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 Th52:
  x in (A*) ^^ A or x in A ^^ (A*) implies x in A*
proof
A1: now
    let x;
    assume x in A ^^ (A*);
    then consider a, b such that
A2: a in A and
A3: b in A* and
A4: x = a ^ b by Def1;
    consider n such that
A5: b in A |^ n by A3,Th41;
    a in A |^ 1 by A2,Th25;
    then a ^ b in A |^ (n + 1) by A5,Th40;
    hence x in A* by A4,Th41;
  end;
  now
    let x;
    assume x in (A*) ^^ A;
    then consider a, b such that
A6: a in A* and
A7: b in A and
A8: x = a ^ b by Def1;
    consider n such that
A9: a in A |^ n by A6,Th41;
    b in A |^ 1 by A7,Th25;
    then a ^ b in A |^ (n + 1) by A9,Th40;
    hence x in A* by A8,Th41;
  end;
  hence thesis by A1;
end;
