$f=\sum_{{\bf w}\in \ZZ^n} c_{\bf w} x^{\bf w}\in\II_{\bf x}$ We assume that $f$ has the smallest number of nonzero summands such that no monomial term $c_{\bf w} x^{\bf w}$ with $c_{\bf w}\neq 0$ is contained in $\II$. It must therefore have at least two monomial terms.

Assume, as above that $c_{\bf w},c_{{\bf w}'}\ne 0$ and denote by ${\bf v}$ the difference ${\bf v}={\bf w}-{\bf w}'$. Since $\Lambda$ has full rank, there exists $i\in [1,n]$ such that $\{x_i,x^{\bf v}\}\ne 0$. Therefore, $\{x_i, x^{\bf w}\}=cx_ix^{\bf w}\ne dx_ix^{{\bf w}'}\{x_i, x^{{\bf w}'}\}$ for some $c,d\in \CC$. Note that $\{x_i,f\}=\sum_{{\bf w}\in \ZZ^n} c_{\bf w} \lambda_{\bf w}x^{\bf w}x_i$ for certain $ \lambda_{\bf w}\in\ZZ$. Clearly, $cx_if-\{x_i,f\}\in \II$ and $$cx_if-\{x_i,f\}=(c-c)c_{\bf w} x^{\bf w}x_i+(c-d)c_{{\bf w}'} x^{{\bf w}'}x_i+\ldots\ .$$

Hence, $cx_if-\{x_i,f\}\ne 0$ and it has fewer monomial summands than $f$ which contradicts our assumption. Therefore, $\II$ contains a monomial. The proposition is proved.

The following fact is an obvious corollary of Proposition \[pr:TPP super toric\].

\[th:intersection\] Let $\AA$ be the cluster algebra defined by a Poisson seed $({\bf x}, B,\Lambda)$ with $rank(\Lambda)=n$. If $\II\subset \AA$ is a non-zero Poisson prime ideal, then $\II$ contains a cluster variable $x_i$.

We also have the following quantum version of Proposition \[pr:TPP super toric\] which 