I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically regular if there is a neighborhood $N\subset M$ of $p$ such that $N \cap S$ is a smooth resp. analytic submanifold of $N$. Then $p$ is smoothly regular if and only if it is analytically regular.
My question is, what is the best reference to this result, in both analytic and semianalytic case? In Mather's 1973 lecture notes "Stratifications and Mappings" he said that the analytic case follows from the result of Malgrange (Sur les fonctions différentiables et les ensembles analytiques) and that the semianalytic case should follow similarly. I wonder if there is any reference containing a clear statement and a proof.
