NL-NWO 1962 Problem 5

There are three kinds of things, which are designated respectively by the words (stripped of all common meaning) notes, staves, and heads. There can be a certain relationship between a note and a head, which is expressed by the saying: they match. Also, a note and a head can match and two different staves can match. Given are the following axioms: (a) If a note and a head each match the same stave, then they match, (b) If two different notes both match with stave B, and also both match with head V, then B and V match, (c) If two staves match, then there is a note that matches both, (d) If a note and a stave are given, then there is a head that matches both. Prove the following theorem, denoting the axiom you apply by its letter. If three staves that differ from each other, each one matches every other, and no note matches any of the three staves, then there is a head that matches all three staves.