(strictly) after $a$. We stress that if a position is not $(\sigma, \tau, k)$-rich than it is not necessarily $(\sigma, \tau, k)$-poor, and vice versa.

\[lemma:richpoor\] If a word has no $(\sigma, \tau, k)$-rich position then it has a $(\sigma, \tau, k+1)$-poor position.

Let $w$ be a word with no $(\sigma, \tau, k)$-rich position. Towards a contradiction assume that $w$ has no $(\sigma, \tau, k+1)$-poor position. Then, by definition, for each position $i$ of $w$ there are at least $k+1$ $\sigma$-labeled positions strictly left of $i$ or at least $k+1$ $\tau$-labeled positions strictly right of $i$.

In particular there are at least $(k+1)$ $\sigma$-labeled positions to the left of the last position of $w$. Let $i$ be the $(k+1)$th $\sigma$-labeled position of $w$. Since $i$ has only $k$ $\sigma$-labeled positions to its left, it must have at least $k+1$ $\tau$-labeled positions to is right. Hence, $i$ is $(\sigma, \tau, k)$-rich; a contradiction.

As outlined above, the proof of Theorem \[theorem:successorandorder\] follows immediately from the following small model property and Lemma \[lemma:formulaToConstraint\]. The proof of the small model property is in the same spirit as the proof of Lemma 2.3 in [@GradelR99].

\[lemma:succordfinitemodel\] If a constraint problem $C$ has a finite 