From result
R5715: Probability of union event equals probability of event plus probability of difference, we have
\begin{equation}
\mathbb{P}(E \cup F)
= \mathbb{P}(F) + \mathbb{P}(E \setminus F)
\end{equation}
From result
R5714: Probability of event equals probability of intersection plus probability of difference, we have
\begin{equation}
\mathbb{P}(E \setminus F)
= \mathbb{P}(E) - \mathbb{P}(E \cap F)
\end{equation}
Combining these, we can write
\begin{equation}
\begin{split}
\mathbb{P}(E \cup F)
= \mathbb{P}(F) + \mathbb{P}(E \setminus F)
= \mathbb{P}(F) + \mathbb{P}(E) - \mathbb{P}(E \cap F)
\end{split}
\end{equation}
$\square$
