Characteristically simple and normal fully normalized implies minimal normal

Statement
If $$H$$ is a fact about::normal fully normalized subgroup of $$G$$ and $$H$$ is characteristically simple as a group, then $$H$$ is a fact about::minimal normal subgroup of $$G$$.

Facts used

 * 1) Normal upper-hook fully normalized implies characteristic: If $$K \le H \le G$$ are such that $$K$$ is normal in $$G$$ and $$H$$ is fully normalized in $$G$$, then $$K$$ is characteristic in $$H$$.

Proof
Given: A group $$G$$, a characteristically simple, normal and fully normalized subgroup $$H$$.

To prove: $$H$$ is a minimal normal subgroup of $$G$$.

Proof: By assumption $$H$$ is normal, so it suffices to show that any normal subgroup $$K$$ of $$G$$ contained in $$H$$ is either trivial or equal to $$H$$. Let's do this.

By fact (1), $$K$$ is characteristic in $$H$$. Since $$H$$ is characteristically simple, we see that $$K$$ must be either equal to $$H$$, or trivial, completing the proof.