For the purposes of this problem, we will define a "generalized context$-$free grammar" as a four$-$tuple $\backslash ($ G $=($N$,$ T$,$ P$,$ x$)\backslash ),$ where $\backslash ($ N $\backslash )$ is the set of non$-$terminal symbols, $\backslash ($ T $\backslash )$ is the set of terminal symbols, $\backslash ($ N $\backslash$cap T $=\backslash$emptyset $\backslash ),\backslash ($ P $\backslash$subseteq N $\backslash$times $($N $\backslash$cup T$)^*\backslash )$ is a finite set of production rules, and $\backslash ($ x $\backslash$in $($N $\backslash$cup T$)^*\backslash )$ is the initial sentence form, known as the axiom. We define the derivation relation $\backslash (\backslash$Rightarrow$\_$G $\backslash )$ in grammar $\backslash ($ G $\backslash )$ in the same way as for context$-$free grammars, and the language generated by grammar $\backslash ($ G $\backslash )$ is given as in phot. Clearly, every context$-$free grammar is also a generalized context$-$free grammar.

Determine whether for every generalized context$-$free grammar $\backslash ($ G $\backslash ),$ there exists a context$-$free grammar $\backslash ($ G$\text{'}\backslash )$ such that $\backslash ($ L$($G$\text{'})=$ L$($G$)\backslash ).$ Prove your claim; in the case of constructing an equivalent grammar $\backslash ($ G$\text{'}\backslash ),$ prove its correctness using good mathematical inductiond$($