Normality satisfies intermediate subloop condition

Statement
Suppose $$L$$ is an algebra loop and $$N$$ is a normal subloop of $$L$$. Then, if $$M$$ is any subloop of $$L$$ containing $$N$$, $$N$$ is a normal subloop of $$M$$.

Related facts

 * Normality satisfies intermediate subgroup condition
 * Normality is upper join-closed (for groups)
 * Normality is not upper join-closed for algebra loops
 * Ideal property satisfies intermediate subring condition