The corollary:
$$H(X, Y|Z)=H(Y|Z) + H(X|Y, Z)$$
My proof:
$\begin{align}
H(X,Y|Z) &= - \sum_zp(z)H(X, Y|Z=z) \\
&= - \sum_zp(z)\sum_x\sum_yp(x, y|z)\log p(x,y|z)\\
&=-\sum_x\sum_y\sum_zp(x,y,z)\log p(x|y, z)p(y|z)\\
&=-\sum_x\sum_y\sum_zp(x,y,z)\log p(x|y, z) - \sum_x\sum_y\sum_z p(x,y,z) \log p(y|z)\\
&=-\sum_y\sum_z p(y, z) \sum_x p(x|y,z)\log p(x|y,z) - \sum_z p(z) \sum_yp(y|z) \log p(y|z)\\
&=-\sum_y\sum_zp(y,z)H(X|Y=y,Z=z) - \sum_z p(z) H(Y|Z=z)\\
&=H(X|Y，Z)+ H(Y|Z)
\end{align}$
I am not sure if my proof is right, especially in the last three steps.