I am having trouble understanding one of the results in the following paper http://arxiv.org/PS_cache/math/pdf/0104/0104175v1.pdf

In proposition 3.1, the author says

Let $(R,\frak{m})$ be a regular local ring, $p,q$ prime ideals s.t. $rad(p+q)= \frak{m}$ and $dim(R/p)+dim(R/q)=dim(R)$. If $R/p$ is regular then $p^{(m)}\cap q^{(n)}\subseteq p^{(m)}\frak{m}^n$.

If I take $q=\frak{m}$ (the condition on the radical is automatically satisfied) and $n=1$, then this would imply $p^{(m)} \subseteq p^{(m)}\frak{m}$ (for any associated prime $p$ of $R$ since this would satisfy the condition on dimension and requiring that $R/p$ is regular). Am I missing something here?