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 Th21:
  A |^ (m, n) = {} iff m > n or m > 0 & A = {}
proof
A1: now
    assume
A2: m > 0 & A = {};
    now
      let x be object;
      assume x in A |^ (m, n);
      then ex k st m <= k & k <= n & x in A |^ k by Th19;
      hence contradiction by A2,FLANG_1:27;
    end;
    hence A |^ (m, n) = {} by XBOOLE_0:def 1;
  end;
  thus A |^ (m, n) = {} implies m > n or m > 0 & A = {}
  proof
    assume that
A3: A |^ (m, n) = {} and
A4: m <= n &( m <= 0 or A <> {});
A5: now
      assume that
A6:   m <= n and
A7:   A <> {};
      A |^ m <> {} by A7,FLANG_1:27;
      then ex a st a in A |^ m by SUBSET_1:4;
      hence contradiction by A3,A6,Th19;
    end;
    now
      assume that
A8:   m <= n and
A9:   m = 0;
      {<%>E} = A |^ m by A9,FLANG_1:29;
      then <%>E in A |^ m by TARSKI:def 1;
      hence contradiction by A3,A8,Th19;
    end;
    hence thesis by A4,A5;
  end;
  now
    assume
A10: m > n;
    now
      let x be object;
      not (ex k st m <= k & k <= n & x in A |^ k) by A10,XXREAL_0:2;
      hence not x in A |^ (m, n) by Th19;
    end;
    hence A |^ (m, n) = {} by XBOOLE_0:def 1;
  end;
  hence thesis by A1;
end;
