reserve r for Real;

theorem
  for M being non empty MetrSpace, p,r,x being Element of M holds
  x in open_dist_Segment(p,r) iff x is_between p,r
proof
  let M be non empty MetrSpace, p,r,x be Element of M;
  x in open_dist_Segment(p,r) implies x is_between p,r
  proof
    assume x in open_dist_Segment(p,r);
    then ex q be Element of M st x = q & q is_between p,r;
    hence thesis;
  end;
  hence thesis;
end;
