
theorem
  for a, b, c, d being Point of I[01] holds [:[.a,b.], [.c,d.]:] is
  compact Subset of [:I[01], I[01]:]
proof
  let a, b, c, d be Point of I[01];
  per cases;
  suppose
    a <= b & c <= d;
    hence thesis by Th9;
  end;
  suppose
    a > b & c <= d;
    then reconsider A = [.a,b.] as empty Subset of I[01] by Th31;
    [:A, [.c,d.]:] = {};
    hence thesis by Th30;
  end;
  suppose
    a > b & c > d;
    then reconsider A = [.c,d.] as empty Subset of I[01] by Th31;
    [:[.a,b.], A:] = {};
    hence thesis by Th30;
  end;
  suppose
    a <= b & c > d;
    then reconsider A = [.c,d.] as empty Subset of I[01] by Th31;
    [:[.a,b.], A:] = {};
    hence thesis by Th30;
  end;
end;
