Contranormality is transitive

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 $$K \le G$$, 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, $$K \le L$$. So, $$L$$ is a normal subgroup of $$G$$ containing $$K$$. Since $$K$$ is contranormal in $$G$$, we get $$L = G$$.