Partitions and (m and n) Sums of Products - Two Cell Partition

We let $SP_m (\langle X_t \rangle^\infty_{t=1})$ denote the sums of $m$ increasing products from a sequence $\langle X_t \rangle^{\infty}_{t=1}$. Given $m \neq n$, we construct a two cell partition of $\mathbb{N}$ so that neither cell contains $SP_n \langle X \rangle^{\infty}_{t=1}) \cup SP_m (\langle y\rangle>^{\infty}_{t = 1})$ for any sequences $\langle X_t \rangle^{\infty}_{t=1}$ and $\langle y \rangle^\infty_{t=1}$.