## CRT Proof help

I'm trying to understand a Chinese Remainder Theorem proof, with the base case $n=2$, and I'm hoping someone can help me in my jam.

Statement
Suppose $gcd(m_1,m_2)=1$ Then the system of congruences
$x\equiv{a} (mod$ $m_1)$
$x\equiv{b} (mod$ $m_2)$

has a unique solution $(mod$ $mn)$

Proof
Under conditions
$x-a=m_1R$ and $x-b=m_2T$. So

$x=a+m_1R=b+m_2T$. So

$m_1R-m_2T=b-a
$

This has a solution $R_0$ and $T_0$ and the full set of solutions set given by
$R=R_0+m_2k$ and $T=T_0+m_1k$

So $x=a+m_1R_0+m_1m_2k=b+m_2T_0+m_1m_2k$

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 $x=a+m_1R_0+m_1m_2k=b+m_2T_0+m_1m_2k$ demonstrates that this x solves the system and is unique $(mod$ $mn)$