 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
  A _/\_ A = A
  proof
A1: A _/\_ A c= A
    proof
      let x be object;
      assume x in A _/\_ A; then
      consider X, Y being set such that
A2:   X in A & Y in A & x = X /\ Y by SETFAM_1:def 5;
A3:   A``1 c= X & X c= A``2 by A2,Th14;
      A``1 c= Y & Y c= A``2 by A2,Th14; then
A4:   A``1 c= X /\ Y by A3,XBOOLE_1:19;
      X /\ Y c= X by XBOOLE_1:17; then
      X /\ Y c= A``2 by A3;
      hence thesis by A4,A2,Th14;
    end;
    A c= A _/\_ A
    proof
      let x be object;
     reconsider xx=x as set by TARSKI:1;
      assume
A5:   x in A;
      x = xx /\ xx;
      hence thesis by A5,SETFAM_1:def 5;
    end;
    hence thesis by A1;
  end;
