I have to prove or give a counter example to the statement if U1, U2, W are subspaces of V such that V=U1 direct sum W and V=U2 direct sum W, then U2=U1. This is what I did: Let v be an element of V. Then v=v1+v2 for v1 an element of U1 and v2 and element of W and v=v3+v2 for v3 an element of U2. So v-v2=v1 and v-v2=v3. Therefore v1=v3. Hence U1=U2 since every vector in each subspace is the same. I just feel like I am missing something to make my small proof 100% airtight. Should I mention somewhere that v is represented in a unique way since V=U1 direct sum W and V=U2 direct sum W?