$B_1$. The element $b'$ is the $(d, \gamma_\ell, \tau)$-witness of $a$ in ${\ensuremath{\mathfrak{A}}}'$.

    2.  If $b \notin B$ is a $\tau$-labeled element that has a $(\bar d, \gamma_\ell, \sigma)$-witness $a \in W(b)$ in ${\ensuremath{\mathfrak{A}}}$ then the binary $T$-type of $(b, a')$ in ${\ensuremath{\mathfrak{A}}}'$ is $\gamma_\ell$ where $a'$ is the $\ell$th element of $A_{1}$. The element $a'$ is the $(d, \gamma_\ell, \tau)$-witness of $b$ in ${\ensuremath{\mathfrak{A}}}'$.

3.  If a tuple $(a, b)$ with $\sigma$-labeled $a$, $\tau$-labeled $b$ and satisfying $d$ has not been assigned a binary $T$-type in ${\ensuremath{\mathfrak{A}}}'$ so far, then it inherits its type from ${\ensuremath{\mathfrak{A}}}$.

Observe that no conflicts arise from (A1) and (A2). Furthermore, for all $\sigma$- and $\tau$-labeled elements $a$ there is a set $W'(a)$ satisfying (1) and (2). Moreover, no conflicts with universal constraints arise since no new types have been created. This concludes the case when there is a $(\sigma, \tau, k)$-rich position in ${\ensuremath{\mathfrak{A}}}$.

If there is no $(\sigma, \tau, k)$-rich position $u$ in ${\ensuremath{\mathfrak{A}}}$ then Conditions (1) and (2) are already satisfied for all $\sigma$- and $\tau$-labeled positions $a$ and $b$ satisfying $d = x < y \wedge {\ensuremath{{S}}}(x,y)$. To see this we argue as follows. By Lemma \[lemma:richpoor\] there is a 