Pronormality satisfies intermediate subgroup condition

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a pronormal subgroup of $$G$$. Then, $$H$$ is also a pronormal subgroup of $$K$$.

Related metaproperty dissatisfactions for pronormality

 * Pronormality does not satisfy transfer condition: We can have a pronormal subgroup $$H$$ of $$G$$ and a subgroup $$K$$ of $$G$$ such that $$H \cap K$$ is not pronormal in $$K$$.
 * Pronormality is not upper join-closed: If $$H$$ is pronormal in intermediate subgroups $$K_1, K_2$$, it is not necessary that $$H$$ is pronormal in $$\langle K_1, K_2$$.

Related properties satisfying the intermediate subgroup condition

 * Weak pronormality satisfies intermediate subgroup condition
 * Paranormality satisfies intermediate subgroup condition
 * Polynormality satisfies intermediate subgroup condition
 * Weak normality satisfies intermediate subgroup condition

Proof
This is direct from the definition.