Proof of the Interchange Lemma

Proof By the Origin Lemma, there is an h-inversion hP interchanging O and P. Suppose that hP(Q) = R, so hP(R) = Q as hP has order 2. Again by the Lemma, there is an h-inversion hR interchanging O and R. Let t = hPohRohP. Then a direct calculation shows that t(P) = Q and t(Q) = P. Also, t is an h-inversion by the the algebraic inversion theorem. If u is a second h-inversion interchanging P and Q, then a calculation shows that hPouohP interchanges O and R, so it must be hR by the uniqueness clause of the Origin Lemma.

We make frequent use of the fact that, for an h-inversion h, we have h-1 = h.

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