Completeness of some axioms of Lukasiewicz's: an exercise in problem-solving

Among many systems of axioms for Propositional Logic, one due to Lukasiewicz is rather hard to prove complete, although it has appeared as a textbook exercise. This note retraces a roundabout route leading to a proof of its completeness. In view of this, the note is much longer than would be needed just to present the proof.