 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 _\_ A = Inter (A``1 \ A``2, A``2 \ A``1) by Th40;
A2: A``1 \ A``2 c= {}
    proof
      let x be object;
      assume x in A``1 \ A``2; then
      x in A``1 & not x in A``2 & A``1 c= A``2 by Th16,XBOOLE_0:def 5;
      hence thesis;
    end;
    {} c= A``2 \ A``1;
    hence thesis by Th1,A2,A1;
  end;
