Permutably complemented satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., permutably complemented subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Statement

Suppose $H$ is a permutably complemented subgroup of $G$: in other words, there exists a subgroup $K\leq G$ such that $H\cap K$ is trivial and $HK=G$. Then, if $H\leq L\leq G$, $H$ is a permutably complemented subgroup of $L$. In fact, the subgroup $K\cap L$ is a permutable complement to $H$ in $L$.

Facts used

1. Modular property of groups: This states that if $H,K,L\leq G$, and $H\leq L$, we have:

$H(K\cap L)=HK\cap L$.

Proof

Given: $H,K\leq G$, $HK=G$ and $H\cap K$ is trivial. $H\leq L\leq G$.

To prove: $H\cap (K\cap L)$ is trivial and $H(K\cap L)=L$.

Proof: Since $H\cap K$ is trivial, $H\cap (K\cap L)$ is also trivial. As for the second part, observe that by fact (1):

$H(K\cap L)=HK\cap L=G\cap L=L$

as required.