Spaces with congruence

(E;L) denotes an aÆne plane, (E;L; ) an ordered plane, and denotes the congruence relation on E E. If we assume (E;L; ) as an hyperbolic plane, there exists the corresponding theorem for hyperbolic planes (cf. [3]). For both proofs we consider rst the group of motions, in particular the line re ections. For the de nition of a motion and a line re ection we need only a congruence relation (cf. [1, 7, 8]), but already for the proof that the line re ection is a motion it seems that additional assumptions on the geometry are necessary. For example K. Sorensen assumes here that for given lines G1; G2 , there exist distinct lines H1; H2 through a common point z which intersect G1; G2 (cf. [8]). Also for the proof that for any line G and any point x we have a unique perpendicular line through x, it seems that one needs additional assumptions. Here we restrict ourselves not to planes. We consider a space with congruence not assuming any geometrical properties, and we deal with the group of motions. We assume that the planes satisfy the exchange property, which can be easily shown for ordered spaces. We also assume an additional property (W4), which is also valid in an ordered space with congruence.