Higher-Order Contingentism, Part 3: Expressive Limitations

Abstract

Two expressive limitations of an infinitary higher-order modal language interpreted on models for higher-order contingentism – the thesis that it is contingent what propositions, properties and relations there are – are established: First, the inexpressibility of certain relations, which leads to the fact that certain model-theoretic existence conditions for relations cannot equivalently be reformulated in terms of being expressible in such a language. Second, the inexpressibility of certain modalized cardinality claims, which shows that in such a language, higher-order contingentists cannot express what is communicated using various instances of talk of ‘possible things’, such as ‘there are uncountably many possible stars’.

Appendices

Appendix: A: Projective Generation

In this appendix, Lemma 2.10 is proven. To do so, two subsidiary lemmas are established.

Lemma A.1

Let $\mathfrak {S}=\langle W,I,D\rangle$ be an homogeneous fis , $\mathfrak {S^{\prime }}=\langle W^{\prime },I^{\prime },D^{\prime }\rangle$ an fis and P a projection from $\mathfrak {S}$ to $\mathfrak {S^{\prime }}$ . For all p,q ∈ P , there are p ^′∈ P and $f\in \text {aut}^{i}(\mathfrak {S})_{(\text {dom}(p\cap q))}$ such that p ⊆ p ^′ and q ⊆ p ^′ f .

Proof

By condition (ii) of projections, there is a p ^′∈ P such that p ⊆ p ^′and im(q) ⊆im(p ^′).Let f be the partial function from I to I mapping every x ∈dom(q)to p ^′−1 q(x).It is routine to show that f is a finite partial permutation of I which respects worlds. Since$\mathfrak {S}$is homogeneous, there is an $f^{\prime }\in \text {aut}^{i}(\mathfrak {S})$such that f ⊆ f ^′.It is routine to show that $f^{\prime }\in \text {aut}^{i}(\mathfrak {S})_{(\text {dom}(p\cap q))}$and q ⊆ p ^′ f ^′. □

Lemma A.2

For any type t, $o\in {D^{t}_{w}}$ , $o^{\prime }\in \iota ^{t}_{\mathfrak {F^{\prime }}}$ , p ∈ P and $f\in \text {aut}^{i}(\mathfrak {S})$ , if $o{Z^{t}_{p}}o^{\prime }$ then $\hat {f}.oZ^{t}_{pf^{-1}}o^{\prime }$ .

Proof

By induction on types. For t = e,note that if $o{Z^{e}_{p}}o^{\prime }$,then p(o) = o ^′,so p f ⁻¹ f(o) = o,whence $\hat {f}.oZ^{e}_{pf^{-1}}o^{\prime }$.Let $t=\bar {t}$.To show that $\hat {f}.oZ^{t}_{pf^{-1}}o^{\prime }$,note that condition (1)follows from $o{Z^{t}_{p}}o^{\prime }$.For condition (2), consider any q ∈ P such that p f ⁻¹ ⊆ q,$v\in \text {dom}(\dot {q})$,$\bar {v}\in \text {dom}(\dot {q})^{n}$,$\bar {o}\in {\Pi }_{i\leq n}D^{t_{i}}_{v_{i}}$,and $\bar {o}^{\prime }\in {\Pi }_{i\leq n}\iota ^{t_{i}}_{\langle W^{\prime },I^{\prime }\rangle }$such that $\bar {o}Z_{q}\bar {o}^{\prime }$.It will be shown that $\bar {o}\in \hat {f}.o(v)$iff $\bar {o}^{\prime }\in o^{\prime }(\dot {q}(v))$.By induction hypothesis, it follows from $\bar {o}Z_{q}\bar {o}^{\prime }$that $\hat {f}^{-1}.\bar {o}Z_{qf}\bar {o}^{\prime }$.Note also that p ⊆ q f(for any o ∈dom(p), p(o) = p f ⁻¹ f(o) = q f(o)).Thus it follows from $o{Z^{t}_{p}}o^{\prime }$that $\hat {f}^{-1}.\bar {o}\in o(\hat {f}^{-1}.v)$iff $\bar {o}^{\prime }\in o^{\prime }(\dot {q}(v))$.Since $\bar {o}\in \hat {f}.o(v)$iff $\hat {f}^{-1}.\bar {o}\in o(\hat {f}^{-1}.v)$,the desired equivalence follows. □

Lemma 2.10 can now be proven, which claims that for all types t and p ∈ P:

• (i) ${Z^{t}_{p}}$ is a bijection from $D^{t}_{\text {dom}(\dot {p})}$ to $(D^{P})^{t}_{\text {im}(\dot {p})}$.

