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