Indices in Definition of External Semidirect Product

Problem

In my notes, the external semidirect product $G_1 \rtimes_\gamma G_2$ of two groups $G_1$ and $G_2$ with respect to a homomorphism $\gamma: G_2 \to \Aut G_1$, is defined as

\begin{multline} \forall\, x_1,y_1 \in G_1, \forall\, x_2,y_2 \in G_2, (x_1,x_2) \times_{G_1 \rtimes_\gamma G_2} (y_1,y_2) \\ = (x_1 \times_{G_1} \gamma(x_2)(y_1), x_2 \times_{G_2} y_2). \end{multline}

Why don’t we write $(x_1,y_1)$ and $(x_2,y_2)$ instead?

Explanation

I don’t think that this question can last for a day on Mathematics Stack Exchange.

If we really do so, we’ll create a chaos of indices: we've $y_1 \in G_2$ and $x_2 \in G_1$, so we substitute $y_{\color{red}{1}}$ in $\gamma: G_{\color{red}{2}} \to \Aut G_1$ and then $x_{\color{blue}{2}}$ in $\gamma(y_1): G_{\color{blue}{1}} \to G_{\color{blue}{1}}$, which belongs to $\Aut G_{\color{blue}{1}}$.