Conditional Entropy of Join as Sum/Corollary 2

Corollary to Conditional Entropy of Join as Sum
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

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

Then:
 * $\AA \subseteq \CC \implies \map H {\AA \mid \DD} \le \map H {\CC \mid \DD} $

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

Proof
Let $\AA \subseteq \CC$.

Then: