reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem
  for p,q,p1,p2 being Point of TOP-REAL n st p in LSeg(p1,p2) & q in
  LSeg(p1,p2) holds LSeg(p1,p2) = LSeg(p1,p) \/ LSeg(p,q) \/ LSeg(q,p2)
proof
  let p,q,p1,p2 be Point of TOP-REAL n;
  assume that
A1: p in LSeg(p1,p2) and
A2: q in LSeg(p1,p2);
A3: LSeg(p,q) c= LSeg(p1,p2) by A1,A2,Th6;
A4: LSeg(p1,p2) = LSeg(p1,q) \/ LSeg(q,p2) by A2,Th5;
A5: now
    per cases;
    suppose
      p in LSeg(p1,q);
      hence LSeg(p1,p2) c= LSeg(p1,p) \/ LSeg(p,q) \/ LSeg(q,p2) by A4,Th5;
    end;
    suppose
      not p in LSeg(p1,q);
      then p in LSeg(q,p2) by A1,A4,XBOOLE_0:def 3;
      then
A6:   LSeg(q,p2) = LSeg(q,p) \/ LSeg(p,p2) by Th5;
      LSeg(p1,p2) = LSeg(p1,p) \/ LSeg(p,p2) by A1,Th5;
      then
A7:   LSeg(p1,p2) c= LSeg(p1,p) \/ LSeg(q,p2) by A6,XBOOLE_1:7,9;
A8:   LSeg(p1,p) \/ LSeg(q,p2) \/ LSeg(p,q) = LSeg(p1,p) \/ LSeg(p,q) \/
      LSeg(q,p2) by XBOOLE_1:4;
      LSeg(p1,p) \/ LSeg(q,p2) c= LSeg(p1,p) \/ LSeg(q,p2) \/ LSeg(p,q)
      by XBOOLE_1:7;
      hence LSeg(p1,p2) c= LSeg(p1,p) \/ LSeg(p,q) \/ LSeg(q,p2) by A7,A8;
    end;
  end;
  p1 in LSeg(p1,p2) by RLTOPSP1:68;
  then LSeg(p1,p) c= LSeg(p1,p2) by A1,Th6;
  then
A9: LSeg(p1,p) \/ LSeg(p,q) c= LSeg(p1,p2) by A3,XBOOLE_1:8;
  p2 in LSeg(p1,p2) by RLTOPSP1:68;
  then LSeg(q,p2) c= LSeg(p1,p2) by A2,Th6;
  then LSeg(p1,p) \/ LSeg(p,q) \/ LSeg(q,p2) c= LSeg(p1,p2) by A9,XBOOLE_1:8;
  hence thesis by A5;
end;
