= Solution
<Hawking's area theorem> states that the area of later cross-sections of a classical future <event horizon> cannot be smaller than that of earlier cross-sections, including the sum over initially disconnected components. Its hypotheses include the <null energy condition>, the Einstein equations, and the global regularity/predictability conditions that exclude future naked pathologies of the horizon generators. A convenient sufficient version assumes those generators are future complete. Without the energy and global hypotheses the assertion is not unconditional.
Let $k^a$ be an affinely parametrized null generator, with expansion $\theta$. Because the horizon is a <null hypersurface>, the generator congruence has zero <null twist>. The <Null Raychaudhuri equation> in four dimensions gives
$$
\frac{d\theta}{d\lambda}=-\frac12\theta^2-\sigma_{ab}\sigma^{ab}-R_{ab}k^ak^b
\le-\frac12\theta^2.
$$
The <null shear> square is nonnegative and the <null energy condition> makes $R_{ab}k^ak^b=8\pi G T_{ab}k^ak^b\ge0$; a cosmological term has zero null contraction. Suppose $\theta(\lambda_0)<0$. Integration of this inequality forces $\theta\to-\infty$ within affine distance at most $2/|\theta(\lambda_0)|$. This produces a focal point to the initial horizon cross-section. Beyond it the <null geodesic> cannot remain on an <achronal boundary>, whereas an <event horizon> is precisely such a boundary. Future completeness, or the corresponding standard predictability condition ensuring continuation over this interval, gives the contradiction. Hence
$$
\theta\ge0,\qquad \frac{d(dA)}{d\lambda}=\theta\,dA\ge0.
$$
This proves the local area increase along every smooth generator bundle. Generators can enter the horizon at past crease points, but cannot leave through future endpoints on a regular horizon under these hypotheses. New generators add area, rather than subtracting it, so the result extends across the nonsmooth merger set and to whole horizon cross-sections. This is the <future-complete horizon focusing proof of the area theorem>.
Choose an early cross-section with the two separate components of areas $A_1,A_2$, and a later cross-section of the settled merged hole. Following the old generators forward maps their area elements injectively into the later horizon; a future crossing/focal point would violate the same achronality argument. Each element has nondecreased area. Any additional generators contribute nonnegative area, giving
$$
\boxed{A_3\ge A_1+A_2.}
$$
This is a comparison of event-horizon cross-sections, not an assertion that set containment alone bounds the areas of arbitrary apparent horizons.
For a <Schwarzschild black hole>, $r_h=2Gm/c^2$ and $A=4\pi r_h^2=16\pi G^2m^2/c^4$. Applying the merger inequality to the two equal initial masses and the final Schwarzschild mass gives
$$
\frac{16\pi G^2M^2}{c^4}\ge\frac{32\pi G^2m^2}{c^4},\qquad
\boxed{M\ge\sqrt2\,m.}
$$
For initially well-separated holes at rest, with no extra incoming energy or external work, energy conservation gives $E_{\rm rad}=(2m-M)c^2$. The <area bound on equal-mass Schwarzschild merger radiation> is consequently
$$
\boxed{E_{\rm rad}\le(2-\sqrt2)mc^2,\qquad
\frac{E_{\rm rad}}{2mc^2}\le1-\frac1{\sqrt2}\simeq29.3\%.}
$$
This is the maximum permitted by the area/energy inequalities, not a demonstration that a dynamical merger attains it. A reversible limiting area change would be needed to saturate the bound. If initial separation, binding energy or external agents change the initial total energy, replace $2mc^2$ by that energy when computing escaped radiation. Quantum evaporation does not contradict the theorem: the classical energy-condition hypotheses are not retained unchanged in that setting.
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