A note on free products of linear groups

For a field $K$, let $\bar K$ denote its algebraic closure. Assume that $\left\vert {\bar K:K} \right\vert = \infty$. Then for any linear groups $G,{\text{ }}H \subseteq {\text{G}}{{\text{L}}_n}(K)$ their free product $G * H$ can be embedded into ${\text{G}}{{\text{L}}_N}(K(t))$. Here $N$ is an integer depending on $K$ only and $t$ stands for an indeterminate.