Total Probability Theorem/Conditional Probabilities

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\set {B_1, B_2, \ldots}$ be a partition of $\Omega$ such that $\forall i: \map \Pr {B_i} > 0$.

Let $C \in \Sigma$ be an event independent to any of the $B_i$.

Let $\map \Pr C > 0$.

Then:
 * $\ds \forall A \in \Sigma: \condprob A C = \sum_i \condprob A {C \cap B_i} \, \map \Pr {B_i}$

Proof
First define $Q_C := \condprob {\, \cdot} C$.

Then, from Conditional Probability Defines Probability Space, $\struct {\Omega, \Sigma, Q_C}$ is a probability space.

Moreover:

Therefore the Total Probability Theorem also holds true.

Hence we have: