theorem Th29:
  k <= d implies for A being Subset of REAL d holds A in cells(k,G) iff
  ex l,r st A = cell(l,r) & ((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)) or
  (k = d & for i holds r.i < l.i & [l.i,r.i] is Gap of G.i))
proof
  assume k <= d;
  then cells(k,G) = { cell(l,r) : ((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)) or
  (k = d & for i holds r.i < l.i & [l.i,r.i] is Gap of G.i)) } by Def7;
  hence thesis;
end;
