Example of a real analytic subset which is not subanalytic?

Question

It is mentioned on page 40 of Shiota's book, "geometry of subanalytic and semialgebraic sets", that a (real) analytic set in $\mathbb{R}^n$ is not necessarily subanalytic. Here a set $S \subseteq \mathbb{R}^n$ is analytic if it is locally the zero locus of an analytic function, and is subanalytic if it is locally of the form $\operatorname{im} (f_1) \setminus \operatorname{im} (f_2)$ for some proper analytic map $f_i: Y_i \rightarrow \mathbb{R}^n$ where $Y_i$ is an analytic manifold for $i = 1, 2$. My questions is what's a first example of an analytic but non-subanalytic set and how one can verify it?