B_3$ with disjoint $B_i$ and $|B_i| = |\Gamma|$ contain the last $k$ $\tau$-labeled elements. Assume that $\Gamma {\ensuremath{\mathrel{\smash{\stackrel{\scriptscriptstyle{
    \text{def}}}{=}}}} \;}\{\gamma_1, \ldots, \gamma_m\}$. Then:

1.  Witnesses for elements in $A \cup B$ are assigned as follows:

    1.  If $a \in A_i$ is a $\sigma$-labeled element that has a $(d, \gamma_\ell, \tau)$-witness $b \in W(a)$ in ${\ensuremath{\mathfrak{A}}}$ then the binary $T$-type of $(a, b')$ is set to $\gamma_\ell$ where $b'$ is the $\ell$th element of $B_i$. The element $b'$ is the intended $(d, \gamma_\ell, \tau)$-witness of $a$ in ${\ensuremath{\mathfrak{A}}}'$.

    2.  If $b \in B_i$ 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 $(b, a')$ is set to $\gamma_\ell$ where $a'$ is the $\ell$th element of $A_{i+1}$ (where $i+1$ is calculated modulo 3). The element $a'$ is the intended $(\bar d, \gamma_\ell, \tau)$-witness of $b$ in ${\ensuremath{\mathfrak{A}}}'$.

2.  Witnesses for all other tuples of $\sigma$- and $\tau$-labeled elements are assigned as follows:

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