= Solution
Since the $B_i$ form a <partition of a set> of the <sample space>, the <events> $A\cap B_i$ are pairwise disjoint and their <set union> is $A$. By <countable additivity> of <probability>, followed by the definition of <conditional probability>,
$$
\mathbb P(A)=\sum_{i=1}^{\infty}\mathbb P(A\cap B_i)
=\boxed{\sum_{i=1}^{\infty}\mathbb P(A\mid B_i)\mathbb P(B_i)}.
$$
Every ratio is legitimate because each $\mathbb P(B_i)>0$. This proves the countable <law of total probability>.
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