  We prove that the area of sections of future event horizons in space-times
satisfying the null energy condition is non-decreasing towards the future under
any one of the following circumstances: 1) the horizon is future geodesically
complete; 2) the horizon is a black hole event horizon in a globally hyperbolic
space-time and there exists a conformal completion with a ``H-regular'' Scri
plus; 3) the horizon is a black hole event horizon in a space-time which has a
globally hyperbolic conformal completion. (Some related results under less
restrictive hypotheses are also established.) This extends a theorem of
Hawking, in which piecewise smoothness of the event horizon seems to have been
assumed. No assumptions about the cosmological constant or its sign are made.
We prove smoothness or analyticity of the relevant part of the event horizon
when equality in the area inequality is attained - this has applications to the
theory of stationary black holes, as well as to the structure of compact Cauchy
horizons. In the course of the proof we establish several new results
concerning the differentiability properties of horizons.
