Proof By the Origin Lemma, there is an hinversion h_{P} interchanging O and P. Suppose that h_{P}(Q) = R, so h_{P}(R) = Q as h_{P} has order 2. Again by the Lemma, there is an hinversion h_{R} interchanging O and R. Let t = h_{P}oh_{R}oh_{P}. Then a direct calculation shows that t(P) = Q and t(Q) = P. Also, t is an hinversion by the the algebraic inversion theorem.
If u is a second hinversion interchanging P and Q, then a calculation

We make frequent use of the fact that, for an hinversion h, we have h^{1} = h.
