Intermediate subgroup condition is right residual-preserved

Statement
Suppose $$p,q,r$$ are subgroup properties, such that the following holds:

A subgroup $$K$$ of $$G$$ satisfies property $$r$$ in $$G$$, if, whenever $$H$$ has property $$q$$ in $$K$$, $$H$$ has property $$p$$ in $$G$$.

Suppose, further, that $$p$$ satisfies the intermediate subgroup condition. Then, $$r$$ satisfies the intermediate subgroup condition.

Examples

 * The fact that normality satisfies intermediate subgroup condition can be used to deduce that transitive normality satisfies intermediate subgroup condition.

Proof
Given: Subgroup properties $$p,q,r$$ such that $$r$$ is the right residual of $$p$$ by $$q$$. $$p$$ satisfies the intermediate subgroup condition.

To prove: $$r$$ satisfies the intermediate subgroup condition. In other words, if $$K \le L \le G$$ such that $$K$$ satisfies property $$r$$ in $$G$$, $$K$$ also satisfies property $$r$$ in $$L$$.

Proof: To show that $$K$$ satisfies property $$r$$ in $$L$$, it suffices to show that whenever $$H \le K$$ has property $$q$$ in $$K$$, $$H$$ has property $$p$$ in $$L$$. We do this in two steps:


 * 1) $$H$$ has property $$p$$ in $$G$$: This is because $$H$$ has property $$q$$ in $$K$$ and $$K$$ has property $$r$$ in $$G$$.
 * 2) $$H$$ has property $$p$$ in $$L$$: This is because $$H$$ has property $$p$$ in $$G$$, $$L$$ is an intermediate subgroup, and $$p$$ satisfies the intermediate subgroup condition.