• (ii) For all q ∈ P, w ∈ W such that w ⊆dom(p ∩ q) and $o\in {D^{t}_{w}}$, ${Z^{t}_{p}}(o)={Z^{t}_{q}}(o)$.

Proof Proof of Lemma 2.10

By induction on types. The case of t = e is trivial for both (i) and (ii). So consider any type$t=\bar {t}$and p ∈ P. (i)will be established first:

Claim 1::

${Z^{t}_{p}}$is functional. Proof. Consider any $o\in D^{t}_{\text {dom}(\dot {p})}$and $o^{\prime },o^{\prime \prime }\in (D^{P})^{t}_{\text {im}(\dot {p})}$such that $o{Z^{t}_{p}}o^{\prime }$and $o{Z^{t}_{p}}o^{\prime \prime }$.Let v ∈ W ^′;it will be shown that o ^′(v) ⊆ o ^″(v)(the other direction follows by symmetry). So consider any$\bar {o}^{\prime }\in o^{\prime }(v)$.Since $D^{P}\boxtimes o^{\prime }$,there are $\bar {v}\in W^{\prime n}$such that $\bar {o}^{\prime }\in {\Pi }_{i\leq n}(D^{P})^{t_{i}}_{v_{i}}$.By condition (ii) of the definition of projections, there is a q ∈ P such that p ⊆ q and $v,v_{1},\dots ,v_{n}\in \text {im}(\dot {q})$.By induction hypothesis (i), there are $\bar {o}\in {\Pi }_{i\leq n}D^{t_{i}}_{\text {dom}(\dot {q})}$such that $\bar {o}Z_{q}\bar {o}^{\prime }$.So it follows from $o{Z^{t}_{p}}o^{\prime }$that $\bar {o}\in o(\dot {q}^{-1}(v))$,and therefore with $o{Z^{t}_{p}}o^{\prime \prime }$that $\bar {o}^{\prime }\in o^{\prime \prime }(v).\checkmark$

Claim 2::

${Z^{t}_{p}}$is total. Proof. Consider any $o\in D^{t}_{\text {dom}(\dot {p})}$.Define $o^{\prime }\in \iota ^{t}_{\mathfrak {F^{\prime }}}$such that for all v ∈ W ^′, o ^′(v)is the set of $\bar {o}^{\prime }\in {\Pi }_{i\leq n}(D^{P})^{t_{i}}_{W^{\prime }}$(if × = +)/ $\bar {o}^{\prime }\in {\Pi }_{i\leq n}(D^{P})^{t_{i}}_{v}$(if × = −)such that there is a q ∈ P such that p ⊆ q and v ⊆im(q),and n-tuple $\bar {o}$such that $\bar {o}Z_{q}\bar {o}^{\prime }$and $\bar {o}\in o(\dot {q}^{-1}(v^{\prime }))$.It will be shown that $o{Z^{t}_{p}}o^{\prime }$.$D^{P}\boxtimes o^{\prime }$is immediate by construction. So consider any q ∈ P such that p ⊆ q,$w\in \text {dom}(\dot {q})$and n-tuples $\bar {o},\bar {o}^{\prime }$such that $\bar {o}Z_{q}\bar {o}^{\prime }$.It will be proven that $\bar {o}\in o(w)$iff $\bar {o}^{\prime }\in o^{\prime }(\dot {q}(w))$.By construction of o ^′,the latter follows from the former. So assume that$\bar {o}^{\prime }\in o^{\prime }(\dot {q}(w))$.Then by construction of o ^′,there is an r ∈ P such that p ⊆ r and $\dot {q}(w)\subseteq \text {im}(r)$,and n-tuple $\bar {o}^{*}$such that $\bar {o}^{*}Z_{r}\bar {o}^{\prime }$and $\bar {o}^{*}\in o(\dot {r}^{-1}\dot {q}(w))$.By Lemma A.1, there are q ^′,r ^′∈ P such that q ⊆ q ^′, r ⊆ r ^′and an $f\in \text {aut}^{i}(\mathfrak {S})_{(\text {dom}(p))}$such that q ^′ f = r ^′.By induction hypothesis (ii), $\bar {o}Z_{q^{\prime }}\bar {o}^{\prime }$and $\bar {o}^{*}Z_{r^{\prime }}\bar {o}^{\prime }$.Thus $\bar {o}^{*}Z_{q^{\prime }f}\bar {o}^{\prime }$,and so with Lemma A.2, $\hat {f}.\bar {o}^{*}Z_{q^{\prime }}\bar {o}^{\prime }$.By induction hypothesis (i), $Z^{t_{i}}_{q^{\prime }}$is a bijection, for each i ≤ n,so $\hat {f}.\bar {o}^{*}=\bar {o}$.Thus from the fact that $\bar {o}^{*}\in o(\dot {r}^{-1}\dot {q}(w))$,we obtain $\bar {o}\in \hat {f}.o(\hat {f}.\dot {r}^{-1}\dot {q}(w))$.Since $f\in \text {aut}^{i}(\mathfrak {S})_{(\text {dom}(p))}$and $o\in D^{t}_{\text {dom}(\dot {p})}$,$\hat {f}.o=o$,and so $\bar {o}\in o(\hat {f}.\dot {r}^{-1}\dot {q}(w))$,from which $\bar {o}\in o(w)$follows with q ^′ f = r ^′as required. $\checkmark$

