A finitary internal characterization of Segal Theta_n-spaces

We show that the Segal condition for Theta_n-spaces can be equivalently restated in terms of internal locality with respect to a finite number of spine inclusions. In particular, a simplicial presheaf X over Theta_n is a Segal Theta_n-space if and only X is internally local with respect to a single horizontal spine inclusion and the Theta_{n-1}-space of morphisms of X is a Segal Theta_{n-1}-space. This extends an analogous result for usual Segal spaces, which already found application in homotopy type theory. Moreover we develop, along the way, several results about presheaves over Reedy categories and over Theta_n that are of independent interest.