Contranormality is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., contranormal subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Verbal statement

A contranormal subgroup of a contranormal subgroup of a group is contranormal in the whole group.

Facts used

1. Normality satisfies transfer condition: If $L$ is normal in $G$, and $K\leq G$, then $K\cap L$ is normal in $K$.

Proof

Given: A group $G$, a contranormal subgroup $K$ of $G$, a contranormal subgroup $H$ of $K$.

To prove: If $L$ is a normal subgroup of $G$ containing $H$, then $L=G$.

Proof: By fact (1), $L\cap K$ is normal in $K$, and contains $H$. Thus, since $H$ is contranormal in $K$, $L\cap K=K$. Thus, $K\leq L$. So, $L$ is a normal subgroup of $G$ containing $K$. Since $K$ is contranormal in $G$, we get $L=G$.