucp_setb x y).