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 Th29:
  A c= B implies A |^ (m, n) c= B |^ (m, n)
proof
  assume
A1: A c= B;
  thus thesis
  proof
    let x be object;
    assume x in A |^ (m, n);
    then consider k such that
A2: m <= k & k <= n & x in A |^ k by Th19;
    A |^ k c= B |^ k by A1,FLANG_1:37;
    hence thesis by A2,Th19;
  end;
end;
