= Solution
The <second law of black-hole mechanics> states that the total area of a classical future <event horizon> cannot decrease toward the future, under the appropriate causal and energy assumptions. Use the <null convergence condition> $R_{ab}U^aU^b\geq0$; in general relativity the <Einstein field equations> imply it from the <null energy condition>. Assume, as allowed here, that the horizon's null generators remain on the horizon and are complete to the future. The horizon is an <achronal boundary>, and its smooth generators are hypersurface-orthogonal, so their <null twist> is zero. The <Null Raychaudhuri equation> therefore gives
$$
\frac{d\theta}{d\lambda}=-\frac12\theta^2-\widehat\sigma^2-R_{ab}U^aU^b
\leq-\frac12\theta^2.
$$
If some smooth horizon point had $\theta_0<0$, then while the optical description remains regular, $\theta$ stays negative and
$$
\frac{d}{d\lambda}\frac1\theta=-\frac{\theta'}{\theta^2}\geq\frac12.
$$
Thus
$$
\theta(\lambda)\leq\frac{\theta_0}{1+\tfrac12\theta_0(\lambda-\lambda_0)},
$$
forcing focusing no later than $\lambda_0+2/|\theta_0|$.
The assumed geometrical result is that this focusing produces a <conjugate point to a spacelike surface>, namely a focal point relative to a small spacelike horizon cross-section through the original point. A null geodesic beyond such a focal point ceases to generate an achronal boundary: a timelike variation connects the cross-section to points just beyond it. Since the cross-section and the generator lie on the achronal horizon, this is impossible. Future completeness ensures that a singular endpoint cannot terminate the generator before the finite focusing parameter. This <future-complete horizon focusing proof of the area theorem> rules out $\theta_0<0$.
Therefore $\theta\geq0$ on smooth horizon pieces, and the transverse-area identity gives
$$
\boxed{\frac{dA}{d\lambda}=\theta A\geq0.}
$$
Integrating over generators proves area nondecrease between later cross-sections. At crease sets, new generators may enter the horizon from the past, adding area; horizon generators do not leave it toward the future under the stated assumptions. Hence the statement also applies to total horizon area through black-hole mergers. <Hawking's area theorem> is classical: quantum stress-energy can violate the null energy condition, so it does not forbid a shrinking evaporating black hole.
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