We have \begin{equation} \begin{split} \{ X \leq b \} = \{ X \in (-\infty, b] \} & = \{ X \in (-\infty, a] \text{ or } x \in (a, b] \} \\ & = \{ X \leq a \text{ or } a < X \leq b \} \\ & = \{ X \leq a \} \cup \{ a < X \leq b \} \\ \end{split} \end{equation} Subtracting $\{ X \leq a \}$ from both sides, mapping both sides in $\mathbb{P}$, and applying R2060: Probability of set difference then gives \begin{equation} \mathbb{P}(a < X \leq b) = \mathbb{P}( \{ X \leq b \} \setminus \{ X \leq a \} ) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a) \end{equation} $\square$