  Let $\mathcal{F}_{n}^*$ be the set of Boolean functions depending on all $n$
variables. We prove that for any $f\in \mathcal{F}_{n}^*$, $f|_{x_i=0}$ or
$f|_{x_i=1}$ depends on the remaining $n-1$ variables, for some variable $x_i$.
This existent result suggests a possible way to deal with general Boolean
functions via its subfunctions of some restrictions.
  As an application, we consider the degree lower bound of representing
polynomials over finite rings. Let $f\in \mathcal{F}_{n}^*$ and denote the
exact representing degree over the ring $\mathbb{Z}_m$ (with the integer $m>2$)
as $d_m(f)$. Let $m=\Pi_{i=1}^{r}p_i^{e_i}$, where $p_i$'s are distinct primes,
and $r$ and $e_i$'s are positive integers. If $f$ is symmetric, then $m\cdot
d_{p_1^{e_1}}(f)... d_{p_r^{e_r}}(f) > n$. If $f$ is non-symmetric, by the
second moment method we prove almost always $m\cdot d_{p_1^{e_1}}(f)...
d_{p_r^{e_r}}(f) > \lg{n}-1$. In particular, as $m=pq$ where $p$ and $q$ are
arbitrary distinct primes, we have $d_p(f)d_q(f)=\Omega(n)$ for symmetric $f$
and $d_p(f)d_q(f)=\Omega(\lg{n}-1)$ almost always for non-symmetric $f$. Hence
any $n$-variate symmetric Boolean function can have exact representing degree
$o(\sqrt{n})$ in at most one finite field, and for non-symmetric functions,
with $o(\sqrt{\lg{n}})$-degree in at most one finite field.
