ca.analysis and odes - Real-analytic manifolds in real-analytic sets

Let $Usubset mathbb{R}^n$ be open, and let $f:Utomathbb{R}$ be real-analytic. We consider the zero set $Z:=f^{-1}({0})$.

For a paper I am writing, I am looking for the best reference to the following basic fact:

If $Z$ has topological dimension equal to $d$, then $Z$ contains a real-analytic manifold of dimension $d$.

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I can get this from Lojasiewicz's theorem or similar results, but that is a slightly unwieldy reference, and something probably needs to be said about how exactly one deduces it. Given that the statement is rather simple, I was wondering if someone knows of a more direct reference to this fact.

And to add a mathematical question: This result is obviously much weaker than Lojasiewicz's theorem. Is there a proof that doesn't require developing the full structure theorem?

Many thanks for any pointers!