Boolean Specification Satisfiability $$ Using the Boolean specification and implementation listed below, prove the following: Boolean Specification Satisfiability Using the Boolean specification and implementation listed below, prove the following: Theorem $1.$ Let X be any non$-$empty set. The $\backslash$Sigma $^$Banl$-$algebra $2^$X of power sets of X satrifies the Boolean algebra axioms; in symbols X T$.$ The proof of this theorem involves demonstrating for Axiom $5$ that for any subsets U$,$ V of X$,$ the equation U $\backslash$cup V$=$X $\backslash ($X $\backslash$ U $\backslash$cap X $\backslash$ V$)$ holds. Demonstrate this equation holds for each of the following cases: a in U$,$ a in V$,$ a in U $\backslash$cup V$,$ a $$ U $\backslash$cup V$.$ Signature $\backslash$Sigma Sorts Bool Constants False : $->$ Bool, True : $->$ Bool Operations and: B o o l$^2->$ B o o l or: Bool$^2->$ Bool not : Bool $->$ Bool Axioms T Let a$,$ b in B o o l $1.$ not $($ true $)=$ false $2.$ not$($f a l s e$)=$ true $3.$ True and b$=$b $4.$ False and b$=$F a l s e $5.$ a or b$=$not$($not$($a$)$ and not$($b$))$