Claim 3::

${Z^{t}_{p}}$is injective. Proof. Consider any $o,o^{*}\in D^{t}_{\text {dom}(\dot {p})}$and $o^{\prime }\in (D^{P})^{t}_{\text {im}(\dot {p})}$such that $o{Z^{t}_{p}}o^{\prime }$and $o^{*}{Z^{t}_{p}}o^{\prime }$.Let w ∈ W;it will be shown that o(w) ⊆ o ^∗(w)(the other direction follows by symmetry). So consider any$\bar {o}\in o(w)$.By condition (i) of the definition of projections, there is a q ∈ P such that p ⊆ q,$w\in \text {dom}(\dot {q})$and $\bar {o}\in {\Pi }_{i\leq n}D^{t_{i}}_{\text {dom}(\dot {q})}$.So by induction hypothesis (i), there are $\bar {o}^{\prime }\in {\Pi }_{i\leq n}(D^{P})^{t_{i}}_{\text {im}(\dot {q})}$such that $\bar {o}^{\prime }\in o^{\prime }(\dot {q}(w))$,and so $\bar {o}\in o^{*}(w)$.$\checkmark$

Claim 4::

${Z^{t}_{p}}$is surjective. Proof. Consider any $o^{\prime }\in (D^{P})^{t}_{\text {im}(\dot {p})}$.By construction of D ^P,there is a $v\in \text {im}(\dot {p})$and q ∈ P such that v ⊆im(q)and $o^{\prime }\in {Z^{t}_{q}}[D^{t}_{\dot {q}^{-1}(v)}]$.Hence there is an $o\in D^{t}_{\dot {q}^{-1}(v)}$such that $o{Z^{t}_{q}}o^{\prime }$.By Lemma A.1, there are p ^′,q ^′∈ P such that p ⊆ p ^′, q ⊆ q ^′,and an $f\in \text {aut}^{i}(\mathfrak {S})$such that p ^′ f = q ^′.So by Lemma A.2, $\hat {f}.oZ^{t}_{q^{\prime }f^{-1}}o^{\prime }$,hence $\hat {f}.oZ^{t}_{p^{\prime }}o^{\prime }$.Since $o\in D^{t}_{\dot {q}^{-1}(v)}$,$\hat {f}.o\in D^{t}_{\hat {f}.q^{-1}(v)}=D^{t}_{p^{-1}(v)}$.So $\hat {f}.o\in \text {dom}({Z^{t}_{p}})$,and thus with induction hypothesis (ii), $\hat {f}.o{Z^{t}_{p}}o^{\prime }$.Hence $o^{\prime }\in \text {im}({Z^{t}_{p}})$as required. $\checkmark$

