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 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;
