Conditional Entropy Decreases if More Given/Corollary

Corollary to Conditional Entropy Decreases if More Given
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

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

Then:
 * $\map H \AA \ge \map H {\AA \mid \DD} $

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

Proof
Let $\NN := \set {\O, \Omega}$ be the trivial $\sigma$-algebra.

Then: