reserve r for Real;

theorem
  for M being non empty MetrStruct, p,r,x being Element of M holds
  x in close_dist_Segment(p,r) iff (x is_between p,r or x = p or x = r)
proof
  let M be non empty MetrStruct, p,r,x be Element of M;
A1: x in close_dist_Segment(p,r) implies (x is_between p,r or x = p or x = r)
  proof
    assume x in close_dist_Segment(p,r);
    then x in {q where q is Element of M: q is_between p,r} or x in {p,r}
      by XBOOLE_0:def 3; then
    (ex q be Element of M st x = q & q is_between p,r) or (x = p or x = r)
    by TARSKI:def 2;
    hence thesis;
  end;
  now
    assume x is_between p,r or x = p or x = r;
    then x in {q where q is Element of M: q is_between p,r} or x in {p,r}
      by TARSKI:def 2;
    hence x in close_dist_Segment(p,r) by XBOOLE_0:def 3;
  end;
  hence thesis by A1;
end;
