= Solution
The <second law of black-hole mechanics> states that, assuming the <null energy condition> and the usual predictability hypotheses, the area of cross-sections of a future <event horizon> never decreases toward the future.
Indeed, the horizon generators form a hypersurface-orthogonal null congruence, so their twist vanishes. If the expansion were negative anywhere, the <null focusing theorem> would make it diverge to $-\infty$ within finite affine parameter, provided the generator extends that far. This produces a <conjugate point>, after which the generator cannot remain on an <achronal set> such as an event horizon. Future completeness rules out the alternative that the generator simply ends first. Therefore $\theta\geq0$ everywhere, and
$$
\frac{dA}{d\lambda}=\theta A\geq0.
$$
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