Theorem 26.1 (Seifert and Van Kampen - version I):

Let $U$, $V$ be open path connected subsets of a topological space such that $U\cap V$ is path connected. Let $x_0\in U\cap V$ and $i_1:U\cap V\longrightarrow U$, $i_2:U\cap V\longrightarrow V$ denote the inclusion maps. Then $\pi_1(U\cup V, x_0)$ is the free product (coproduct) of $\pi_1(U, x_0)$ and $\pi_1(V, x_0)$ amalgamated along $\pi_1(U\cap V, x_0)$ with respect to the maps ${i_1}_*$ and ${i_2}_*$. That is to say if $N$ is the normal subgroup

$$N = \langle {i_1}_*[\gamma]({i_2}_*[\gamma])^{-1}:[\gamma] \in \pi_1(U\cap V, x_0) \rangle \eqno(26.1)$$

then the fundamental group of $U\cup V$ is given by

$$\pi_1(U\cup V, x_0) = \pi_1(U, x_0)*\pi_1(V, x_0)/N. \eqno(26.2)$$

Considering $\pi_1(U, x_0)$ and $\pi_1(V, x_0)$ as subgroups of $\pi_1(U)*\pi_1(V)$, their images in the quotient group generate $\pi_1(U\cup V, x_0)$.

The result may be elegantly stated using a push-out diagram namely,