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