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 Th56:
  del 0_(k + 1,G) = 0_(k,G)
proof
  now
    let A be Cell of k,G;
    card(star A /\ 0_(k + 1,G)) = 2* 0;
    hence A in del 0_(k + 1,G) iff A in 0_(k,G) by Th48;
  end;
  hence thesis by SUBSET_1:3;
end;
