The space of $r$-immersions of a union of discs in $\mathbb R^n$

Abstract: For a manifold $M$ and an integer $r>1$, the space of $r$-immersions of $M$ in $\mathbb R^n$ is defined to be the space of immersions of $M$ in $\mathbb R^n$ such that the preimage of every point in $\mathbb R^n$ contains fewer than $r$ points. We consider the space of $r$-immersions when $M$ is a disjoint union of $k$ $m$-dimensional discs, and prove that it is equivalent to the product of the $r$-configuration space of $k$ points in $\mathbb R^n$ and the $k^{\text{th}}$ power of the space of injective linear maps from $\mathbb R^m$ to $\mathbb R^n$. This result is needed in order to apply Michael Weiss's manifold calculus to the study of $r$-immersions. The analogous statement for spaces of embeddings is ``well-known'', but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for $r$-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of $r$-immersions.