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 Th13:
  A ^^ {<%>E} = A & {<%>E} ^^ A = A
proof
A1: for x being object holds x in A ^^ {<%>E} implies x in A
  proof let x be object;
    assume x in A ^^ {<%>E};
    then consider a, d such that
A2: a in A and
A3: d in {<%>E} & x = a ^ d by Def1;
    x = a ^ {} by A3,TARSKI:def 1;
    hence thesis by A2;
  end;
A4: for x being object holds x in A implies x in {<%>E} ^^ A
  proof let x be object;
    assume
A5: x in A;
    ex d, a st d in {<%>E} & a in A & x = d ^ a
    proof
      reconsider a = x as Element of E^omega by A5;
      consider d such that
A6:   d = <%>E;
      take d, a;
      thus thesis by A5,A6,TARSKI:def 1;
    end;
    hence thesis by Def1;
  end;
A7: for x being object holds x in A implies x in A ^^ {<%>E}
  proof let x be object;
    assume
A8: x in A;
    ex a, d st a in A & d in {<%>E} & x = a ^ d
    proof
      reconsider a = x as Element of E^omega by A8;
      set d = <%>E;
      take a, d;
      thus thesis by A8,TARSKI:def 1;
    end;
    hence thesis by Def1;
  end;
  for x being object holds x in {<%>E} ^^ A implies x in A
  proof let x be object;
    assume x in {<%>E} ^^ A;
    then consider d, a such that
A9: d in {<%>E} and
A10: a in A and
A11: x = d ^ a by Def1;
    x = {} ^ a by A9,A11,TARSKI:def 1;
    hence thesis by A10;
  end;
  hence thesis by A1,A4,A7,TARSKI:2;
end;
