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 Th30:
  k <= d implies (cell(l,r) in cells(k,G) iff
  ((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
A1: k <= d;
  hereby
    assume cell(l,r) in cells(k,G);
    then consider l9,r9 such that
A2: cell(l,r) = cell(l9,r9) and
A3: (ex X being Subset of Seg d st card X = k & for i holds (i in X &
l9.i < r9.i & [l9.i,r9.i] is Gap of G.i) or (not i in X & l9.i = r9.i & l9.i in
    G.i)) or (k = d & for i holds r9.i < l9.i & [l9.i,r9.i] is Gap of G.i)
    by A1,Th29;
    l = l9 & r = r9
    proof
      per cases by A3;
      suppose ex X being Subset of Seg d st card X = k &
        for i holds i in X & l9.i < r9.i & [l9.i,r9.i] is Gap of G.i or
        not i in X & l9.i = r9.i & l9.i in G.i;
        then for i holds l9.i <= r9.i;
        hence thesis by A2,Th28;
      end;
      suppose for i holds r9.i < l9.i & [l9.i,r9.i] is Gap of G.i;
        hence thesis by A2,Th28;
      end;
    end;
    hence (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 A3;
  end;
  thus thesis by A1,Th29;
end;
