proof of lemma 3

Lemma 3
If tεA(2) fixes every point of the line L, and sεA(2), then u = sotos-1 fixes
every point of s(L).

Proof
For Pεs(L), P = s(Q) for some Q on L, and so t(Q) = Q.
Then u(P) = sotos-1os(Q) = sot(Q) = s(Q) = P.

fixed point theorems page