 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
A1: [#]U in Inter ({}U,[#]U); then
    consider A being non empty IntervalSet of U such that
A2: A = Inter ({}U,[#]U);
A3: A^ = Inter (([#]U)`,({}U)`) by A2,Th46
    .= Inter ({}U,[#]U);
    A^ = Inter (A^``1,A^``2) by Th15; then
    A^``1 = {}U & A^``2 = [#]U by Th6,A3; then
A4: A _/\_ A^ = Inter ({}U /\ {}U, [#]U /\ [#]U) by A2,Th18,A3
    .= Inter ({}U, [#]U);
    not [#]U c= {}U; then
    not [#]U in Inter ({}U,{}U) by Th1;
    hence thesis by A4,A1;
  end;
