  We show that the second order field equations characterizing extremal
solutions for spherically symmetric, stationary black holes are in fact implied
by a system of first order equations given in terms of a prepotential W. This
confirms and generalizes the results in [14]. Moreover we prove that the
squared prepotential function shares the same properties of a c-function and
that it interpolates between M^2_{ADM} and M^2_{BR}, the parameter of the
near-horizon Bertotti-Robinson geometry. When the black holes are solutions of
extended supergravities we are able to find an explicit expression for the
prepotentials, valid at any radial distance from the horizon, which reproduces
all the attractors of the four dimensional N>2 theories. Far from the horizon,
however, for N-even our ansatz poses a constraint on one of the U-duality
invariants for the non-BPS solutions with Z \neq 0. We discuss a possible
extension of our considerations to the non extremal case.
