 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
  ex A being non empty IntervalSet of U st A _\/_ A^ <> Inter ([#]U,[#]U)
  proof
    not [#]U c= {}U; then
A1: not {}U in Inter ([#]U,[#]U) by Th1;
A2: {}U in Inter ({}U,[#]U); then
    consider A being non empty IntervalSet of U such that
A3: A = Inter ({}U,[#]U);
A4: A^ = Inter (([#]U)`,({}U)`) by Th46,A3
      .= Inter ({}U,[#]U);
    A^ = Inter (A^``1,A^``2) by Th15; then
    A^``1 = {}U & A^``2 = [#]U by Th6,A4; then
    A _\/_ A^ = Inter ({}U \/ {}U, [#]U \/ [#]U) by Th17,A3,A4
             .= Inter ({}U,[#]U);
    hence thesis by A2,A1;
  end;
