If $\kappa$ is a regular, uncountable cardinal, $S$ is a stationary subset of $\kappa$ and $f:\kappa\rightarrow\kappa$ is regressive on $S$ (that is, $f(\alpha)<\alpha$ for any $\alpha\in S$ then there is some $\gamma$ and some stationary $S_0\subseteq S$ such that $f(\alpha)=\gamma$ for any $\alpha\in S_0$