reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, a1, a2, b for Element of E^omega;
reserve i, k, l, m, n for Nat;

theorem
  (A+)+ = A+
proof
  now
    let x be object;
    assume that
A1: x in (A+)+;
    per cases;
    suppose
      x = <%>E;
      hence x in A+ by A1,Th56;
    end;
    suppose
A2:   x <> <%>E;
      (A+)+ c= A* by Th55,Th61;
      then x in A* by A1;
      then
A3:   x in (A+) \/ {<%>E} by Th53;
      not x in {<%>E} by A2,TARSKI:def 1;
      hence x in A+ by A3,XBOOLE_0:def 3;
    end;
  end;
  then
A4: (A+)+ c= A+;
  A+ c= (A+)+ by Th59;
  hence thesis by A4,XBOOLE_0:def 10;
end;
