i'm sry, please delete.. or please inform me how to..?
Let $\displaystyle x\in H \mbox{ and }x\in K$. Let $\displaystyle d = \mbox{ord}(x)$. Then by Lagrange's theorem $\displaystyle d|(|H|)$ and $\displaystyle d|(|K|)$. Since these are relatively prime it means $\displaystyle d=1$. So $\displaystyle x$ must be $\displaystyle \mbox{identity}$.
Q.E.D.