Order-topological separable complete modular ortholattices admit order continuous faithful valuations

We prove that on every separable complete atomic modular ortholattice (i.e.order topological) there exists an order continuous faithful valuation. We also give a construction of the existing order continuous faithful valuation. For separable atomic modular ortholattices we give a necessary and sufficient condition to admit an order continuous faithful valuation and we show that it is equivalent with the condition to have a modular MacNeille completion. We improve one statement on complete metric lattices from Birkhoff's Lattice Theory.