; U\cap V is nonempty and path-connected; and w\in U\cap V; then \pi_1(X,w) is the free product with amalgamation of \pi_1(U,w) and \pi_1(V,w), with respect to the (not necessarily injective) homomorphisms I:\pi_1(U\cap V, w)\to \pi_1(U,w) and J:\pi_1(U\cap V, w)\to \pi_1(V,w). Given group presentations:
