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<br />
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)