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))}.