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 Th20:
  LSeg(|[0,0]|,|[0,1]|) misses LSeg(|[1,0]|,|[1,1]|)
proof
  set x = the Element of LSeg(|[0,0]|,|[0,1]|) /\ LSeg(|[1,0]|,|[1,1]|);
  assume
A1: LSeg(|[0,0]|,|[0,1]|) /\ LSeg(|[1,0]|,|[1,1]|) <> {};
  then x in LSeg(|[0,0]|,|[0,1]|) by XBOOLE_0:def 4;
  then
A2: ex p st p = x & p`1 = 0 & p`2 <= 1 & p`2 >= 0 by Th13;
  x in LSeg(|[1,0]|,|[1,1]|) by A1,XBOOLE_0:def 4;
  then ex p2 st p2 = x & p2`1 = 1 & p2`2 <= 1 & p2`2 >= 0 by Th13;
  hence contradiction by A2;
end;
