Claim. Let $(\zeta_{i})_{i\in I}$ be a family of cycles of a finite set E whose supports are pairwise disjoint. Let $\sigma=\Pi_{i\in I}\zeta_i$. Then $\sigma(x)=\zeta_i(x)$ for $x\in supp(\zeta_i)$ and $i\in I$.
Note that $supp(\zeta_i)=\{x\in E\ |\ \zeta_i(x)\ne x\}$.
My attempt:
I know that, for $i\in I$ and $x\in supp(\zeta_i)$, $\zeta_i(x)\in supp(\zeta_i)$; therefore, $\zeta_i(x)\notin supp(\zeta_j)$ for $j\ne i$. Hence, $\zeta_j\zeta_i(x)=\zeta_i(x)$ for $j\ne i$. I also know that $\zeta_i\zeta_j=\zeta_j\zeta_i$ for $i,j\in I$.
I can see why this equality holds. I just don't know how to prove it.
Any hints will be appreciated.