(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. [...

It is well known [4] that the non-Archimedean residue class fields K of the ring of continuous real valued functions on a space are realclosed and 7/i-sets. It does not appear to be known that the exponential function in the reals induces an exponential function in K (definitions to follow) ; thus K is exponentially closed. The property of being exponentially closed is a new invariant which wil...