(ii): Since (i) has been established, ${Z^{t}_{p}}$must be a bijection from $D^{t}_{\text {dom}(\dot {p})}$to $(D^{P})^{t}_{\text {im}(\dot {p})}$, for all o ∈ P. Note first that it isroutine to show that for all p,q ∈ P,if p ⊆ q then${Z^{t}_{p}}\subseteq {Z^{t}_{q}}$. Now considerany p,q ∈ P, w ∈ W such that w ⊆dom(p ∩ q)and$o\in {D^{t}_{w}}$; it will be shown that${Z^{t}_{p}}(o)={Z^{t}_{q}}(o)$. By Lemma A.1,there are p ^′,q ^′∈ P and $f\in \text {aut}^{i}(\mathfrak {S})_{(w)}$suchthat p ⊆ p ^′, q ⊆ q ^′and p ^′ f = q ^′. Since$Z_{p}\subseteq Z_{p^{\prime }}$,${Z^{t}_{p}}(o)=Z^{t}_{p^{\prime }}(o)$, which byLemma A.2 is $Z^{t}_{p^{\prime }f}(\hat {f}^{-1}.o)$.Since $o\in {D^{t}_{w}}$,$\hat {f}^{-1}.o=o$, so${Z^{t}_{p}}(o)=Z^{t}_{p^{\prime }f}(o)$, which is$Z^{t}_{q^{\prime }}(o)$. With thefact that ${Z^{t}_{q}}\subseteq Z^{t}_{q^{\prime }}$, itfollows that ${Z^{t}_{p}}(o)={Z^{t}_{q}}(o)$. □

Appendix: B: FFISs and Bi-Projections

This appendix proves Theorem 3.5. For the following, let $\mathfrak {S}=\langle W,I,B\rangle$ and $\mathfrak {S^{\prime }}=\langle W^{\prime },I^{\prime },B^{\prime }\rangle$ be ffis, and P a bi-projection from $\mathfrak {S}$ to $\mathfrak {S^{\prime }}$. Let D and D ^′ be the domain assignments of $\otimes \mathfrak {S}$ and $\otimes \mathfrak {S^{\prime }}$, Z the extension of P as defined above, and $D^{P^{-1}}$ and D ^P the domain assignments of the structures projectively generated by P ⁻¹ and P, respectively.

Definition B.1

For any p ∈ P,define a relation ${Z^{p}_{p}}\subseteq \text {aut}^{i}_{\omega }(\mathfrak {S})\times \text {aut}^{i}_{\omega }(\mathfrak {S^{\prime }})$such that for all $f\in \text {aut}^{i}_{\omega }(\mathfrak {S})$and $g\in \text {aut}^{i}_{\omega }(\mathfrak {S^{\prime }})$,$f{Z^{p}_{p}}g$iff

• supp(f) ⊆dom(p),

• supp(g) ⊆im(p),and

• p f(o) = g p(o)for all o ∈supp(f).

Lemma B.2

For any p ∈ P , $w\in \text {dom}(\dot {p})$ and $f\in \text {fix}^{i}_{\omega }(\mathfrak {S^{\prime }},\dot {p}(w))$ such that supp(f) ⊆im(p), there is a $g\in \text {fix}^{i}_{\omega }(\mathfrak {S},w)$ such that $g{Z^{p}_{p}}f$ .

Proof

Define g : I → I suchthat for all o ∈ I,

• $g(o)=\left \{\begin {array}{ll} p^{-1}fp(o) & \text { if } o\in \text {dom}(p) \\ o & \text { otherwise} \end {array}\right .$

It is routine to check that $g\in \text {fix}^{i}_{\omega }(\mathfrak {S},w)$and $g{Z^{p}_{p}}f$. □

Lemma B.3

Let p,q ∈ P , $f\in \text {aut}^{i}_{\omega }(\mathfrak {S})$ and $g\in \text {aut}^{i}_{\omega }(\mathfrak {S^{\prime }})$ . If $f{Z^{p}_{p}}g$ and p ⊆ q then $f^{-1}{Z^{p}_{q}}g^{-1}$ .

Proof

Routine. □

Lemma B.4

For all types t:

• (i) For all p ∈ P , $f\in \text {aut}^{i}_{\omega }(\mathfrak {S})$ , $g\in \text {aut}^{i}_{\omega }(\mathfrak {S^{\prime }})$ such that $f{Z^{p}_{p}}g$ and $\langle o,o^{\prime }\rangle \in {Z^{t}_{p}}$ , $\hat {f}.o{Z^{t}_{p}}\hat {g}.o^{\prime }$ .

• (ii) For all p ∈ P , $({Z^{t}_{p}})^{-1}=Z^{t}_{p^{-1}}$ .

