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 Th60:
  (A*)+ = A* & (A+)* = A*
proof
A1: A* c= (A+)* by Th59,FLANG_1:61;
  now
    let x be object;
    assume x in (A*)+;
    then consider k such that
    0 < k and
A2: x in (A*) |^ k by Th48;
    (A*) |^ k c= A* by FLANG_1:65;
    hence x in A* by A2;
  end;
  then
A3: (A*)+ c= A*;
  now
    let x be object;
    assume x in (A+)*;
    then consider k such that
A4: x in (A+) |^ k by FLANG_1:41;
    (A+) |^ k c= A* by Th55,FLANG_1:59;
    hence x in A* by A4;
  end;
  then
A5: (A+)* c= A*;
  A* c= (A*)+ by Th59;
  hence thesis by A1,A3,A5,XBOOLE_0:def 10;
end;
