Conditional Entropy Decreases if More Given

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

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

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

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