 reserve U for set,
         X, Y for Subset of U;
 reserve U for non empty set,
         A, B, C for non empty IntervalSet of U;

theorem
  for A being non empty IntervalSet of U holds {} in A _/\_ (A^)
    proof
    let A be non empty IntervalSet of U;
A1: A``1 /\ (A``2)` c= {}
    proof
      let x be object;
A2:   A``1 c= A``2 by Th16;
      assume x in  A``1 /\ (A``2)`; then
      x in A``1 & x in (A``2)` by XBOOLE_0:def 4;
      hence thesis by A2,XBOOLE_0:def 5;
    end;
A3: A^ = Inter ((A``2)`,(A``1)`) by Th45;
    A^ = Inter ((A^)``1, (A^)``2) by Th15; then
 A^``1 = (A``2)` & A^``2 = (A``1)` by Th6,A3;
then A4: A _/\_ (A^) = Inter (A``1 /\ (A``2)`, A``2 /\ (A``1)`) by Th18;
    A``1 /\ (A``2)` c= {} & {} c= A``2 /\ (A``1)` by A1;
    hence thesis by A4,Th1;
  end;
