Using the definition of an
D5811: Event-conditional probability and the result
R3646: Countable partition additivity of probability measure, we have
\begin{equation}
\mathbb{P}(E)
= \sum_{n \in \mathbb{N}} \mathbb{P} (E \cap F_n)
= \sum_{n \in \mathbb{N}} \frac{\mathbb{P} (E \cap F_n)}{\mathbb{P}(F_n)} \mathbb{P}(F_n)
= \sum_{n \in \mathbb{N}} \mathbb{P}(E \mid F_n) \mathbb{P}(F_n)
\end{equation}
$\square$
