there exist $a,b\in A$ with $x\in\operatorname{acl}(ab)$.

We may assume that $A$ is finite. By induction it suffices to prove that if $d\in\operatorname{acl}(bc),g\in\operatorname{acl}(ad)$, then $g\in\operatorname{acl}(ab)\cup\operatorname{acl}(bc)\cup\operatorname{acl}(ac)$.

By Corollary \[r:flagcl\] there is a reduced path $\gamma_1=(b,\ldots,d,\ldots, c)$ changing direction in $d$ and a reduced path $\gamma_2=(d,\ldots ,g,\ldots, a)$ changing direction in $g$. If $\gamma_1\cup\gamma_2\in V_i\cup V_{i+1}$ for some $i$, then clearly either $(a,\ldots,g\ldots,d,\ldots,b)$ or $(a,\ldots,g\ldots,d,\ldots,c)$ is reduced.

Now assume that $\gamma_1\cup\gamma_2\in V_i\cup\ldots\cup V_{i+k}$. Clearly we may assume that $d$ is not contained in every reduced path from $a$ to $b$ or in every reduced path from $a$ to $c$. Furthermore, we may reduce to the case where no vertex of $\gamma_1$ is contained in every reduced path from $b$ to $c$ and no vertex of $\gamma_2$ is contained in every reduced path from $a$ to $d$.

By Remark \[r:reducedpath\] we may therefore replace any element $x\in V_{i+k}$ in the interior of $\gamma_1$ and $\gamma_2$ by an $E_{i+k-1}$ path between its neighbours. If $d\in V_{i+k}$, we may replace $d$ as follows: the neighbour $w$ of $d$ in $\gamma_2$ either appears in the $E_{i+k-1}$ path between the neighbours of $d$ in $\gamma_1$ or that path extends to $w$ on one of its ends. We may 