Finite-relative-intersection-closed implies transitive

Statement
Suppose $$p$$ is a subgroup property that is a finite-relative-intersection-closed subgroup property. Explicitly, this means that whenever $$H,K,L \le G$$ are such that $$H,K$$ are both contained in $$L$$, $$H$$ satisfies $$p$$ in $$G$$, and $$K$$ satisfies $$p$$ in $$L$$, then $$H \cap K$$ satisfies $$p$$ in $$G$$.

Then, $$p$$ is a transitive subgroup property: if $$K \le H \le G$$ are groups such that $$K$$ satisfies $$p$$ in $$H$$ and $$H$$ satisfies $$p$$ in $$G$$, then $$K$$ satisfies $$p$$ in $$G$$.

Related facts

 * Characteristicity is transitive, characteristicity is not finite-relative-intersection-closed

Proof
We can set $$L = G$$ with the notation used in the definitions to complete the proof.