Finite Sequences in Set Form Acyclic Graph

Theorem
Let $S$ be a set.

Let $V$ be the set of finite sequences in $S$.

Let $E$ be the set of unordered pairs $\{p, q\}$ of elements of $V$ such that either:


 * $q$ is formed by extending $p$ by one element or
 * $p$ is formed by extending $q$ by one element.

That is:
 * $| \operatorname{Dom}(p) * \operatorname{Dom}(q) | = 1$, where $*$ is symmetric difference and
 * $p \restriction D = q \restriction D$, where $D = \operatorname{Dom}(p) \cap \operatorname{Dom}(q)$

Then $T = (V, E)$ is an acyclic graph.