= Solution
The <second law of black-hole mechanics> states that the area of spatial cross-sections of a future <event horizon> cannot decrease toward the future, provided the <null energy condition> and the standard global assumptions hold.
Let $k^a$ be an affinely parametrized horizon generator. It is hypersurface-orthogonal, so its <null twist> vanishes. The <Null Raychaudhuri equation> in $D$ dimensions and the Einstein equation give
$$
\frac{d\theta}{d\lambda}
=-\frac{\theta^2}{D-2}
-\widehat\sigma_{ab}\widehat\sigma^{ab}
-R_{ab}k^ak^b
\leq-\frac{\theta^2}{D-2}.
$$
If $\theta(\lambda_0)<0$, integration implies that $\theta$ diverges to $-\infty$ within affine distance at most $(D-2)/|\theta(\lambda_0)|$. The resulting focal point would make the generator leave the achronal boundary that defines the event horizon, contradicting its assumed future completeness. Hence $\theta\geq0$ everywhere. Since <null expansion> obeys
$$
\frac{dA}{d\lambda}=\theta A,
$$
every horizon area element, and therefore every complete cross-section area, is nondecreasing.
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