theorem Th31:
  k <= d & cell(l,r) in cells(k,G) implies
  (for i holds (l.i < r.i & [l.i,r.i] is Gap of G.i) or
  (l.i = r.i & l.i in G.i)) or
  for i holds r.i < l.i & [l.i,r.i] is Gap of G.i
proof
  assume that
A1: k <= d and
A2: cell(l,r) in cells(k,G);
  per cases by A1,A2,Th30;
  suppose ex X being Subset of Seg d st card X = k &
    for i holds i in X & l.i < r.i & [l.i,r.i] is Gap of G.i or
    not i in X & l.i = r.i & l.i in G.i;
    hence thesis;
  end;
  suppose k = d & for i holds r.i < l.i & [l.i,r.i] is Gap of G.i;
    hence thesis;
  end;
end;
