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 Th33:
  <%>E in A |^ (m, n) iff m = 0 or m <= n & <%>E in A
proof
  thus <%>E in A |^ (m, n) implies m = 0 or m <= n & <%>E in A
  proof
    assume that
A1: <%>E in A |^ (m, n) and
A2: m <> 0 &( m > n or not <%>E in A);
    per cases by A2;
    suppose
      m <> 0 & m > n;
      hence contradiction by A1,Th21;
    end;
    suppose
A3:   m <> 0 & not <%>E in A;
      consider k such that
A4:   m <= k and
      k <= n and
A5:   <%>E in A |^ k by A1,Th19;
      k > 0 by A3,A4;
      hence contradiction by A3,A5,FLANG_1:31;
    end;
  end;
  assume
A6: m = 0 or m <= n & <%>E in A;
  per cases by A6;
  suppose
A7: m = 0;
    {<%>E} = A |^ 0 by FLANG_1:29;
    then
A8: {<%>E} c= A |^ (0, n) by Th20;
    <%>E in {<%>E} by TARSKI:def 1;
    hence thesis by A7,A8;
  end;
  suppose
    m <= n & <%>E in A;
    then <%>E in A |^ m & A |^ m c= A |^ (m, n) by Th20,FLANG_1:30;
    hence thesis;
  end;
end;
