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 Th50:
  A+ = A |^.. 1
proof
A1: now
    let x be object;
    assume x in A+;
    then consider k such that
A2: 0 < k and
A3: x in A |^ k by Th48;
    0 + 1 <= k by A2,NAT_1:13;
    hence x in A |^.. 1 by A3,Th2;
  end;
  now
    let x be object;
    assume x in A |^.. 1;
    then ex k st 1 <= k & x in A |^ k by Th2;
    hence x in A+ by Th48;
  end;
  hence thesis by A1,TARSKI:2;
end;
