 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 4;
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:   X \/ Y c= A``2 by A3,XBOOLE_1:8;
      X c= X \/ Y by XBOOLE_1:7; then
      A``1 c= X \/ Y by A3;
      hence thesis by Th14,A4,A2;
    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 4;
    end;
    hence thesis by A1;
  end;
