Since $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is continuous, $A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is compact, and $\Delta({\mathcal{X}})$ is Hausdorff, it suffices to prove that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ is bijective.

Using Lemma 4.6 (c)–(d) and Lemma 4.7, one proves by induction on the number of blowups that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ is bijective when ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is a composition of simple blowups.

Now consider the general case. Using Lemma 4.1 we find an snc model ${\mathcal{X}}^{\prime\prime}$ dominating both ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ and such that the morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is a composition of simple blowups. Thus $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is bijective. By Lemma 4.6 (a), it follows that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is surjective. Since ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ were arbitrary snc models with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$, it follows that $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is also surjective. It now follows from Lemma 4.6 (b) that $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is injective, which completes the proof. ∎