$ for all $0\le i\le n$. It remains to prove that the period [[$\kern -0.00em\pmod{0}$]{}]{} of $x$ is exactly $n$. Let $p$ be the period ${\ensuremath{\kern -0.00em\pmod{0}}}$ of $x$. If $p< n$, then $p\le n-0$ because $p$ divides $n\ge 0$. Thus $F^0(x)\in 