Solution
= Solution
Apply the definitions of <conditional entropy> and insert $H(X,Y)$:
$$
\begin{aligned}
H(Z,X\mid Y)&=H(Z,X,Y)-H(Y)\\
&=[H(X,Y)-H(Y)]+[H(Z,X,Y)-H(X,Y)]\\
&=H(X\mid Y)+H(Z\mid X,Y).
\end{aligned}
$$
Inserting $H(Z,Y)$ instead gives the alternative <chain rule for conditional entropy>
$$
\boxed{H(Z,X\mid Y)=H(Z\mid Y)+H(X\mid Z,Y).}
$$
Both are expansions of the same joint conditional uncertainty, with the variables exposed in opposite orders.