 reserve U for set,
         X, Y for Subset of U;

theorem Th8:
  for A being Subset of U holds Inter (A, A) = { A }
  proof
    let A be Subset of U;
    thus Inter (A, A) c= { A }
    proof
      let x be object;
     reconsider xx=x as set by TARSKI:1;
      assume x in Inter (A,A); then
      A c= xx & xx c= A by Th1; then
      A = x by XBOOLE_0:def 10;
      hence thesis by TARSKI:def 1;
    end;
    let x be object;
    assume x in { A }; then
    x = A by TARSKI:def 1;
    hence thesis;
  end;
