This is formal consequence of the relations $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}=r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\circ r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}$ and $A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}=A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\cap r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}^{-1}(A_{{\mathcal{X}}{\mathcal{X}}^{\prime}})$. For example, let us prove (a). Pick any point $w\in\Delta({\mathcal{X}})$. The assumption implies that we can find $w^{\prime\prime}\in A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}$ with $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}(w^{\prime\prime})=w$. Then $w^{\prime}:=r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}(w^{\prime\prime})\in A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ and $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}(w^{\prime})=w$. Thus (a) holds. The proofs of (b)–(d) are similar and left to the reader. ∎