  A future holographic screen is a hypersurface of indefinite signature,
foliated by marginally trapped surfaces with area $A(r)$. We prove that $A(r)$
grows strictly monotonically. Future holographic screens arise in gravitational
collapse. Past holographic screens exist in our own universe; they obey an
analogous area law. Both exist more broadly than event horizons or dynamical
horizons. Working within classical General Relativity, we assume the null
curvature condition and certain generiticity conditions. We establish several
nontrivial intermediate results. If a surface $\sigma$ divides a Cauchy surface
into two disjoint regions, then a null hypersurface $N$ that contains $\sigma$
splits the entire spacetime into two disjoint portions: the
future-and-interior, $K^+$; and the past-and-exterior, $K^-$. If a family of
surfaces $\sigma(r)$ foliate a hypersurface, while flowing everywhere to the
past or exterior, then the future-and-interior $K^+(r)$ grows monotonically
under inclusion. If the surfaces $\sigma(r)$ are marginally trapped, we prove
that the evolution must be everywhere to the past or exterior, and the area
theorem follows. A thermodynamic interpretation as a Second Law is suggested by
the Bousso bound, which relates $A(r)$ to the entropy on the null slices $N(r)$
foliating the spacetime. In a companion letter, we summarize the proof and
discuss further implications.
