= Solution
The <chain rule for conditional entropy> gives
$$
H(X,Y\mid Z)-H(X\mid Z)=H(Y\mid X,Z)\geq0.
$$
Indeed, for every conditioning value $(x,z)$ of positive <probability>, each summand $-p(y\mid x,z)\log_2p(y\mid x,z)$ is nonnegative. Its sum is zero exactly when the <conditional probability mass function> is concentrated at one value. Averaging over $(X,Z)$ therefore proves both the inequality and its equality criterion: \b[equality holds exactly when $Y=f(X,Z)$ <almost surely> for some function $f$.] Conditioning values of <probability> zero impose no restriction. This is a statement about <discrete random variables>; a <differential entropy> need not be nonnegative.
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