A permutation \(\sigma\) of \(\{1, 2, \ldots, 2p\}\) is defined to have a crossing just when there are \(a\), \(b\), \(c\), \(d\) in \(\{1, 2, \ldots, 2p\}\) such that \(a &lt; b &lt; c &lt; d\) and \(\sigma(a) = c\) and \(\sigma(b) = d\). The focus is on the proof of a theorem which employs this notion. (The theorem is that when a permutation of \(\{1, 2, \ldots, 2p\}\) of the relevant kind is non-crossing, there will be exactly \(p+1\) R-equivalence classes, where \(R\) is a certain equivalence relation on \(\{1, 2, \ldots, 2p\}\) defined in terms of the permutation.) Carter says that the proofs of some lemmas “rely on a visualization of the setup”, in that to grasp the correctness of one or more of the steps one needs to visualize the situation. There is also a nice example of some reasoning in terms of a diagram which gives the idea for a proof (“suggests a proof strategy”) for the lemma that every non-crossing permutation has a neighbouring pair. Reflection on a diagram such as Figure 9 does the work.