proof of theorem PI1

Theorem PI1
If A,B,C,D,E are collinear p-points such that (A,B,C,D) = (A,B,C,E), then D = E.

Observe that, as the cross-ratios are defined, we must have A,B,C,D,E collinear.
Suppose that A = [a], B =[b], C = [c], D = [d] and E = [e]. As before,
we have real constants such that c = αa + βb, d = γa + δb and d = λa + μb.
Then, by the definition of cross ratio, (A,B,C,D) = βγ/αδ and (A,B,C,E) = βλ/αμ.
As the cross-ratios are equal, βγ/αδ = βλ/αμ, so λ = γμ/δ.
Then e = (γ&mu/δ)a + μb = (μ/δ)(γa + δb) = (μ/δ)d.
Since e is a multiple of d, E = D, as required.

projective cross-ratio page