Problem
Is the operation of addition on the subspaces of $V$ associative? In other words, if $U_1, U_2, U_3$ are subspaces of $V$, is $$(U_1 + U_2) + U_3 = U_1 + (U_2 + U_3) ?$$
Solution
Yes, Proof: $$(U_1 + U_2) + U_3 \stackrel{D1.36}{=} \lbrace u_1 + u_2 | u_1 \in U_1, u_2 \in U_2 \rbrace + U_3$$ $$\stackrel{D1.36 \land D1.39}{=} \lbrace (u_1 + u_2) + u_3 | u_1 \in U_1, u_2 \in U_2, u_3 \in U_3 \rbrace$$ $$\stackrel{D1.19 \land D1.39}{=} \lbrace u_1 + (u_2 + u_3) | u_1 \in U_1, u_2 \in U_2, u_3 \in U_3\rbrace$$ $$\stackrel{D1.36 \land D1.39}{=} U_1 + \lbrace u_2 + u_3 | u_2 \in U_2, u_3 \in U_3 \rbrace \stackrel{D1.36}{=} U_1 + (U_2 + U_3)$$