reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, a1, a2, b for Element of E^omega;
reserve i, k, l, m, n for Nat;

theorem Th30:
  A* ^^ A = A |^.. 1
proof
A1: now
    let x be object;
    assume x in (A*) ^^ A;
    then consider a1, a2 such that
A2: a1 in A* and
A3: a2 in A and
A4: x = a1 ^ a2 by FLANG_1:def 1;
    consider n such that
A5: a1 in A |^ n by A2,FLANG_1:41;
A6: n + 1 >= 1 by NAT_1:11;
    a2 in A |^ 1 by A3,FLANG_1:25;
    then a1 ^ a2 in A |^ (n + 1) by A5,FLANG_1:40;
    hence x in A |^.. 1 by A4,A6,Th2;
  end;
  now
    let x be object;
    assume x in A |^.. 1;
    then consider n such that
A7: n >= 1 and
A8: x in A |^ n by Th2;
    consider m such that
A9: m + 1 = n by A7,NAT_1:6;
    A |^ (m + 1) c= (A*) ^^ A by FLANG_1:51;
    hence x in (A*) ^^ A by A8,A9;
  end;
  hence thesis by A1,TARSKI:2;
end;
