Editing Chinese remainder theorem (section)

===Existence (first proof)===
The map
:<math>x \bmod N \mapsto (x \bmod n_1, \ldots, x\bmod n_k)</math>
maps [[congruence class]]es modulo {{math|''N''}} to sequences of congruence classes modulo {{math|''n<sub>i</sub>''}}. The proof of uniqueness shows that this map is [[injective]]. As the [[domain of a function|domain]] and the [[codomain]] of this map have the same number of elements, the map is also [[surjective]], which proves the existence of the solution.

This proof is very simple but does not provide any direct way for computing a solution. Moreover, it cannot be generalized to other situations where the following proof can.