 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
    Inter ({}U,[#]U) <> {} by Th1; then
    consider A being non empty IntervalSet of U such that
A1: A = Inter ({}U,[#]U);
A2: A _\_ A = Inter ({}U \ [#]U, [#]U \ {}U) by Th42,A1
    .= Inter ({}U, [#]U);
    not U c= {}; then
    [#]U in Inter ({}U,[#]U) & not [#]U in Inter ({}U,{}U) by Th1;
    hence thesis by A2;
  end;
