Permutable and subnormal implies join-transitively subnormal

Statement with symbols
Suppose $$H, K \le G$$ are subgroups such that $$H$$ is a permutable subnormal subgroup -- it is both subnormal and permutable, and $$K$$ is a subnormal subgroup. Then the join $$\langle H, K \rangle$$ is also subnormal, and its subnormal depth is bounded by a function of the subnormal depths of $$H$$ and $$K$$.

Facts used

 * 1) uses::Subnormality is permuting join-closed