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
  m <= k & k <= n implies A |^ (m, n) = A |^ (m, k) \/ A |^ (k, n)
proof
A1: A |^ (m, n) c= A |^ (m, k) \/ A |^ (k, n)
  proof
    let x be object;
    assume x in A |^ (m, n);
    then consider l such that
A2: m <= l & l <= n & x in A |^ l by Th19;
    l <= k or l > k;
    then x in A |^ (m, k) or x in A |^ (k, n) by A2,Th19;
    hence thesis by XBOOLE_0:def 3;
  end;
  assume m <= k & k <= n;
  then A |^ (m, k) c= A |^ (m, n) & A |^ (k, n) c= A |^ (m, n) by Th23;
  then A |^ (m, k) \/ A |^ (k, n) c= A |^ (m, n) by XBOOLE_1:8;
  hence thesis by A1,XBOOLE_0:def 10;
end;
