Suppose that Bold w in V has the property that Bold uplusBold wequalsBold wplusBold uequalsBold u for all Bold u in V. In​ particular, Bold 0plusBold wequalsBold 0. But Bold 0plusBold wequalsBold wplusBold 0 by Axiom nothing​, and Bold wplusBold 0equalsBold w by Axiom nothing. ​Hence, Bold wequalsBold wplusBold 0equalsBold 0plusBold wequalsBold 0.