Definition CoPrime (a b : nat) := gcd (Z_of_nat a) (Z_of_nat b) 1.

Lemma chineseRemainderTheorem : forall x1 x2 : nat, CoPrime x1 x2 -> forall (a b : nat) (pa : a < x1) (pb : b < x2), {y : nat | y < x1 * x2 /\ a = snd (proj1_sig (modulo x1 (ltgt1 _ _ pa) y)) /\ b = snd (proj1_sig (modulo x2 (ltgt1 _ _ pb) y))}.