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How to prove the gluing-condition for a pseudogroup induced by an étale Lie groupoid?

Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let \begin{equation} \Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is …
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