  Sz\H ucs proved in 2000 that the $r$-tuple-point manifold of a generic
immersion is cobordant to the $\Sigma^{1_{r-1}}$-point manifold of its generic
projection. Here we slightly extend this by showing that the natural mappings
of these manifolds are bordant to each other. The main novelty of our approach
is that we construct the bordism explicitly.
