We will only sketch the fourth bullet point of this proof; the remainder is direct (but quite long to write down on paper). Moreover, we will show this for atoms; the result for near-litters is identical. The specifications \( \operatorname {\mathsf{spec}}(S) \) and \( \operatorname {\mathsf{spec}}(T) \) give rise to the same trees \( R^{\mathcal A}\) precisely when

We must show that this holds if and only if

This can be concluded by appealing to the basic behaviour of \( \rho \) and noting the coinjectivity of \( \operatorname {\mathsf{supp}}(t)_B^{\mathcal A}\).

We appeal to proposition 5.12. Clearly the coimage condition holds.

For the atom condition, we must check that if \( (i, a) \in S_A^{\mathcal A}\), we have

and

both of which are trivial.

For the flexible condition, we must check that if \( (i, N) \in S_A^{\mathcal N}\) and \( N^\circ \) is \( A \)-flexible, then \( \rho _A(N)^\circ \) is also \( A \)-flexible (which is direct, and should already be its own lemma), and that

which is similarly trivial.

Finally, for the inflexible condition, suppose that \( I = (\gamma ,\delta ,\varepsilon ) \) is an inflexible \( \beta \)-path and \( t : \mathsf{Tang}_\delta \). Let \( (i, N) \in S_{{A_\varepsilon }_\bot }^{\mathcal N}\) be such that \( N^\circ = f_{\delta ,\varepsilon }(t) \). Then by the coherence condition,

We must check that

which again is trivial.

If \( (a_1, a_2), (a_1, a_3) \in \operatorname {\mathsf{conv}}_{S_A, T_A}^{\mathcal A}\), then there are \( i, j \) such that \( (i, a_1), (j, a_1) \in S_A^{\mathcal A}\) and \( (i, a_2), (j, a_3) \in T_A^{\mathcal A}\). By proposition 5.12, we deduce \( (j, a_1) \in S_A^{\mathcal A}\leftrightarrow (j, a_2) \in T_A^{\mathcal A}\), so by coinjectivity of \( T_A^{\mathcal A}\), we deduce \( a_2 = a_3 \). Hence \( \operatorname {\mathsf{conv}}_{S_A, T_A}^{\mathcal A}\) is coinjective. By symmetry, \( \operatorname {\mathsf{conv}}_{S_A, T_A}^{\mathcal A}\) is one-to-one.

As \( (a_1, a_2) \in \operatorname {\mathsf{conv}}_{S_A, T_A}^{\mathcal A}\), there is \( i \) such that \( (i, a_1) \in S_A^{\mathcal A}\) and \( (i, a_2) \in T_A^{\mathcal A}\), and as \( (N_1, N_2) \in \operatorname {\mathsf{conv}}_{S_A, T_A}^{\mathcal N}\), there is \( j \) such that \( (j, N_1) \in S_A^{\mathcal N}\) and \( (j, N_2) \in T_A^{\mathcal N}\). By proposition 5.12, we deduce that

If \( a_1 \in N_1 \), then as \( (j, N_1) \in S_A^{\mathcal N}\), we deduce that there is a near-litter \( N' \) such that \( (j, N') \in T_A^{\mathcal N}\) and \( a \in N' \). But \( T_A^{\mathcal N}\) is coinjective, so \( N' = N_2 \), giving \( a \in N_2 \). The converse holds by symmetry.

There are \( i, j \) such that \( (i, N_1), (j, N_2) \in S_A^{\mathcal N}\) and \( (i, N_3), (j, N_4) \in T_A^{\mathcal N}\). Suppose that \( N_1^\circ = N_2^\circ \); we show that \( N_3^\circ = N_4^\circ \).

First, suppose that \( N_1^\circ \) is \( A \)-flexible. Then by proposition 5.12, we have

So as \( (j, N_2) \in S_A^{\mathcal N}\) and \( N_1^\circ = N_2^\circ \), there is \( N' \) with \( (j, N') \in T_A^{\mathcal N}\) and \( N_3^\circ = {N’}^\circ \), but clearly \( N' = N_4 \), giving the result.

