2-subnormal implies join-transitively subnormal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., 2-subnormal subgroup) must also satisfy the second subgroup property (i.e., join-transitively subnormal subgroup)
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Statement

Statement with symbols

Suppose $H,K\leq G$ are subgroups such that $H$ is a 2-subnormal subgroup of $G$ and $K$ is a subnormal subgroup of $G$. Then, the join $\langle H,K\rangle$ is a subnormal subgroup of $G$, and its subnormal depth in $G$ is at most twice the subnormal depth of $K$.

Related facts

• Join of normal and subnormal implies subnormal of same depth
• Permutable and subnormal implies join-transitively subnormal
• Subnormality is normalizing join-closed
• Subnormality is permuting join-closed