Let $E, F \in \mathcal{F}$ such that $E \subseteq F$. Result
R977: Ambient set is union of subset and complement of subset yields the decomposition $F = E \cup (F \setminus E)$. Applying
R976: Finite disjoint additivity of unsigned basic measure to this union we then obtain
\begin{equation}
\mu(F) = \mu(E \cup (F \setminus E)) = \mu(E) + \mu(F \setminus E)
\end{equation}
Since $\mu \geq 0$, we conclude that
\begin{equation}
\mu(F) = \mu(E) + \mu(F \setminus E) \geq \mu(E)
\end{equation}
$\square$
