Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E, F \in \mathcal{F}$ are each an D1716: Event in $P$

This result is a particular case of R4318: Inclusion-exclusion principle for unsigned basic measure of binary union. $\square$

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E, F \in \mathcal{F}$ are each an D1716: Event in $P$

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} $\square$