Section Binary.
Variable (u v a b : nat) (Hu : u <> 0) (Hv : v <> 0) (Huv : is_gcd u v 1).
End Binary.
End Informative_Chinese_Remainder_theorems.

Theorem CRT u v a b : u <> 0 -> v <> 0 -> is_gcd u v 1 -> exists w, rem w u = rem a u /\ rem w v = rem b v /\ 2 < w.