  In view of the above-mentioned one-one correspondence between closed subspaces and projections, we may impose upon the set \(L(\mathbf{H})\) the structure of a complete orthocomplemented lattice, defining \(P\le Q\), where \(\rran (P) \subseteq \rran (Q)\) and \(P' = 1 - P\) (so that \(\rran (P') = \rran (P)^{\bot})\). It is straightforward that \(P\le Q\) just in case \(PQ = QP = P\). More generally, if PQ = QP, then \(PQ = P\wedge Q\), the meet of \(P\) and \(Q\) in \(L(\mathbf{H})\); also in this case their join is given by \(P\vee Q = P+Q - PQ\).