By definition, $E \triangle F : = (E \setminus F) \cup (F \setminus E)$. Since $E \setminus F$ and $F \setminus E$ are disjoint and since $E$ is contained in $F$ with the measure of $E$ finite, we may use results
to obtain
\begin{equation}
\begin{split}
\mu(E \triangle F)
= \mu(E \setminus F) + \mu(F \setminus E)
= \mu(\emptyset) + \mu(F) - \mu(E)
= \mu(F) - \mu(E)
\end{split}
\end{equation}
$\square$
