Normality satisfies intermediate subloop condition

ANALOGY: This is an analogue in algebra loops of a fact encountered in group. The old fact is: normality satisfies intermediate subgroup condition.
View other analogues of normality satisfies intermediate subgroup condition|View other analogues from group to algebra loop (OR, View as a tabulated list)

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