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 Th11:
  A |^.. n = A* iff <%>E in A or n = 0
proof
  thus A |^.. n = A* implies <%>E in A or n = 0
  by FLANG_1:48,Th10;
  assume
A1: <%>E in A or n = 0;
    now
      let x be object;
      assume x in A*;
      then consider k such that
A2:   x in A |^ k by FLANG_1:41;
      per cases;
      suppose
        n <= k;
        hence x in A |^.. n by A2,Th2;
      end;
      suppose
        k < n;
        then A |^ k c= A |^ n by A1,FLANG_1:36;
        hence x in A |^.. n by A2,Th2;
      end;
    end;
    then
A3: A* c= A |^.. n;
    A |^.. n c= A* by Th9;
    hence thesis by A3,XBOOLE_0:def 10;
end;
