Complete the proof of Theorem 6.1 by showing that
$S\overset{*}{\Rightarrow}_{\widehat{G}} w$
implies
$S\overset{*}{\Rightarrow}_{G} w$.
Theorem 6.1
Let $G = (V, T, S, P)$ be a context-free grammar. Suppose that $P$ contains a production of the form $A → x_1 Bx_2.$
Assume that $A$ and $B$ are different variables and that $B → y_1 |y_2 |…|y_n$ is the set of all productions in $P$ that have $B$ as the left side. Let $\widehat{G}= (V, T, S,\widehat{P} )$ be the grammar in which $\widehat{P}$ is constructed by deleting $A\rightarrow x_1Bx_2$ from $P$, and adding to it $A\rightarrow x_1y_1x_2|x_1y_2x_2|...|x_1y_nx_2.$
Then, $L(\widehat{G})=L(G)$.
