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 Th63:
  for C being Chain of 0,G holds C is Cycle of 0,G iff card C is even
proof
  let C be Chain of 0,G;
  hereby
    assume C is Cycle of 0,G;
    then 0 = 0 & card C is even or
    ex k9 st 0 = k9 + 1 & ex C9 being Chain of (k9 + 1),G st C9 = C &
    del C9 = 0_(k9,G) by Def14;
    hence card C is even;
  end;
  thus thesis by Def14;
end;
