The crossing number of complete graphs

We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph K${}_n$ is $\frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor \frac{n-3}{2}\rfloor .$