Tangential approximation of analytic sets

Two subanalytic subsets of Rn 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 Rn 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≥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).