  In earlier work, the author described various stratification conditions for a
complex analytic set X in terms of the theory of integral closure of modules.
However, even if an analytic set has a reduced structure, often geometric
operations like intersection give a non-reduced structure on the resulting
analytic space. In this note we show that it is not necessary for X to have a
reduced structure to apply the earlier integral closure results. As an
application we give a simple proof of a necessary and sufficient condition for
the Whitney conditions to pass to the intersection of X with a hyperplane
containing the smooth stratum Y, and for this intersection to be "typical". In
the case where X is a family of ICIS singularities, we show the condition on H
at a point y of Y depends only on the fiber of the family over y.
