reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p, p1, p2 for Path of G,
  vs, vs1, vs2 for FinSequence of the carrier of G,
  e, X for set,
  n, m for Nat;

theorem Th10:
  for p being cyclic Path of G st n in dom p ex p9 being cyclic
  Path of G st p9.1 = p.n & len p9 = len p & rng p9 = rng p
proof
  let p be cyclic Path of G;
  assume
A1: n in dom p;
  then
A2: 1 <= n by FINSEQ_3:25;
  per cases by A2,XXREAL_0:1;
  suppose
A3: n = 1;
    take p;
    thus thesis by A3;
  end;
  suppose
    1 < n;
    then 1+1 <= n by NAT_1:13;
    then consider n1 being Nat such that
    1<=n1 and
    n1<n and
A4: n=n1+1 by FINSEQ_6:127;
    reconsider r = ((n, len p)-cutp)^((1,n1)-cut p) as cyclic Path of G by A4
,Th8;
    take r;
    thus thesis by A1,A4,Th9;
  end;
end;
