= Solution
The <second law of black-hole mechanics>, or <Hawking's area theorem>, states that the total area of future-event-horizon cross-sections cannot decrease toward the future under the <null energy condition> and the usual predictability assumptions. If any horizon generator had $\theta<0$, the twist-free <Null Raychaudhuri equation> and the <null focusing theorem> would force a conjugate point at finite affine parameter. A generator complete to the future cannot pass through such a point while remaining on the achronal boundary of $J^-(\mathcal I^+)$. Hence $\theta\geq0$ everywhere on the regular future horizon, and $dA/d\lambda=\theta A\geq0$ proves the area law.
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