Join of 3-subnormal and finite subnormal implies subnormal

Statement with symbols
Suppose $$G$$ is a group, $$H$$ is a 3-subnormal subgroup of $$G$$ and $$K$$ is a finite subnormal subgroup of $$G$$. Then, the join $$\langle H, K \rangle$$ is a subnormal subgroup of $$G$$.

Further, if $$k$$ is the subnormal depth of $$K$$ and $$r$$ is its order, the subnormal depth of $$\langle H, K \rangle$$ is at most:

$$2^rk + 1$$

Facts used

 * 1) uses::2-subnormal implies join-transitively subnormal: In fact, the join of a 2-subnormal subgroup and a $$s$$-subnormal subgroup is $$2s$$-subnormal.
 * 2) uses::Join of subnormal subgroups is subnormal iff their commutator is subnormal: