I would like to get an explanation of how to proof the uniqueness of the solution of s system of congruences. I have already read 4 books about it but none of them does not explain me an specific part of the reasoning. The following image has also this part
I don't understand why $lcm(m_1,\dots,m_r)\mid (X_1-X_2)$.
I tried to prove it in a simpler case, considering that $m_1$ and $m_2$ (relatively prime) are divisors $a$. Then
$$a=m_1x=m_2y$$
For some integers $x$ and $y$. But I couldn't combine this and the fact of $m_1r+m_2s=1$, with $r, s\in \mathbb{Z}$, to conclude that $m_1m_2\mid a$.
Lemma: If $m_1,m_2,\dots,m_k$ are non-zero integers, then let $M=[m_1,\dots,m_k],$ the LCM. If $N$ is an integer such that $m_i\mid N$ for all $i=1,\dots,k,$ then $M\mid N.$
Proof: By division algorithm, there are integers $q,r$ with $N=Mq+r,$ and $0\leq r <M.$
But then $r=N-Mq$ is divisible by each $m_i.$ So, $r=0,$ or else we’ve found a smaller positive common multiple, contradicting that $M$ is the least common multiple