Problem $24($Homework $3$ Problem $4).$ Prove rigorously $($by induction on an appropriate quantity$)$ that the grammar with a single non$-$terminal $E$ and production rules $E0E1E,E$ generates the language

$\{xin\{0,1{\}}^{**}|$#$0\text{'}sinx=$#$1\text{'}sinx,$ and for every prefix $x^{\text{'}}ofx,$#$0\text{'}sin{x}^{\text{'}}$#$1\text{'}sin{x}^{\text{'}}\}.$

Also prove that the grammar is unambiguous.$\{\begin{array}{c}|\end{array}$