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

(i)	$E, F_0, F_1, F_2, \dots \in \mathcal{F}$ are each an D1716: Event in $P$
(ii)	$F_0, F_1, F_2, \ldots$ is a D5143: Set partition of $\Omega$
(iii)	\begin{equation} \forall \, n \in \mathbb{N} : \mathbb{P}(F_n) > 0 \end{equation}

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} $\square$