Exercise 5, § 4

Let G be a Lie group operating properly on an unlaceable and completely regular topological space X. Put $Y = X/G$ and denote by $f: X\rightarrow Y$ the canonical mapping. Prove that the canonical groupoid morphism $\varpi (X)/G\rightarrow$ Coeg($f$) is an isomorphism. (Use LIE, IX, §9, exerc. 17 to adapt the proof of Theorem 3 of IV, p. 403.)