proof of the main theorem.

\[lem:small-perturbations-are-time-change-equivalent\] Let $ \mathbb{R}^{d} {\curvearrowright}X $ be a free Borel flow, let $ \mathcal{C}, \mathcal{D} \subseteq X $ be Borel cross sections, and let $ \phi : \mathcal{C} \to \mathcal{D} $ be an orbit equivalence. If $ \mathcal{C} $ is $ B(\delta) $-lacunary for some $ \delta > 0 $ and $$\bigl|\bigl|\rho\bigl(x, \phi(x)\bigr)\bigr|\bigr| < \delta \quad \textrm{for all} \quad x \in
    \mathcal{C},$$ then there exists a time change equivalence $ \psi : X \to X $ that extends $ \phi $.

For any $ \vec{r} $ in $ B(\delta) \subset \mathbb{R}^{d} $ there exists a diffeomorphism $ f_{\vec{r}} : B(\delta) \to B(\delta) $ with compact support such that $ f_{\vec{r}}(\vec{0}) = \vec{r} $. The proof of this assertion is sufficiently concrete (see e.g., [@MR1487640 p. 22]), so that the dependence on $ \vec{r} $ is Borel, i.e., one can pick a Borel map $ f : B(\delta) \times B(\delta) \to B(\delta) $ such that $ f(\vec{r}, \, \cdot \,) : B(\delta) \to B(\delta) $ is a compactly supported diffeomorphism, and $ f(\vec{r},\vec{0}) = \vec{r} $ for all $ \vec{r} \in B(\delta) $.

Let $ \xi : \mathcal{C} + B(\delta) \to \mathcal{C} $ be the map 