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

(i)	$E_0, E_1, \ldots, E_N \in \mathcal{F}$ are each an D1716: Event in $P$
(ii)	\begin{equation} \mathbb{P}(E_0) = 1 \end{equation}

Since we can write \begin{equation} \bigcap_{n = 0}^N E_n = E_0 \cap \left( \bigcap_{n = 1}^N E_n \right) = E_0 \cap F \end{equation} with $F : = \bigcap_{n = 1}^N E_n$, then this result is a consequence of R4331: Probability of binary intersection with an almost sure event. $\square$