ILL 1992 Problem 56

A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If $x, u,$ and $v$ are three distinct vertices such that $x \to u$ and $x \to v$ , then $u \to w$ and $v \to w$ for some vertex $w$ . Suppose that $x \to u \to y \to\cdots \to z$ is a path of length $n$ , that cannot be extended to the right (no arrow goes away from $z$ ). Prove that every path beginning at $x$ arrives after $n$ steps at $z.$