Suppose that $A,B$ are real analytic subsets of $\Omega\subseteq \mathbb{R}^n$ and $p\in A\cap B \neq \emptyset$. Does the intersection inequality from complex analysis still hold, i.e. does the following inequality hold:
$\mathrm{dim}^{\mathbb{R}}_p(A\cap B) \geq \mathrm{dim}^{\mathbb{R}}_p(A)+\mathrm{dim}^{\mathbb{R}}_p( B)-n$?
By real analytic subset, I mean sets that are locally given as the zero set of finitely manly real analytic functions and by dimension, I mean the maximal dimension of regular points near $p$ as manifolds. The complex analytic proofs that I know use the "Active Lemma", i.e. $\mathrm{dim}(\left\{f=0\right\})=n-1$, for non-zero $f$, which does not hold over $\mathbb{R}$.
If not, are there explicit counterexamples?
