  Past holographic screens are codimension-one surfaces of indefinite signature
that are foliated by marginally anti-trapped surfaces called leaves. Future
holographic screens are defined similarly except with marginally trapped
leaves. Bousso and Engelhardt recently showed that the leaves of past and
future holographic screens have monotonic area. We prove a stronger area law
that shows that subregions of leaves also have monotonic area. For every past
and future holographic screen, there exists a family of leaf-orthogonal curves
called the fibration of the screen. Any region in a leaf can be translated
along the fibration to a leaf of larger area. Our result states that the area
of the subregion grows as it is translated.
