  Let $\gamma: [-1, 1]\to \mathbb{R}^n$ be a smooth curve that is
non-degenerate. Take $m\le n$ and a Borel set $E\subset [0, 1]^n$. We prove
that the orthogonal projection of $E$ to the $m$-th order tangent space of
$\gamma$ at $\theta\in [-1, 1]$ has Hausdorff dimension $\min\{m, \dim(E)\}$
for almost every $\theta\in [-1, 1]$.
