Regularity and Tresse's theorem for geometric structures

For any non-special bundle $P\to X$ of geometric structures we prove that the $k$-jet space $J^k$ of this bundle with an appropriate $k$ contains an open dense domain $U_k$ on which Tresse's theorem holds. For every $s\geqslant k$ we prove that the pre-image $\pi^{-1}(k,s)(U_k)$ of $U_k$ under the natural projection $\pi(k,s)\colon J^s\to J^k$ consists of regular points. (A point of $J^s$ is said to be regular if the orbits of the group of diffeomorphisms induced from $X$ have locally constant dimension in a neighbourhood of this point.)