  Two subanalytic subsets of $ \mathbb R^n$ are called $s$-equivalent at a
common point $P$ if the Hausdorff distance between their intersections with the
sphere centered at $P$ of radius $r$ vanishes to order $>s$ as $r$ tends to
$0$. In this work we strengthen this notion in the case of real subanalytic
subsets of $\mathbb R^n$ with isolated singular points, introducing the notion
of tangential $s$-equivalence at a common singular point which considers also
the distance between the tangent planes to the sets near the point. We prove
that, if $V(f)$ is the zero-set of an analytic map $f$ and if we assume that
$V(f)$ has an isolated singularity, say at the origin $O$, then for any $s\geq
1$ the truncation of the Taylor series of $f$ of sufficiently high order
defines an algebraic set with isolated singularity at $O$ which is tangentially
$s$-equivalent to $V(f)$.
