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

theorem Th2:
  Inter (X,Y) <> {} implies X in Inter (X,Y) & Y in Inter (X,Y)
  proof
    assume Inter (X,Y) <> {}; then
    consider x being object such that
A1: x in Inter (X,Y) by XBOOLE_0:def 1;
    reconsider x as set by TARSKI:1;
    X c= x & x c= Y by A1,Th1; then
    X c= Y;
    hence thesis;
  end;
