Proof, for clarity, let's change one of the sets of parentheses () to brackets []: [S v T] v (U v W) S v [T v (U v W)] associative law <--moving the []s S v [(T v U) v W)] associative law <--moving the ()'s S v [(U v T) v W)] commutative law <--swapping U, T [(U v T) v W)] v S commutative law <--swapping the [], S (U v T) v [W v S] associative law <--moving the []'s (U v T) v [S v W] commutative law <--swapping S,W Edwin