If \(a ≡ b \bmod m\) and \(c ≡ d \bmod m\):
$$\begin{gathered} a = b + mk \\ c = d + ml\end{gathered}$$
If we add them:
$$\begin{gathered} a + c = b + d + mk + ml \\ a + c = b + d + m(k + l) \end{gathered}$$
This proves that \(a +c ≡ b + d \bmod m\). Using a similar line of reason, we can show \(a -c ≡ b - d \bmod m\).