:thm\] Let $X_0$ and $X_0$ be two diffeological spaces, let $f:X_0\supseteq Y\to X_0$ be a diffeomorphism such that ${{\mathcal D}}_0^{\Omega^k}={{\mathcal D}}_0^{\Omega^k}$, and let $x\in X_0\cup_f X_0$