Paranormal implies polynormal

Statement
Any paranormal subgroup of a group is also a polynormal subgroup.

Definitions used
For these definitions, $$H^g = g^{-1}Hg$$ denotes the conjugate of $$H$$ by $$g$$, using the right-action convention (the action convention doesn't really matter). For subgroups $$H,K \le G$$, $$H^K$$ is the smallest subgroup of $$G$$ containing $$H$$ and invariant under the action of $$K$$ by conjugation.

Paranormal subgroup
A subgroup $$H$$ of a group $$G$$ is termed paranormal in $$G$$ if, for any $$g \in G$$, $$H$$ is a contranormal subgroup of $$\langle H, H^g \rangle$$.

Polynormal subgroup
A subgroup $$H$$ of a group $$G$$ is termed polynormal in $$G$$ if, for any $$g \in G$$, $$H$$ is a contranormal subgroup of $$H^{\langle g \rangle}$$.

Facts used

 * 1) uses::Contranormality is upper join-closed

Proof
Given: A paranormal subgroup $$H$$ of a group $$G$$.

To prove: For any $$g \in G$$, $$H$$ is contranormal in $$H^{\langle g \rangle}$$.

Proof: Clearly, $$H^{\langle g \rangle}$$ is generated by $$H^k$$ for all $$k \in \langle g \rangle$$, which in turn means that it is generated by the subgroups $$\langle H, H^k \rangle$$. $$H$$ is contranormal in each of these by definition of paranormality, so by fact (1), $$H$$ is contranormal in $$H^{\langle g \rangle}$$.