Coxeter- Theorem 5.21

5.21 The product of two parabolic projectivies having the same invariant point is another such parabolic projectivity (if it is not merely the identity). PROOF. Clearly, the common invariant point C of the projectivies is still invariant for the product, which is therefore either parabolic (having one invariant point) or hyperbolic (having two invariant points). The latter possbility is excluded by the following argument. If any other point A were invariant for the product, the first parabolic projectivity would take A to some different point B, and the second would take B back to A. Thus the first would be ACC projective to BCC, the second would be its inverse BCC projective to ACC, and the product would not be properly hyperbolic but merely the identity. So the product of ACC projective to A1CC and A1CC projective to A2CC is ACC projective to A2CC with a continuing string of such parabolic relationships. (Note: the relationship is written out as "ACC" because this format shows the pair of corresponding points A,C with the associated invariant point C)