between appropriate elements of $\gamma'$.

To fix notation let us assume that the neighbours of $x$ in $\gamma$ are $y,z\in V_{k-1}$. By Lemma \[l:cycle\] we may replace the path $(y,x,z)$ by an $E_k$-path. This path is obviously still reduced and $x$ is on the $E_{k+1}$-path from $y$ to $z$.

\[l:nicepath\] Suppose $D$ is a nice set containing an $E_j$-path from $a$ to $b$. Suppose furthermore that $\gamma=(a=y_0,\ldots, y_t=b)$ is a reduced path such that the composition with the $E_j$-path is a simple cycle. If $\gamma$ changes direction in $c$, then $c\in D$.

Using the fact that $M_n$ is $\omega$-saturated and homogeneous for strong substructures we can now describe the algebraic closure:

\[l:acl2\] The algebraic closure $\operatorname{acl}(a,b)$ is the intersection of all nice sets containing $a,b$.

Since there are finite nice sets containing $a,b$, the intersection of all nice sets containing $a,b$ is certainly finite and invariant over $a,b$, hence contained in $\operatorname{acl}(a,b)$.

Conversely let $D$ be a nice set containing $\{a,b\}$ and $c\notin D$. If there is no reduced path between $a$ and $b$ changing direction in $c$, then $c$ has infinitely many conjugates over $ab$, so $c\notin\operatorname{acl}(ab)$.

So suppose there is a reduced path $\gamma'$ from $a$ to $b$ 