thus replace the paths $(b,\ldots,d,\ldots, c)$ and $(a,\ldots g,\ldots, d)$ by reduced paths $(b,b_1,\ldots,w, \ldots,c_1,c)$ and $(a,a_1,\ldots w)$ contained in $V_i\cup\ldots\cup V_{i+k-1}$, except possibly for the endpoints. We may assume that at most $a\in V_{i+k}$ since we may exchange the roles of $V_i$ and $V_{i+k}$ and replace the elements of $V_i$ instead.

By induction assumption, at least one of $(a_1,\ldots w,\ldots b_1)$ and $(a_1,\ldots w,\ldots c_1)$ is reduced, and replacing the new $E_{i+k-1}$-paths by the old $E_{i+k}$-paths, this path remains reduced and changes direction in $g$. If $a,b\in V_{i+k}$, then we may exchange the roles of $V_i$ and $V_{i+k}$ and replace the elements of $V_i$ instead. Thus we see that the path remains reduced when adding the endpoint.

\[r:acl\] For any set $A$, the algebraic closure $\operatorname{acl}(A)$ is the intersection of all nice sets containing it.

Clearly, we may restrict ourselves to finite sets $A$. As before we see that the intersection of all nice sets containing $A$ is contained in $\operatorname{acl}(A)$.

For the converse, assume that $c$ is not in the intersection of all nice sets containing $A$. If $c$ was in $\operatorname{acl}(A)$, then by Lemma \[l:treeacl\] there are $a,b\in A$ with $c\in\operatorname{acl}(ab)$. Then by Lemma \[l:acl2\] $c\in\bigcap_{a,b\in D}D\subset\bigcap_{A\subseteq 