Given prime monomial ideals $P_i$ (which are generated by sets of variables) in $K[X_1,\ldots,X_n]$ and another variable $x$ (that may or not be a generator in some ideals $P_i$) then I would need to show that $\langle x \rangle + \cap P_i = \cap (\langle x \rangle + P_i)$ where intersections are finite.
The inclusion $\subseteq$ is clear. For the other one I'm having more problems. The intuition is that each $\cap P_i$ is computed by taking $lcm(X_i,X_j) = X_iX_j$ and therefore consists of monomials. Can you help me to prove the other inclusion?