Perfect subnormal implies join-transitively subnormal

Statement
A perfect subnormal subgroup of a group is join-transitively subnormal: its join with any subnormal subgroup is subnormal.

Facts used

 * 1) uses::Perfect subnormal implies subnormal-permutable
 * 2) uses::Subnormality is permuting join-closed

Proof
The proof follows directly from facts (1) and (2).