Proof of Zassenhaus Lemma
Solution 1:
Suppose $ax$ is in the kernel of $\phi$. This means that $x\in D=(A\cap B^*)(A^*\cap B)$. Hence, we can write $x=a'x'$ where $a'\in A\cap B^*\subset A$ and $x'\in A^*\cap B$. It follows that $ax=aa'x'\in A(A^*\cap B)$.