Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$E, F \in \mathcal{F}$ are each a D1109: Measurable set in $M$
(ii)	$E \subseteq F$ is a D78: Subset of $F$
(iii)	\begin{equation} \mu(E) < \infty \end{equation}

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

(i)	R4321: Set and set complement are disjoint
(ii)	R976: Finite disjoint additivity of unsigned basic measure

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 $\mu(F) = \infty$. $\square$