  We prove that the reduced Kurosh rank of the intersection of two subgroups
$H$ and $K$ of a free product of right-orderable groups is bounded above by the
product of the reduced Kurosh ranks of $H$ and $K$.
  In particular, taking the fundamental group of a graph of groups with trivial
vertex and edge groups, and its Bass-Serre tree, our Theorem becomes the
desired inequality of the usual Strengthened Hanna Neumann conjecture for free
groups.
