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
  LSeg(|[0,0]|,|[1,0]|) /\ LSeg(|[1,0]|,|[1,1]|) = {|[1,0]|}
proof
  for a being object holds a in LSeg(|[0,0]|,|[1,0]|) /\ LSeg(|[1,0]|,|[1,1]|
  ) iff a = |[1,0]|
  proof
    set p00 = |[0,0]|, p10 = |[1,0]|, p11 = |[1,1]|;
    let a be object;
    thus a in LSeg(p00,p10) /\ LSeg(p10,p11) implies a = p10
    proof
      assume
A1:   a in LSeg(p00,p10) /\ LSeg(p10,p11);
      then a in { p : p`1 <= 1 & p`1 >= 0 & p`2 = 0} by Th13,XBOOLE_0:def 4;
      then
A2:   ex p st p = a & p`1 <= 1 & p`1 >= 0 & p`2 = 0;
      a in LSeg(p10,p11) by A1,XBOOLE_0:def 4;
      then ex p2 st p2=a & p2`1 = 1 & p2`2 <= 1 & p2`2 >= 0 by Th13;
      hence thesis by A2,EUCLID:53;
    end;
    assume
A3: a = |[1,0]|;
    then
A4: a in LSeg(p10,p11) by RLTOPSP1:68;
    a in LSeg(p00,p10) by A3,RLTOPSP1:68;
    hence thesis by A4,XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:def 1;
end;