Now suppose that \( N_1^\circ \) is \( A \)-inflexible, so there is an inflexible \( \beta \)-path \( (\gamma ,\delta ,\varepsilon ,B) \) and tangle \( t : \mathsf{Tang}_\delta \) such that

Then by proposition 5.12, there is some \( \delta \)-allowable \( \rho \) such that \( N_3^\circ = f_{\delta ,\varepsilon }(\rho (t)) \) and

But as \( N_1^\circ = N_2^\circ \), we draw the same conclusion about \( N_2 \) and \( N_4 \), giving a \( \delta \)-allowable permutation \( \rho ' \) such that \( N_4^\circ = f_{\delta ,\varepsilon }(\rho '(t)) \); note that the inflexible path and tangle in question will be the same for both pairs. We also have

Combining these, we obtain

We claim that \( \rho (\operatorname {\mathsf{supp}}(t)) = \rho '(\operatorname {\mathsf{supp}}(t)) \). As \( T \) is strong, for each atom \( a \) such that \( (j, a) \in \operatorname {\mathsf{supp}}(t)_B^{\mathcal A}\), there is some \( k \) such that \( (i, \rho _B(a)) \in T_{{A_\delta }_B}^{\mathcal A}\). Thus \( \rho _B(a) = \rho '_B(a) \). The same conclusion may be drawn for near-litters. Thus \( \rho (\operatorname {\mathsf{supp}}(t)) = \rho '(\operatorname {\mathsf{supp}}(t)) \), giving \( \rho (t) = \rho '(t) \), and hence \( N_3^\circ = N_4^\circ \).

As \( S \) is strong, we have \( \operatorname {\mathsf{interf}}(N_1, N_2) \subseteq \operatorname {\mathsf{im}}S_A^{\mathcal A}\). But \( \operatorname {\mathsf{coim}}\operatorname {\mathsf{conv}}_{S_A, T_A}^{\mathcal A}= \operatorname {\mathsf{im}}S_A^{\mathcal A}\), as required. ³ The result for \( \operatorname {\mathsf{interf}}(N_3, N_4) \) then follows by symmetry.

Suppose that \( (N_1, N_2) \in \operatorname {\mathsf{conv}}_{S_A,T_A}^{\mathcal N}\), so there is \( i \) such that \( (i, N_1) \in S_A^{\mathcal N}\) and \( (i, N_2) \in T_A^{\mathcal N}\).

Suppose that \( N_1^\circ \) is \( A \)-flexible. By proposition 5.12, we immediately conclude that \( N_2^\circ \) is \( A \)-flexible as required.

Now suppose that \( N_1^\circ \) is \( A \)-inflexible with inflexible \( \beta \)-path \( I = (\gamma ,\delta ,\varepsilon ,B) \) and tangle \( t : \mathsf{Tang}_\delta \). By proposition 5.12, there is some \( \delta \)-allowable permutation \( \rho \) such that \( N_2^\circ = f_{\delta ,\varepsilon }(\rho (t)) \) and

We must show that \( ((\operatorname {\mathsf{conv}}_{S,T})_B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho (\operatorname {\mathsf{supp}}(t)) \). We will show the result for atoms; the result for near-litters is identical. Let \( (j, a) \in \operatorname {\mathsf{supp}}(t)_C^{\mathcal A}\). Then as \( S \) is strong, there is \( k \) such that \( (k, a) \in S_{{B_\delta }_C}^{\mathcal A}\). Then by the equation above, \( (k, \rho _C(a)) \in T_{{B_\delta }_C}^{\mathcal A}\). Hence \( (a, \rho _C(a)) \in (((\operatorname {\mathsf{conv}}_{S,T})_B)_\delta )_C \) as required.

By proposition 5.20, we may apply theorem 4.38 to \( \operatorname {\mathsf{conv}}_{S,T} \) to obtain an allowable permutation \( \rho \) that \( \operatorname {\mathsf{conv}}_{S,T} \) approximates, which directly gives \( \rho (S) = T \) as required.