Corollary 3

If the groups $\pi_1(X, a)$ and $\pi_1(A, a)$ are trivial, the same is true of the groups $\pi_1(B, a)$ and $\pi_1(C, a)$.

By corollary 2, the set A is arcwise connected. The group $\pi_1(X, a)$ is therefore isomorphic to the free product of the groups $\pi_1(B, a)$ and $\pi_1(C, a)$. In particular it contains subgroups isomorphic to the groups $\pi_1(B, a)$ and $\pi_1(C, a)$ (A, I, p. 83). These two groups are therefore trivial if $\pi_1(X, a)$ is.