1202 Then there exist

  1202 Then there exist

  1203 \begin{itemize}

  1203 \begin{itemize}

  1204 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$;

  1204 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$;

  1205 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and

  1205 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and

  1206 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, 

  1206 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, 

  1208 \end{itemize}

  1208 \end{itemize}

  1209 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$.

  1209 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$.

  1210 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same

  1210 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same

  1211 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity.

  1211 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity.

  1212 

  1212