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

theorem Th30:
  x in A & x <> <%>E & (m > 0 or n > 0) implies A |^ (m, n) <> { <%>E}
proof
  assume that
A1: x in A and
A2: x <> <%>E and
A3: m > 0 or n > 0;
  per cases by A3;
  suppose
    m > n;
    hence thesis by Th21;
  end;
  suppose
A4: m <= n & m > 0;
A5: A |^ m <> {} by A1,FLANG_1:27;
    A |^ m <> {<%>E} by A1,A2,A4,Th7;
    then
A6: ex x being object st x in A |^ m & x <> <%>E by A5,ZFMISC_1:35;
    A |^ m c= A |^ (m, n) by A4,Th20;
    hence thesis by A6,TARSKI:def 1;
  end;
  suppose
A7: m <= n & n > 0;
A8: A |^ n <> {} by A1,FLANG_1:27;
    A |^ n <> {<%>E} by A1,A2,A7,Th7;
    then
A9: ex x being object st x in A |^ n & x <> <%>E by A8,ZFMISC_1:35;
    A |^ n c= A |^ (m, n) by A7,Th20;
    hence thesis by A9,TARSKI:def 1;
  end;
end;
