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 Th14:
  x in A & x <> <%>E implies A |^.. n <> {<%>E}
proof
  assume that
A1: x in A and
A2: x <> <%>E;
  per cases;
  suppose
A3: n = 0;
    x in A |^ 1 by A1,FLANG_1:25;
    then x in A |^.. n by A3,Th2;
    hence thesis by A2,TARSKI:def 1;
  end;
  suppose
A4: n > 0;
A5: A |^ n <> {} by A1,FLANG_1:27;
    A |^ n <> {<%>E} by A1,A2,A4,FLANG_2:7;
    then consider y being object such that
A6: y in A |^ n and
A7: y <> <%>E by A5,ZFMISC_1:35;
    y in A |^.. n by A6,Th2;
    hence thesis by A7,TARSKI:def 1;
  end;
end;
