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 $P$
(ii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}
(iii)	\begin{equation} \mathbb{P}(F \mid E) = 1 \end{equation}

By R2556: Probability calculus expression for probability conditioned on event of nonzero probability, we have \begin{equation} 1 = \mathbb{P}(F \mid E) = \frac{\mathbb{P}(E \cap F)}{\mathbb{P}(E)} = \mathbb{P}(E \cap F) \end{equation} Whence, from R4331: Probability of binary intersection with an almost sure event, we have \begin{equation} 1 = \mathbb{P}(E \cap F) = \mathbb{P}(F) \end{equation} $\square$