= Solution
The <second law of black-hole mechanics> says that the total area of future <event horizon> cross-sections cannot decrease toward the future, assuming the classical <Einstein field equations>, the <null energy condition>, suitable global predictability and regularity, and here the stipulated future completeness of its generators.
A future <event horizon> is an <achronal boundary> generated by <null geodesics>. If a smooth generator had $\theta<0$, the <null focusing theorem> would produce a <conjugate point to a spacelike surface> within finite <affine parameter>. Future completeness lets the generator reach it, but beyond such a point it could no longer remain on the <achronal boundary>, a contradiction. Hence $\theta\ge0$ wherever the <null expansion> is defined. A transported area element obeys
$$
\frac{d}{d\lambda}\log dA=\theta\ge0.
$$
Every generator present on an early cut continues to a later cut without decreasing its area contribution. New generators can join at past endpoints, adding area; future endpoints in a regular <spacetime> are excluded. Therefore
$$
\boxed{A_{\mathrm{later}}\ge A_{\mathrm{earlier}}.}
$$
This argument also covers the sum of initially disconnected <event horizons> of the <black holes> in a merger. Nonsmooth crease sets require the standard regularity treatment; they do not provide a classical area-decrease loophole under these hypotheses. Quantum negative-energy flux can violate the classical <null energy condition>, so this is a classical <black-hole area theorem>, not an unconditional law for evaporating <black holes>.
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