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
  for C being Cycle of d,G holds C` is Cycle of d,G
proof
  let C be Cycle of d,G;
  consider d9 being Nat such that
A1: d = d9 + 1 by NAT_1:6;
  reconsider d9 as Element of NAT by ORDINAL1:def 12;
  reconsider G as Grating of d9 + 1 by A1;
  reconsider C as Cycle of (d9 + 1),G by A1;
  del C = 0_(d9,G);
  then del C` = 0_(d9,G) by Th59;
  hence thesis by A1,Th61;
end;
