Corollary 1

The homomorphism $\mu$ is injective. More precisely, there exists a retraction associated with $\mu$ which is a group homomorphism.

Let $\rho$ be the unique homomorphism of $\pi_1(B, a)*\pi_1(C, a)*F(\pi_0(A)$ $\{j_0\})$ into $F(\pi_0(A)$ - $\{j_0\})$ which induces the trivial homomorphism on $\pi_1(B, a)$ and $\pi_1(C, a)$ and the identity on $F(\pi_0(A)-\{j_0\})$. Let N be the kernel of the homomorphism $\mathsf{M}$. By proposition $2,\rho (N)$ is reduced to the identity element. There therefore exists a unique homomorphism $\mathsf{r}$ of $\pi_1(X, a)$ into $F(\pi_0(A)-\{j_0\})$ such that $\rho =\mathsf{r}\circ \mathsf{M}$. For every $v\in F(\pi_0(A)-\{j_0\})$, one has $\mathsf{M}(v) =\mu(v)$, hence $\mathsf{r}\circ \mu$ is the identity homomorphism. The corollary follows.