1. ## well-defined proof

$A$, $A^{*}$, $B$, $B^{*}$ be four subgroups of a group $G$ with $A \vartriangleleft A^{*}$ and $B \vartriangleleft B^{*}$.
Let $D=(A^{*} \cap B)(A \cap B^{*})$.
I have shown the following:
(1) $B \vartriangleleft B(A^{*} \cap B^{*})$
(2) $(A^{*} \cap B) \vartriangleleft (A^{*} \cap B^{*})$
(3) $(A \cap B^{*}) \vartriangleleft (A^{*} \cap B^{*})$
(4) $D \vartriangleleft (A^{*} \cap B^{*})$

If $x \in B(A^{*} \cap B^{*})$, then $x=bc$ for $b \in B$ and $c\in (A^{*} \cap B^{*})$;
define $f:B(A^{*} \cap B^{*}) \rightarrow (A^{*} \cap B^{*})/D$ by $f(x)=Dc$.

I'm facing problem to show that $f$ is well-defined.
I started with $x_{1}=x_{2}$;
implies that $b_{1}c_{1}=b_{2}c_{2}$. ----------( $*$)
I'm stuck here.
I know I need to get $c_{1}c_{2}^{-1} \in D$; ------------( $**$)
implies that $Dc_{1}=Dc_{2}$;
finally $f(x_{1})=f(x_{2})$.

How to continue from ( $*$) to ( $**$)?

2. I have another question.

How to show that $Ker\:f=B(A \cap B^{*})$ ?