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 Th50:
  x in A |^ (m + 1) implies x in (A*) ^^ A & x in A ^^ (A*)
proof
  assume x in A |^ (m + 1);
  then
A1: x in (A |^ m) ^^ A by Th23;
  then consider a, b such that
A2: a in A |^ m and
A3: b in A & x = a ^ b by Def1;
  a in A* by A2,Th41;
  hence x in (A*) ^^ A by A3,Def1;
  x in A ^^ (A |^ m) by A1,Th32;
  then consider a, b such that
A4: a in A and
A5: b in A |^ m and
A6: x = a ^ b by Def1;
  b in A* by A5,Th41;
  hence thesis by A4,A6,Def1;
end;
