Prove that $N P$^\{$N P $\backslash$cap $\backslash$operatorname$\{$coNP$\}\}=$N P$$.$ Be sure you understand what this means. A language is in $N P$^\{$N P $\backslash$cap c o N P$\}$$ if it can be written in the form $$\backslash$left$\backslash \{$x $\backslash$mid $\backslash$exists w: M$^\{$A$\}($x$,$ w$)=1\backslash$right$\backslash \}$$$,$ where $M$ is a poly$-$time oracle TM$,$ and $A$ is a language in NP $$\backslash$cap$ coNP. In other words, $A$ has both an $"$NP$-$style" algorithm and a "coNP$-$style" one. More formally, there are polytime TMs $M$\_\{1\}$$ and $M$\_\{2\}$$ such that $A$=\backslash$left$\backslash \{$a $\backslash$mid $\backslash$exists w$\_\{1\}$: M$\_\{1\}\backslash$left$($a$,$ w$\_\{1\}\backslash$right$)=1\backslash$right$\backslash \}=\backslash$left$\backslash \{$a $\backslash$mid $\backslash$forall w$\_\{2\}$: M$\_\{2\}\backslash$left$($a$,$ w$\_\{2\}\backslash$right$)=1\backslash$right$\backslash \}$$$.$

Hint: compare the situation to $N P$^\{$N P$\}$$ or $N P$^\{\backslash$text $\{$coNP $\}\}$$$,$ both of which are equal to $$\backslash$Sigma$\_\{2\}$$$.$ Think about where the additional quantifier comes from in these cases, and why you might not need one in our problem.