Proof of the Interchange Lemma

Interchange Lemma
If P, Q ε D, there is an h-inversion mapping P to Q and Q to P.
If P ≠ Q then the h-inversion is unique.

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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