My question is about a proposition that I found on Munkres book: Topology (page 419). My definitions are based on this book.
I give you the definition of free product that he uses.
$\textbf {Definition}$ Let $G$ be a group, let $\{G_\alpha\}_{\alpha\in J}$ a family of subgroups of $G$ that generates $G$. Suppose that $G_\alpha\cap G_\beta$ consists of the identity element alone whenever $\alpha\not=\beta$. We say that $G$ is the $\textbf{free product}$ of the groups $G_\alpha$ if for each $x\in G$, there is only one reduced word in the groups $G_\alpha$ that represents $x$. In this case, we write: $G=\prod_{\alpha \in J}^*G_\alpha$.
The following is the proposition that I don't understand:
$ \textbf {Proposition}$ Let $G=G_1*G_2$, where $G_1$ is the free product of the subgroups $\{H_\alpha\}_{\alpha\in J}$ and $G_2$ is the free product of the subgroups $\{H_\beta\}_{\beta\in K}$. $\textbf{If the index sets $J$ and $K$ are disjoint}$, then $G$ is the free product of the subgroups $\{H_\gamma\}_{\gamma\in J\cup K}$
So we have a group $G$ and subgroups $G_1$ and $G_2$ with the properties in the definition. Moreover, $G_1$ is the free product of some its subgroups and by transitivity these $H_\alpha$ are subgroups of $G$ too. Analogous for $G_2$.
I don't understand why he requires the sets $J$ and $K$ have to be disjoint. I mean, for me they $\textbf{always}$ have this property. Indeed, suppose they have a common index, say $c$. Then we have that $H_c$ is a subgroup of $G_1$ and at the same time $H_c$ is a subgroup of $G_2$ too, so their intersection cannot consist of the identity element alone and so they cannot be factors of the free product.
He makes the same request also to prove the analogous proposition about direct sums.
Where is my mistake?
