  We discuss a formula of S. Spodzieja and generalize it for the isolated
improper Achilles-Tworzewski-Winiarski intersection index. As an application we
give a simple proof of a result of P. Ebenfelt and L. Rothschild: if $F\colon
(\mathbb{C}^m,0)\to (\mathbb{C}^m,0)$ is a finite holomorphic map, $W$ a germ
of a complex variety at zero such that $F^{-1}(W)$ is a smooth germ and the
Jacobian of $F$ does not vanish identically on it, then $W$ is smooth too.
