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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 theproduct of the reduced Kurosh ranks of $H$ and $K$.  In particular, taking the fundamental group of a graph of groups with trivialvertex and edge groups, and its Bass-Serre tree, our Theorem becomes thedesired inequality of the usual Strengthened Hanna Neumann conjecture for freegroups.

Kurosh rank of intersections of subgroups of free products of orderable groups