reserve r for Real;

theorem
  for S being MetrSpace, p,q,r being Element of S st q is_between p,r
  holds (not p is_between q,r) & not r is_between p,q
proof
  let S be MetrSpace, p,q,r be Element of S;
  assume
A1: q is_between p,r; then
A2: dist(p,r) = dist(p,q) + dist(q,r);
  thus not p is_between q,r
  by A2,Th7;
  assume r is_between p,q; then
A3: dist(p,q) = dist(p,q) + dist(q,r) + dist(r,q) by A2;
  q <> r by A1;
  hence contradiction by A3,Th7;
end;
