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 Th54:
  <%>E in A implies A* = (A*) ^^ A & A* = A ^^ (A*)
proof
  assume
A1: <%>E in A;
A2: <%>E in A* by Th48;
A3: now
    let x;
    assume x in A*;
    then consider n such that
A4: x in A |^ n by Th41;
A5: now
      assume n = 0;
      then x in {<%>E} by A4,Th24;
      then x = <%>E by TARSKI:def 1;
      hence x in (A*) ^^ A & x in A ^^ (A*) by A1,A2,Th15;
    end;
A6: now
      assume n > 0;
      then ex m st m + 1 = n by NAT_1:6;
      hence x in A ^^ (A*) by A4,Th50;
    end;
    now
      assume n > 0;
      then ex m st m + 1 = n by NAT_1:6;
      hence x in (A*) ^^ A by A4,Th50;
    end;
    hence x in (A*) ^^ A & x in A ^^ (A*) by A5,A6;
  end;
  then
A7: for x being object holds x in A* implies x in (A*) ^^ A;
A8: for x being object holds x in A* implies x in A ^^ (A*) by A3;
  for x being object
   holds (x in (A*) ^^ A implies x in A*) & (x in A ^^ (A*) implies x
  in A*) by Th52;
  hence thesis by A7,A8,TARSKI:2;
end;
