theorem Th22:
  (ex i st r.i < l.i) implies
  (x in cell(l,r) iff ex i st r.i < l.i & (x.i <= r.i or l.i <= x.i))
proof
  given i0 such that
A1: r.i0 < l.i0;
  x.i0 < l.i0 or r.i0 < x.i0 by A1,XXREAL_0:2;
  hence
  x in cell(l,r) implies ex i st r.i < l.i & (x.i <= r.i or l.i <= x.i)
  by Th20;
  thus thesis;
end;
