If the equilibrium constants of the following equilibria $\mathrm{SO_2 + \frac{1}{2}O_2 \rightleftharpoons SO_3}$ and $\mathrm{2SO_3 \rightleftharpoons 2SO_2 + O_2}$ are given by K$_1$ and K$_2$ respectively, which of the following relations is correct?