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 Th61:
  for C being Chain of (k + 1),G holds
  C is Cycle of (k + 1),G iff del C = 0_(k,G)
proof
  let C be Chain of (k + 1),G;
  hereby
    assume C is Cycle of (k + 1),G;
    then ex k9 st ( k + 1 = k9 + 1)&( ex C9 being Chain of (k9 + 1),
    G st C9 = C & del C9 = 0_(k9,G)) by Def14;
    hence del C = 0_(k,G);
  end;
  thus thesis by Def14;
end;
