reserve X,x,y,z for set;
reserve n,m,k,k9,d9 for Nat;
reserve d for non zero Nat;
reserve i,i0,i1 for Element of Seg d;
reserve l,r,l9,r9,l99,r99,x,x9,l1,r1,l2,r2 for Element of REAL d;
reserve Gi for non trivial finite Subset of REAL;
reserve li,ri,li9,ri9,xi,xi9 for Real;
reserve G for Grating of d;

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;
