reserve r for Real;

theorem
  for S being MetrSpace, p,q,r,s being Element of S st q is_between p,r
  & r is_between p,s holds q is_between p,s & r is_between q,s
proof
  let S be MetrSpace, p,q,r,s be Element of S;
  assume
A1: q is_between p,r; then
A2: p<>q;
  assume
A3: r is_between p,s; then
A4: p <> s & r <> s;
  dist(p,r) = dist(p,q) + dist(q,r) by A1; then
A5: dist(p,s) = dist(p,q) + dist(q,r) + dist(r,s) by A3;
  dist(p,s) <= dist(p,q) + dist(q,s) & dist(p,q) + dist(q,s) <=
  (dist(q,r) + dist(r,s)) + dist(p,q) by Th4,XREAL_1:6; then
A6: dist(p,q) + dist(q,s) = dist(p,q) + (dist(q,r) + dist(r,s)) by A5,
XXREAL_0:1;
A7: q<>r by A1;
  then q <> s by A5,Th7;
  hence thesis by A2,A7,A4,A5,A6;
end;
