If $a_{1} \equiv a_{2}\ (\textrm{mod}\ m)$ and $b_{1} \equiv b_{2}\ (\textrm{mod}\ m)$, then

#### Proof

The addition and the subtraction

By the definition of modular congruence, we know that when $a_{1} \equiv a_{2}\ (\textrm{mod}\ m)$, we have

And similarly,

Let (1) - (2), we have

Since $c_{1} - c_{2}$ must be an integer, by the defination, we get

Similarly, let (1) + (2), we will prove that

The multiplication

According to (1) and (2), we can say

Therefore,

So,

Since, $c_1,\ c_2,\ m$ are all integers, the coefficient of $(c_{1} c_{2}m + a_2 c_2 + b_2 c_1) m$ must be an integer. By the definition,