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 Th34:
  for A being Subset of REAL d holds
  A in cells(0,G) iff ex x st A = cell(x,x) & for i holds x.i in G.i
proof
  let A be Subset of REAL d;
  hereby
    assume A in cells(0,G);
    then consider l,r such that
A1: A = cell(l,r) and
A2: (ex X being Subset of Seg d st card X = 0 & 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 (0 = d & for i holds r.i < l.i & [l.i,r.i] is Gap of G.i)
    by Th29;
    consider X being Subset of Seg d such that
A3: card X = 0 and
A4: 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
    by A2;
    reconsider l9 = l, r9 = r as Function of Seg d,REAL by Def3;
    X = {} by A3;
    then
A5: for i holds l9.i = r9.i & l.i in G.i by A4;
    then l9 = r9 by FUNCT_2:63;
    hence ex x st A = cell(x,x) & for i holds x.i in G.i by A1,A5;
  end;
  given x such that
A6: A = cell(x,x) and
A7: for i holds x.i in G.i;
  ex X being Subset of Seg d st card X = 0 &
  for i holds i in X & x.i < x.i & [x.i,x.i] is Gap of G.i or
  not i in X & x.i = x.i & x.i in G.i
  proof
    reconsider X = {} as Subset of Seg d by XBOOLE_1:2;
    take X;
    thus thesis by A7;
  end;
  hence thesis by A6,Th29;
end;
