On the intersection of finitely generated subgroups in free products of groups

A subgroup H of a free product Pi(alpha is an element of I)(*) G(alpha) of groups G(alpha), alpha is an element of I, is called factor free if for every S is an element of Pi(alpha is an element of I)(*) and beta is an element of I one has SHS(-1)boolean AND G(beta) = {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote (r) over bar(K) = max(r(K) - 1, 0), where r(K) is the rank of K. It is proven that if H, K are finitely generated factor free subgroups of a free product Pi(alpha is an element of I)(*) G(alpha) then (r) over bar(H boolean AND K) less than or equal to 6 (r) over bar(H)(r) over bar(K). It is also shown that the inequality (r) over bar(H boolean AND K) less than or equal to (r) over bar(H)(r) over bar(K) of Hanna Neumann conjecture on subgroups of free groups does not hold for factor free subgroups of free products.