Corollary 1

Suppose that the group K is generated by the stabilisers of the vertices of G. Then the group homomorphism $\gamma_o: G_o\rightarrow (G/K)_{\gamma(o)}$ is surjective. Moreover, if the groupoid G is simply transitive, the same is true of the groupoid $G/K$.

The relation ($R''_3$) implies $\lambda ([e]) =\lambda (c_e) =\gamma_o(c_e)$. The relations ($R''_3$) then imply that the set of $k\in K$ such that $\lambda ([k])$ belong to the image of $\gamma_o$ is a subgroup of K. Finally, the relations ($R'_3$) show that, for every element $k\in K$ whose stabiliser is not empty, $\lambda ([k])$ belong to the image of $\gamma_o$. The corollary follows.