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 Th23:
  m <= k & l <= n implies A |^ (k, l) c= A |^ (m, n)
proof
  assume
A1: m <= k & l <= n;
  thus thesis
  proof
    let x be object;
    assume x in A |^ (k, l);
    then consider kl such that
A2: k <= kl & kl <= l and
A3: x in A |^ kl by Th19;
    m <= kl & kl <= n by A1,A2,XXREAL_0:2;
    hence thesis by A3,Th19;
  end;
end;
