hen given a set of properties or conditions (say, three) that are claimed to be equivalent, the claim e s can be verified by supplying what we call a circle of proofs. In the case in point, one proves th econd property or condition from the first, the third from the second, and the first from the third. If r s the proof that 1 implies 2 does not rely on 3, then we say that the proof is pure wit...