Normal implies join-transitively subnormal

Verbal statement

 * 1) Any normal subgroup of a group is a join-transitively subnormal subgroup.
 * 2) The join of a normal subgroup and a subnormal subgroup is subnormal.

Statement with symbols

 * 1) If $$H$$ is a normal subgroup of $$G$$, $$H$$ is also join-transitively subnormal in $$G$$.
 * 2) If $$H$$ is a normal subgroup of $$G$$ and $$K$$ is a subnormal subgroup of $$G$$, $$HK$$ is a subnormal subgroup of $$G$$.

Related facts

 * Join of normal and subnormal implies subnormal of same depth
 * Normal implies join-transitively 2-subnormal
 * 2-subnormal implies join-transitively subnormal
 * Permutable and subnormal implies join-transitively subnormal

Facts used

 * 1) uses::Join of normal and subnormal implies subnormal of same depth

Proof
The proof is direct from fact (1).