  In \cite{Mil}, Milnor posed the {\em Monotonicity Conjecture} that the set of
parameters within a family of real multimodal polynomial interval maps, for
which the topological entropy is constant, is connected. This conjecture was
proved for quadratic by Milnor & Thurston \cite{MT} and for cubic maps by
Milnor & Tresser, see \cite{MTr} and also \cite{DGMT}. In this paper we will
prove the general case.
