X $, indexed by finite sequences of natural numbers that satisfies the following two conditions:

1.  $ X = \bigsqcup_{s \in \mathbb{N}^{n}} \Omega_{s} $ for each $ n \in \mathbb{N} $; in particular, $ \Omega_{{\varnothing}} = X $.

2.  $ s \subseteq t \implies \Omega_{t} \subseteq \Omega_{s} $.

\[thm:tce-up-to-cocompressible\] Let $ \mathbb{R}^{d} {\curvearrowright}X_{1} $ and $ \mathbb{R}^{d} {\curvearrowright}X_{2} $ be free Borel flows on standard Borel spaces, let for $ i=1,2 $, $ Q_{i} \subseteq X_{i} $ be rational grids, let $ \mathcal{D}_{i} \subseteq Q_{i} $ be cocompact cross sections on these grids, and let $ \zeta : \mathcal{D}_{1} \to \mathcal{D}_{2} $ be an orbit equivalence between them. There are Borel invariant cocompressible sets $ Z_{i} \subseteq X_{i} $, and a time change equivalence $ \psi : Z_{1} \to Z_{2} $ that extends $ \zeta $, i.e., $ \zeta|_{Z_{1} \cap \mathcal{D}_{1}} = \psi|_{Z_{1} \cap \mathcal{D}_{1}} $.

The proof relies on a back-and-forth argument similar to the one used in the proof of Theorem 1 in [@MR1113569]. For start, let us apply the Uniform Rokhlin Lemma (Lemma \[lem:uniform-rokhlin\]) to both flows yielding Borel invariant cocompressible sets $ \tilde{Z}_{i} \subseteq X_{i} $, as well as cross sections $ \mathcal{C}_{i,n} \subseteq \tilde{Z}_{i} $ 