I know that if you sample $K$ random variables $(X_1, X_2, \dots, X_K)$ from Gamma distributions using shape parameters $(\alpha_1, \alpha_2, \dots \alpha_K)$ and a scale parameter $\theta = 1$ such that $X_i \sim \Gamma(\alpha_i,\theta) = \Gamma(\alpha_i,1)$ then $\left(\frac{X_1}{\sum_{i = 1}^K X_i}, \frac{X_2}{\sum_{i = 1}^K X_i}, \dots, \frac{X_K}{\sum_{i = 1}^K X_i}\right) \sim \textrm{Dir}(\alpha_1, \alpha_2, \dots, \alpha_K)$ where $\textrm{Dir}(\alpha_1, \alpha_2, \dots, \alpha_K)$ is a Dirichlet distribution with a concentration parameter $(\alpha_1, \alpha_2, \dots, \alpha_K)$.
My question is does this result hold for any scale parameter $\theta > 0$? The proofs I've read (e.g. http://mayagupta.org/publications/FrigyikKapilaGuptaIntroToDirichlet.pdf), which use the change-of-variables formula, seem to set $\theta = 1$.