Intersection of finite-dominating subgroup with any subgroup whose product with it is the whole group is finite-dominating in it

Statement
Suppose $$G$$ is a group, $$H$$ is a fact about::finite-dominating subgroup of $$G$$, and $$K$$ is any subgroup of $$G$$ such that $$HK = G$$. Then, $$H \cap K$$ is a finite-dominating subgroup of $$K$$.

This statement can be interpreted as saying that the property of being a finite-dominating subgroup satisfies a weak variant of the transfer condition.

Proof
Given: $$H \le G$$ is a finite-dominating subgroup, $$HK = G$$.

To prove: $$K \cap H$$ is finite-dominating in $$H$$. In other words, if $$F$$ is a finite subgroup of $$K$$, $$F$$ is conjugate in $$K$$ to a subgroup of $$K \cap H$$.

Proof: Since $$H$$ is finite-dominating in $$G$$, there exists $$g \in G$$ such that $$gFg^{-1} \le H$$. Since $$G = HK$$, we can write $$g = hk$$, with $$h \in H, k \in K$$. Then, $$hkFk^{-1}h^{-1} \le H$$. Since $$h \in H$$, this yields $$kFk^{-1} \le H$$.

By assumption, $$k \in K$$, so $$kFk^{-1} \le H \cap K$$. Thus, we have found $$k \in K$$ conjugating $$F$$ to a subgroup of $$H \cap K$$.