Conditional Entropy of Join as Sum/Corollary 3

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

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

Then:
 * $\AA \subseteq \CC \implies \map H \AA \le \map H \CC $

where:
 * $\map H \cdot$ denotes the entropy

Proof
Let $\AA \subseteq \CC$.

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

Then: