  This paper uses the theory of integral closure of modules to study the
sections of both real and complex analytic spaces. The stratification
conditions used are the (t^) conditions introduced by Thom and Trotman.
  Our results include a new simple proof showing how the (t^r) conditions
improve under Grassman modification, and a characterization of the (t^r)
conditions using the multiplicity of a submodule of the Jacobian module of the
singularity. This gives numerical criteria for Verdier Equisingularity of
families of sections of the space.
