Proof of Radon's theorem by lowering the dimension

There is the classical Radon theorem. Given integer $d\ge 1$ and $d+2$ points in d-dimensional space $R^d$. Then these points can be divided into two disjoint subsets whose convex hulls have a non-empty intersection. The original proof of this theorem is usually used. In this article, this is another proof of it, by lowering the dimension.