  In a previous paper we have introduced the notion of geometric directional
bundle of a singular space, in order to introduce global bi-Lipschitz
invariants. Then we have posed the question of whether or not the geometric
directional bundle is stabilised as an operation acting on singular spaces. In
this paper we give a positive answer in the case where the singular spaces are
subanalytic sets, thus providing a new invariant associated with the
subanalytic sets.
