I'm trying to understand a Chinese Remainder Theorem proof, with the base case , and I'm hoping someone can help me in my jam.

Statement

Suppose Then the system of congruences

has a unique solution

Proof

Under conditions

and . So

. So

This has a solution and and the full set of solutions set given by

and

So

This apparently immediately demonstrates that this x solves our system of equations and also shows the solution is unique (modulo mn).

That final deduction confuses me. Can someone explain how demonstrates that this x solves the system and is unique