Problem
Is the operation of addition on the subspaces of $V$ commutative? In other words, if $U$ and $W$ are subspaces of $V$, is $U + W = W + U$ ?
Solution
Yes, proof: $$U + W \stackrel{D1.36}{=} \lbrace u + w | u \in U, w \in W \rbrace = \stackrel{D1.19}{=} \lbrace w + u | w \in W, u \in U \rbrace \stackrel{D1.36}{=} W + U$$