By definition, a
D84: Sigma-algebra is closed under complements so that $F \setminus E \in \mathcal{F}$. Since $E \subseteq F$, we can use result
R977: Ambient set is union of subset and complement of subset to artition $F$ into
\begin{equation}
F = E \cup (F \setminus E)
\end{equation}
We can now use results
and the previous equality to obtain
\begin{equation}
\mu(F)
= \mu(E \cup (F \setminus E))
= \mu(E) + \mu(F \setminus E)
\end{equation}
Subtracting $\mu(E)$ from both sides now gives
\begin{equation}
\mu(F \setminus E)
= \mu(F) - \mu(E)
\end{equation}
Since $\mu(E) < \infty$, then the difference $\mu(F) - \mu(E)$ is well-defined even if $\mu(F) = \infty$. $\square$