Coxeter- Theorem 2.21

2.21 - Any two distinct lines have at most one common point. PROOF. Suppose, if possible, that two given lines have two common points A and B. Axiom 2.13 tells us that each line is determined by these two points. Thus the two lines coincide, contradicting our assumption that they are distinct. In the example provided, if line QR and PE both contain the points A and B, a line can be drawn AB which is equivalent to QR and PE.