• (iii) For all v ∈ W ^′ , $D^{\prime t}_{v}=(D^{P})^{t}_{v}$ .

Proof

By induction on types. Let t = e.(i): Let p ∈ P,$f\in \text {aut}^{i}_{\omega }(\mathfrak {S})$,$g\in \text {aut}^{i}_{\omega }(\mathfrak {S^{\prime }})$suchthat $f{Z^{p}_{p}}g$and $\langle o,o^{\prime }\rangle \in {Z^{e}_{p}}$.Then o ∈dom(p),so p(o) = o ^′.Since $f{Z^{p}_{p}}g$, p f(o) = g p(o), so p f(o) = g(o ^′), i.e.,$\hat {f}.o{Z^{e}_{p}}\hat {g}.o^{\prime }$. (ii) and (iii) areimmediate. So let $t=\bar {t}$.

(i)::

Consider any p ∈ P,$f\in \text {aut}^{i}_{\omega }(\mathfrak {S})$,$g\in \text {aut}^{i}_{\omega }(\mathfrak {S^{\prime }})$such that $f{Z^{p}_{p}}g$and $\langle o,o^{\prime }\rangle \in {Z^{t}_{p}}$.Then $\hat {f}.o\in D^{t}_{\hat {f}.\text {dom}(\dot {p})}$.Since supp(f) ⊆dom(p),$\hat {f}.\text {dom}(\dot {p})=\text {dom}(\dot {p})$,so $\hat {f}.o\in D^{t}_{\text {dom}(\dot {p})}$.It will be shown that $\hat {f}.o{Z^{t}_{p}}\hat {g}.o^{\prime }$by checking conditions (1) and (2) of the construction of Z.

(1): Consider any v ∈ W ^′and $\bar {o}^{\prime }\in \hat {g}.o^{\prime }(v)$.Then $\hat {g}^{-1}.\bar {o}^{\prime }\in o^{\prime }(\hat {g}^{-1}.v)$.Since $o{Z^{t}_{p}}o^{\prime }$,$D^{P}\boxtimes o^{\prime }$,so there are $\bar {v}\in W^{\prime n}$such that $\hat {g}^{-1}.\bar {o}^{\prime }\in {\Pi }_{i\leq n}(D^{P})^{t_{i}}_{v_{i}}$.$g\in \text {aut}^{i}_{\omega }(\mathfrak {S^{\prime }})$,so by Part 1, Lemma 3 (i), $\hat {g}\in \text {aut}(\otimes \mathfrak {S^{\prime }})$.Since by IH (iii), $(D^{P})^{t_{i}}_{v_{i}}=D^{\prime t_{i}}_{v_{i}}$for all i ≤ n,it follows that $\bar {o}^{\prime }\in {\Pi }_{i\leq n}(D^{P})^{t_{i}}_{\hat {g}.v_{i}}$.If × = −,it can be assumed that v _i = v for all i < n,and therefore $\bar {o}^{\prime }\in {\Pi }_{i\leq n}(D^{P})^{t_{i}}_{\hat {g}.v}$.Thus $D^{P}\boxtimes \bar {o}^{\prime }$.

(2): Consider any q ∈ P such that p ⊆ q,$w\in \text {dom}(\dot {q})$and n-tuples $\bar {o},\bar {o}^{\prime }$such that $\bar {o}Z_{q}\bar {o}^{\prime }$.It will be proven that $\bar {o}\in \hat {f}.o(w)$iff $\bar {o}^{\prime }\in \hat {g}.o^{\prime }(\dot {q}(w))$.By Lemma B.3, $f^{-1}{Z^{p}_{q}}g^{-1}$,so by IH (i), $\hat {f}^{-1}.\bar {o}Z_{q}\hat {g}^{-1}.\bar {o}^{\prime }$.Since $\text {supp}(f)\subseteq \text {dom}(\dot {q})$,$\hat {f}^{-1}.w\in \text {dom}(\dot {q})$,so $\hat {f}^{-1}.\bar {o}\in o(\hat {f}^{-1}.w)$iff $\hat {g}^{-1}.\bar {o}^{\prime }\in o^{\prime }(\dot {q}(\hat {f}^{-1}.w))$,and so $\bar {o}\in \hat {f}.o(w)$iff $\bar {o}^{\prime }\in \hat {g}.o^{\prime }(\hat {g}.\dot {q}(\hat {f}^{-1}.w))$. q f = g q,so $\hat {g}.\dot {q}(\hat {f}^{-1}.w)=\dot {q}(w)$,from which the desired claim follows.

