2-subnormal implies conjugate-join-closed subnormal

Statement
A 2-subnormal subgroup of a group is a conjugate-join-closed subnormal subgroup: the join of an arbitrary number of conjugate subgroups to a 2-subnormal subgroup is subnormal.

Facts used

 * 1) uses::2-subnormality is conjugate-join-closed

Proof
The proof follows directly from fact (1), which states that an arbitrary join of conjugate 2-subnormal subgroups is 2-subnormal. Since 2-subnormal subgroups are subnormal, the conclusion follows.