  We introduce the natural and fairly general notion of a subanalytic bundle
(with a finite dimensional vector space $P$ of sections) on a subanalytic
subset $X$ of a real analytic manifold $M$, and prove that when $M$ is compact,
there is a Baire subset $U$ of sections in $P$ whose zero-loci in $X$ have
tubular neighbourhoods, homeomorphic to the restriction of the given bundle to
these zero-loci.