(ii)::

Let p ∈ P.By symmetry, it suffices to show that if $\langle o,o^{\prime }\rangle \in {Z^{t}_{p}}$then $\langle o^{\prime },o\rangle \in Z^{t}_{p^{-1}}$.So assume $\langle o,o^{\prime }\rangle \in {Z^{t}_{p}}$.Then there is a $w\in \text {dom}(\dot {p})$such that $o\in {D^{t}_{w}}$and $o^{\prime }\in (D^{P})^{t}_{\dot {p}(w)}$.It will first be shown that $o^{\prime }\in D^{\prime t}_{\dot {p}(w)}$.By Lemma 3.3, it suffices to show that $D^{\prime }\boxtimes o^{\prime }$and $\hat {f}.o^{\prime }=o^{\prime }$for all $f\in \text {fix}^{i}_{\omega }(\mathfrak {S^{\prime }},\dot {p}(w))$.By Lemma 2.10 (ii), $o^{\prime }\in (D^{P})^{t}_{\text {im}(\dot {p})}$,so with the fact that the structure projectively generated by P is a ×structure,it follows that $D^{P}\boxtimes o^{\prime }$.Therefore by IH (iii), $D^{\prime }\boxtimes o^{\prime }$.So consider any $f\in \text {fix}^{i}_{\omega }(\mathfrak {S^{\prime }},\dot {p}(w))$.Since supp(f)is finite, there is a q ∈ P such that p ⊆ q and supp(f) ⊆im(q).So by Lemma B.2, there is a $g\in \text {fix}^{i}_{\omega }(\mathfrak {S},w)$such that $g{Z^{p}_{q}}f$.By Lemma 2.10 (ii), $o{Z^{t}_{q}}o^{\prime }$,so with claim (i) of the present lemma, $\hat {g}.o{Z^{t}_{q}}\hat {f}.o^{\prime }$.As $o\in {D^{t}_{w}}$,$\hat {g}.o=o$,so $o{Z^{t}_{q}}\hat {f}.o^{\prime }$.Hence by the functionality of ${Z^{t}_{q}}$,established in Lemma 2.10 (i), $\hat {f}.o^{\prime }=o^{\prime }$.

With $o^{\prime }\in D^{\prime t}_{\dot {p}(w)}$established, it can be proven that $o^{\prime }Z^{t}_{p^{-1}}o$by checking conditions (1) and (2) of the definition of Z. Since$o\in {D^{t}_{w}}$,$D\boxtimes o$,so by IH, $D^{P^{-1}}\boxtimes o$.For condition (2), consider any q ∈ P ⁻¹such that p ⁻¹ ⊆ q,$v\in \text {dom}(\dot {q})$and n-tuples $\bar {o},\bar {o}^{\prime }$such that $\bar {o}Z_{q}\bar {o}^{\prime }$.By IH, $\bar {o}^{\prime }Z_{q^{-1}}\bar {o}$;also q ⁻¹ ∈ P, p ⊆ q ⁻¹and $\dot {q}(v)\in \text {dom}(\dot {q}^{-1})$.So by $o{Z^{t}_{p}}o^{\prime }$,$\bar {o}^{\prime }\in o(\dot {q}(v))$iff $\bar {o}\in o^{\prime }(v)$,as required for condition (2). So $o^{\prime }Z^{t}_{p^{-1}}o$.

(iii)::

Let v ∈ W ^′.By condition (iii) of the definition of projections, there is a p ∈ P such that $v\in \text {im}(\dot {p})$.By Lemma 2.11, $D^{\prime t}_{v}=\text {dom}\left (Z^{t}_{p^{-1}|v}\right )=\text {dom}\left (Z^{t}_{(p|p^{-1}(v))^{-1}}\right )$.By (ii), this is $\text {im}\left (Z^{t}_{p|p^{-1}(v)}\right )$,which by Lemma 2.11 again is $\text {im}\left ({Z^{t}_{p}}|D^{t}_{p^{-1}(v)}\right )$.By Lemma 2.10 (i), this is $(D^{P})^{t}_{v}$.

□

Proof Proof of Theorem 3.5

Immediate by Lemma B.4. □