Projection of Euclidean Space onto Euclidean Subspace is Riemannian Submersion

Theorem
Let $\struct {\R^{n + k}, g^E_{n + k}}$, $\struct {\R^n, g^E_n}$ be the real vector spaces with Euclidean metrics.

Let $\pi : \R^{n + k} \to \R^n$ be the projection onto the first $n$ coordinates.

Then $\pi$ is a Riemannian submersion.