Stable Pontryagin-Thom construction for proper maps

We will present proofs for two conjectures stated in [T. O. Rot, Homotopy classes of
proper maps out of vector bundles]. The first one is that for an arbitrary manifold W, the homotopy classes of proper maps 
W x R^n \to R^{k+n} stabilise as n goes to infinity. The second one is that there is a stable Pontryagin-Thom 
construction for proper maps   W x R^n \to R^{k+n} again  as n goes to infinity. The second one actually implies the first one and we shall
prove the secnd one by giving an explicit construction.