Let $ F $ be a non-empty subset of $ \{ 1,2,\dots,n\} $ and $ P_{F}=(\{x_{i}:i\in F\}) $. Let $ F_{1},F_{2},\dots,F_{m} $ be pair-wise distinct non-empty subsets of $ \{1,2,...,n\} $ and $$ I=\prod_{i=1}^{m}P_{F_{i}}^{a_{i}}$$ for some integers $ a_{1},a_{2},\dots,a_{k}\geq 1 $. Prove that $ I $ can be written uniquely in such a way.
My attempt: I know that the ideals $ P_{F_{i}} $ are prime ideals of $ k[x_{1},...,x_{n}] $ and that $ I $ admits a primary decomposition $ I=\cap_{j=1}^{r}Q_{j} $ where the $ Q_{j}$'s are primary ideals, but the $ P_{F_{i}}^{a_{i}}$'s are not comaximal so I can not use this to write $I$ as their intersection and somehow relate this to the primary decomposition. More than this, I don't know how to proceed.
I would really appreciate any ideas, hints or solutions. Thank you very much!