Commutator of two subgroups is normal in join

Statement
Suppose $$H, K$$ are subgroups of a group $$G$$. Then, the fact about::commutator of two subgroups $$[H,K]$$ is a normal subgroup of the fact about::join of subgroups $$\langle H, K \rangle$$.

Facts used

 * 1) uses::Subgroup normalizes its commutator with any subset: If $$H \le G$$ is a subgroup and $$A$$ is a subset of $$G$$, then $$H$$ normalizes the following subgroup:

$$[A,H] = \langle [a,h] \mid a \in A, h \in H \rangle$$

Here, $$[a,h] = a^{-1}h^{-1}ah$$ is the commutator of the two elements.

Proof
Given: Two subgroups $$H, K \le G$$.

To prove: $$[H, K ] \triangleleft \langle H, K \rangle$$.

Proof: By fact (1), $$H$$ normalizes $$[K, H]$$, which is the same as $$[H,K]$$. Also, $$K$$ normalizes $$[H,K]$$. Thus, the normalizer of $$[H,K]$$ in $$G$$ contains both $$H$$ and $$K$$, hence it contains $$\langle H, K \rangle$$, proving that $$[H,K]$$ is normal in $$\langle H, K \rangle$$.