reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;

theorem Th9:
  for f being FinSequence of X st len f>=1 holds ex g being
  FinSequence of X st g is_Shortcut_of f
proof
  let f be FinSequence of X;
  reconsider f1=f as FinSequence of the carrier of PGraph(X);
  assume len f>=1;
  then consider fc being Subset of PairF(f), fvs being Subset of f1, sc being
  oriented simple Chain of PGraph(X), vs1 being FinSequence of the carrier of
  PGraph(X) such that
A1: Seq fc = sc & Seq fvs = vs1 & vs1 is_oriented_vertex_seq_of sc & f1.
  1 = vs1.1 & f1.len f1 = vs1.len vs1 by Th7,GRAPH_4:21;
  reconsider g1=vs1 as FinSequence of X;
  g1 is_Shortcut_of f by A1;
  hence thesis;
end;
