Compatibility of Atlases is Equivalence Relation

Theorem
Let $M$ be a locally Euclidean space of dimension $d$.

Let $\mathcal F$ denote the set of all atlases of class $\mathcal C^k$ on $M$.

Define a relation $\sim$ on $\mathcal F$ by putting, for any two $\mathcal C^k$-atlases $\mathscr F$ and $\mathscr G$:


 * $\mathscr F \sim \mathscr G$ iff $\mathscr F$ and $\mathscr G$ are compatible.

Then $\sim$ is an equivalence relation on $\mathcal F$.