reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem
  A* = {<%>E} iff A = {} or A = {<%>E}
proof
  thus A* = {<%>E} implies A = {} or A = {<%>E}
  proof
A1: A c= A* by Th43;
    assume that
A2: A* = {<%>E} and
A3: A <> {} & A <> {<%>E};
    ex x being object st x in A & x <> <%>E by A3,ZFMISC_1:35;
    hence contradiction by A2,A1,TARSKI:def 1;
  end;
A4: now
    assume
A5: A = {};
A6: now
      let x be object;
      assume x in A*;
      then consider n such that
A7:   x in A |^ n by Th41;
      n = 0 implies x in {<%>E} by A7,Th24;
      hence x in {<%>E} by A5,A7,Th27;
    end;
    now
      let x be object;
      assume x in {<%>E};
      then x in A |^ 0 by Th24;
      hence x in A* by Th41;
    end;
    hence A* = {<%>E} by A6,TARSKI:2;
  end;
  now
    assume
A8: A = {<%>E};
A9: A* c= {<%>E}
    proof
      let x be object;
      assume x in A*;
      then ex n st x in A |^ n by Th41;
      hence thesis by A8,Th28;
    end;
    {<%>E} c= A*
    proof
      let x be object;
      assume x in {<%>E};
      then x in A |^ 0 by Th24;
      hence thesis by Th41;
    end;
    hence A* = {<%>E} by A9,XBOOLE_0:def 10;
  end;
  hence thesis by A4;
end;
