A simple proof for a theorem of Luxemburg and Zaanen

In this paper a simple proof for the following theorem, due to Luxemburg and Zaanen is given: an Archimedean vector lattice A is Dedekind σ-complete if and only if A has the principal projection property and A is uniformly complete. As an application, we give a new and short proof for the following version of Freudenthal's spectral theorem: let A be a uniformly complete vector lattice with the principal projection property and let 0<u∈A. For any element w in A such that 0?w?u there exists a sequence in A which satisfies , where each element sn is of the form , with real numbers α₁,…,αk such that 0?αi?1 (i=1,…,k) and mutually disjoint components p₁,…,pk of u.