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 Th34:
  <%>E in A & m <= n implies A |^ (m, n) = A |^ n
proof
  assume that
A1: <%>E in A and
A2: m <= n;
A3: A |^ (m, n) c= A |^ n
  proof
A4: now
      let k such that
A5:   k <= n;
      per cases by A5,XXREAL_0:1;
      suppose
        k < n;
        hence A |^ k c= A |^ n by A1,FLANG_1:36;
      end;
      suppose
        k = n;
        hence A |^ k c= A |^ n;
      end;
    end;
    let x be object;
    assume x in A |^ (m, n);
    then consider k such that
    m <= k and
A6: k <= n and
A7: x in A |^ k by Th19;
    A |^ k c= A |^ n by A4,A6;
    hence thesis by A7;
  end;
  A |^ n c= A |^ (m, n) by A2,Th20;
  hence thesis by A3,XBOOLE_0:def 10;
end;
