I am trying to prove the following.
Theorem? (Seifert-Van Kampen theorem for Lie-groupoids, a partial statement). Let $G_1\substack{\to \\ \to}G_0$ (denoted simpy by $G$) be a Lie-groupoid. Let $x_0\in G_0$. If $G_0$ is the union of open sets $A_\lambda$ each containing $x_0$ and if each restriction $G|_{A_\lambda}$ and $G|_{A_\lambda\cap A_\mu}$ are connected, then the obvious group-homomorphism $\Phi:*_\lambda\left(\pi_1\left(G|_{A_\lambda},x_0\right)\right)\to\pi_1(G,x_0)$ is surjective.
Terminologies: The fundamental group $\pi_1(H,x)$ of a Lie-groupoid $H$ is defined as the set of all homotopy-classes of $H$-paths from $x$ to $x$, equipped with the obvious multiplication. $H$ is said to be connected if and only if for all $x,y\in H_0$, there exists an $H$-path from $x$ to $y$.
I tried to mimic the proof of Lemma 1.15 of Hatcher's "Algebraic topology." Let $[\alpha]=[g_0,\alpha_1,g_1,\ldots,\alpha_m,g_m]\in\pi_1(G,x_0)$ be arbitrary, where $g_i$'s are elements of $G_1$, and $\alpha_i$'s are continuous paths in $G_0$. In Hatcher's proof, the path is subdivided so that each segment lies in a single $A_\lambda$, and then the end-points of the segments are connected to $x_0$ with a path lying in the intersections $A_\lambda\cap A_\mu$. But that does not seem to work for Lie-groupoids. For example, consider $[\alpha]=[1_{x_0},\alpha_1,g_1,\alpha_2,1_{x_0}]\in\pi_1(G,x_0)$ such that $\alpha_1$ is a continuous path from $x_0$ to a point in $A_1-A_2$ and $\alpha_2$ is a continuous path from a point in $A_2-A_1$ to $x_0$ (see the image below). Then how should I subdivide $[\alpha]$ so that the end-points of the segments lie in $A_1\cap A_2$?
