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 Th22:
  A |^ (m, m) = A |^ m
proof
A1: now
    let x be object;
    assume x in A |^ (m, m);
    then ex k st m <= k & k <= m & x in A |^ k by Th19;
    hence x in A |^ m by XXREAL_0:1;
  end;
  for x being object holds x in A |^ m implies x in A |^ (m, m) by Th19;
  hence thesis by A1,TARSKI:2;
end;
