Nilpotent join of pronormal subgroups is pronormal

Statement
Suppose $$H, K \le G$$ are fact about::pronormal subgroups, such that $$\langle H, K \rangle$$ is a fact about::nilpotent group. Then:


 * $$H$$ and $$K$$ normalize each other.
 * $$\langle H,K \rangle = HK$$ is also a pronormal subgroup of $$G$$.

Facts used

 * 1) uses::Pronormality satisfies intermediate subgroup condition
 * 2) uses::Nilpotent implies every subgroup is subnormal
 * 3) uses::Pronormal and subnormal implies normal
 * 4) uses::Pronormality is normalizing join-closed

Proof
Given: $$H, K \le G$$ are pronormal subgroups, and $$ \langle H, K \rangle$$ is nilpotent.

To prove: $$H$$ and $$K$$ normalize each other, and $$HK = \langle H, K \rangle$$ is also pronormal.

Proof: By fact (1), $$H,K$$ are both pronormal in $$\langle H, K \rangle$$. By fact (2), they are also both subnormal in $$\langle H, K \rangle$$. By fact (3), $$H, K$$ are both normal in $$\langle H, K$$. Thus, we have $$HK = \langle H, K \rangle$$, and that $$H$$ and $$K$$ normalize each other. By fact (4), we obtain that $$\langle H, K \rangle = HK$$ is also pronormal.