$\gamma\$$ and GNF $G$ contains production $S \rightarrow \alpha A \gamma$.

Induction

:   The induction hypotheses states that $S \stackrel{n}{\Rightarrow} \alpha A
    \gamma$ in GNF $G$ iff $S \stackrel{n}{\Rightarrow} \alpha A$ with stack $\gamma\$$ in R-AVG $G'$. Next, we distinguish three cases.

    1.  GNF $G$ contains a production $A \rightarrow a A_1A_2\ldots A_n$. Hence there is a left-most derivation $S \stackrel{n+1}{\Rightarrow}
        \alpha a A_1A_2 \ldots A_n\gamma$. GNF $G$ contains the production $A \rightarrow
        a A_1A_2 \ldots A_n$ iff R-AVG $G'$ contains a production $A \rightarrow a A_1$ with equation ($A_2\ldots A_n$). Since the induction hypotheses states that there is a derivation $S \stackrel{n}{\Rightarrow} \alpha A$ with stack $\gamma\$$, there is a derivation $S \stackrel{n+1}{\Rightarrow} \alpha a A_1$ with stack $A_2 \ldots
        A_n\gamma\$$.

    2.  GNF $G$ contains a production $A \rightarrow a$ and $\gamma = B'\gamma'$. Hence there is a left-most derivation $S \stackrel{n+1}{\Rightarrow}
        \alpha a B' \gamma'$. GNF $G$ contains the production $A \rightarrow a$ iff R-AVG $G'$ contains productions $A \rightarrow a B$ with equation (B), for all $B \in N \setminus \{S\}$. Hence by the induction hypotheses, there is a derivation $S \stackrel{n+1}{\Rightarrow} \alpha a B'$ with stack $\gamma'\$$.

    3.  GNF $G$ contains a production $A \rightarrow a$ 