Conditional Entropy of Join as Sum

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

Let $\AA, \CC, \DD \subseteq \Sigma$ be finite sub-$\sigma$-algebras.

Then:
 * $\ds \map H {\AA \vee \CC \mid \DD} = \map H {\AA \mid \DD} + \map H {\CC \mid \AA \vee \DD} $

where:
 * $\map H {\cdot \mid \cdot}$ denotes the conditional entropy
 * $\vee$ denotes the join