On the Kurosh rank of the intersection of subgroups in free products of groups

The Kurosh rank r_K (H) of a subgroup H of a free product ∏_{α ∈ I}^* G_α of groups G_α, α ∈ I, is defined accordingly to the classic Kurosh subgroup theorem as the number of free factors of H. We prove that if H₁, H₂ are subgroups of ∏_{α ∈ I}^* G_α and H₁, H₂ have finite Kurosh rank, then over(r, ̄)_K (H₁ ∩ H₂) ≤ 2 frac(q_*, q^* - 2) over(r, ̄)_K (H₁) over(r, ̄)_K (H₂) ≤ 6 over(r, ̄)_K (H₁) over(r, ̄)_K (H₂), where over(r, ̄)_K (H) = max (r_K (H) - 1, 0), q^* is the minimum of orders >2 of finite subgroups of groups G_α, α ∈ I, q^* : = ∞ if there are no such subgroups, and frac(q^*, q^* - 2) : = 1 if q^* = ∞. In particular, if the factors G_α, α ∈ I, are torsion-free groups, then over(r, ̄)_K (H₁ ∩ H₂) ≤ 2 over(r, ̄)_K (H₁) over(r, ̄)_K (H₂).