If \( \xi \leq \zeta \) and \( \psi \) is an \( A \)-flexible approximation for \( \zeta \), then \( \psi \) is an \( A \)-flexible approximation for \( \xi \). So it suffices to prove the result for nice base actions \( \xi \) by proposition 4.30.

Define the permutative relation \( R : {\mathcal L}\to {\mathcal L}\to \mathsf{Prop}\) to be a permutative extension of \( \xi ^{\mathcal L}\), which exists by proposition A.5. Let \( \pi \) be the permutation of litters defined by \( R \), or the identity on any litter not in \( \operatorname {\mathsf{coim}}R \).

Define an orbit restriction \( (t, f, \pi ) \) (definition A.3) for \( \operatorname {\mathsf{field}}\xi ^{\mathcal A}\) by

with function \( f : {\mathcal A}\to {\mathcal L}\) defined by \( f(a) = a^\circ \), and litter permutation \( \pi \). We must check that for each litter \( L \), the set \( t \cap \operatorname {\mathsf{LS}}(L) \) has cardinality at least \( \max (\aleph _0, \# \operatorname {\mathsf{field}}\xi ^{\mathcal A}) \). But we can write

where the set being removed from \( \operatorname {\mathsf{LS}}(L) \) is small, so \( t \cap \operatorname {\mathsf{LS}}(L) \) is a large set, and \( \aleph _0 \) and \( \# \operatorname {\mathsf{field}}\xi ^{\mathcal A}\) are less than \( \# \kappa \), as required. Then by proposition A.4, there is a permutative relation \( S \geq \xi ^{\mathcal A}\) defined on a small set and contained in \( \operatorname {\mathsf{field}}\xi ^{\mathcal A}\cup t = \operatorname {\mathsf{field}}\xi ^{\mathcal A}\cup u \), such that if

Let \( T \) be a permutative extension of the restriction of \( \xi ^{\mathcal L}\) to the \( A \)-flexible litters, with coimage contained entirely in the set of \( A \)-flexible litters, given by proposition A.5. From this, we define a base approximation \( \psi = (S, T) \).

It remains to check that \( \psi \) is an \( A \)-flexible approximation of \( \xi \). Conditions 1–3 are trivial, and condition 4 follows from the fact that we assumed \( \xi \) was nice.

We first show an auxiliary result. Let \( (N_1, N_2) \in \xi ^{\mathcal N}\), and let \( (a_1, a_2) \in S \); we will show that \( a_1 \in N_1 \leftrightarrow a_2 \in N_2 \). Suppose first that \( (a_1, a_2) \in \xi ^{\mathcal A}\), in which case we are done as \( \xi \) is a base action. Instead, we have \( a_1 \notin \operatorname {\mathsf{coim}}\xi ^{\mathcal A}, a_2 \notin \operatorname {\mathsf{im}}\xi ^{\mathcal A}\) and \( \pi (a_1^\circ ) = a_2^\circ \). As \( \xi \) is nice, we must have \( a_1 \in N_1 \leftrightarrow a_1^\circ = N_1^\circ \). Similarly, \( a_2 \in N_2 \leftrightarrow a_2^\circ = N_2^\circ \). So if \( a_1 \in N_1 \), we conclude that \( a_2^\circ = \pi (a_1^\circ ) = \pi (N_1^\circ ) = N_2^\circ \) giving \( a_2 \in N_2 \), and if \( a_2 \in N_2 \), we find \( \pi (a_1^\circ ) = a_2^\circ = N_2^\circ \) so \( a_1^\circ = N_1^\circ \), giving \( a_1 \in N_1 \).

We now prove condition 5, which is the equation

where \( (N_1, N_2) \in \xi ^{\mathcal N}\). Consider the first the case where \( (a_1, a_2) \in S \) and \( a_1 \in N_1 \). The auxiliary lemma shows that \( a_2 \in N_2 \) as required. Now consider the case where \( a_2 \notin \operatorname {\mathsf{coim}}S \) and \( a_2^\circ = N_2^\circ \). If \( a_2 \notin N_2 \), then \( a_2 \in N_2 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_2^\circ ) \subseteq \operatorname {\mathsf{im}}\xi ^{\mathcal A}\), a contradiction. Finally suppose that neither holds, so

If \( a_2 \in \operatorname {\mathsf{coim}}S \), then there is \( a_1 \) such that \( (a_1, a_2) \in S \), and we have \( a_1 \notin N_1 \), giving \( a_2 \notin N_2 \) by the auxiliary lemma. Finally, if \( a_2 \notin \operatorname {\mathsf{coim}}S \) and \( a_2^\circ \neq N_2^\circ \), then \( a_2 \notin N_2 \), since \( a_2 \in N_2 \) would imply \( a_2 \in N_2 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_2^\circ ) \), contradicting the fact that \( \xi \) is nice.

First, note that

As \( \psi \) exactly approximates \( \pi \) and \( \pi (N_1^\circ ) = N_2^\circ \), we have the equation

Combining this with the fact that \( \psi ^{E{\mathcal A}} \leq \pi ^{\mathcal A}\), and that \( N_1 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_1^\circ ) \subseteq \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}} \), we obtain

which is equal to \( N_2 \) by part of definition 4.33.

First, note that \( \xi _A^{\mathcal A}\leq \psi _A^{E{\mathcal A}} \) and \( \psi _A^{\mathcal A}\leq \rho _A^{\mathcal A}\) give the required result for atoms. Now suppose that \( (N_1, N_2) \in \xi _A^{\mathcal N}\); we must show that \( \rho _A(N_1) = N_2 \). We prove this by induction on \( \iota (N_1) \), generalising over all \( A \).

By proposition 4.36, it suffices to show that \( \rho _A(N_1^\circ ) = N_2^\circ \). Suppose that \( N_1^\circ \) is \( A \)-flexible. Then by definition 4.33, \( (N_1^\circ , N_2^\circ ) \in \psi _A^{\mathcal L}\). Hence \( \rho _A(N_1^\circ ) = N_2^\circ \) as required.

Suppose not, so \( N_1^\circ \) is \( A \)-inflexible with path \( (\gamma ,\delta ,\varepsilon ,B) \) and tangle \( t : \mathsf{Tang}_\delta \). By coherence of \( \xi \), we know that \( (\xi _B)_\delta \) is defined on \( \operatorname {\mathsf{supp}}(t) \), and it suffices to show that

Let \( C : \delta \rightsquigarrow \bot \) and \( a \) be an atom such that \( (i, a) \in \operatorname {\mathsf{supp}}(t)_C^{\mathcal A}\) for some \( i \). Then \( (a, ((\rho _B)_\delta )_C(a)) \in ((\xi _B)_\delta )_C^{\mathcal A}\) by the result for atoms. Now suppose \( N \) is a near-litter such that \( (i, N) \in \operatorname {\mathsf{supp}}(t)_C^{\mathcal N}\). Then

So we may apply the inductive hypothesis, giving \( (N, ((\rho _B)_\delta )_C(N)) \in ((\xi _B)_\delta )_C^{\mathcal N}\) as required.

Let \( \xi \) be a coherent \( \beta \)-action, and let \( \psi \) be a flexible approximation for it, which exists by proposition 4.34. Then apply theorem 4.24 (freedom of action) to \( \psi \) to obtain a \( \beta \)-allowable permutation \( \rho \) that \( \psi \) exactly approximates. Finally, appeal to proposition 4.37 to conclude that \( \xi \) approximates \( \rho \).