= Apparent-horizon area comparison
{title2=$A_{\rm app}\leq A_{\rm EH,initial}$}
The heuristic derivation of the <Penrose inequality> from <Hawking's area theorem> needs an initial area comparison, in addition to enclosure of the <apparent horizon> by the <event horizon>. Set inclusion alone is not an area inequality. An <outer area-minimizing surface> or appropriate enclosing-area hypotheses supply the needed comparison in formulations where it is valid; arbitrary slice-dependent apparent-horizon area need not do so.
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