 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 U in A _\/_ (A^)
  proof
    let A be non empty IntervalSet of U;
A1: U c= A``2 \/ (A``1)`
    proof
      let x be object;
      assume A2: x in U;
      A``1 c= A``2 by Th16; then
A3:   (A``2)` c= (A``1)` by SUBSET_1:12;
      x in A``2 or x in (A``2)` by A2,XBOOLE_0:def 5;
      hence thesis by A3,XBOOLE_0:def 3;
    end;
A4: 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,A4;
    then A _\/_ (A^) = Inter (A``1 \/ (A``2)`, A``2 \/ (A``1)`) by Th17;
    hence thesis by A1;
  end;
