How can I prove this law of “generalized” total probability?

Given that $X_i$ is a partition of the probability space, and $Y,Z$ are events in the probability space such that $P[Y \cap X_i] > 0$ for all $i$, how can I prove that $$P[\,Z\;|\;Y\,] = \sum_{i=1}^{\infty} P[\,Z\;|\;Y \cap X_i\,] \cdot P[\,X_i\; |\; Y\,]\quad?$$

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Because $X_i$ are a partition, $$P(Z \cap Y) = \sum_i P(Z \cap Y \cap X_i)$$ Now apply the definition of conditional probability, once to the left-hand side and twice to the right-hand side, to get your result.