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 Th8:
  for f,g being FinSequence of X st g is_Shortcut_of f holds 1<=
  len g & len g <= len f
proof
  let f,g be FinSequence of X;
  reconsider df=dom f as finite set;
A1: card df=card (Seg len f) by FINSEQ_1:def 3
    .=len f by FINSEQ_1:57;
  assume g is_Shortcut_of f;
  then consider fc being Subset of PairF(f),fvs being Subset of f, sc being
  oriented simple Chain of PGraph(X), g1 being FinSequence of the carrier of
  PGraph(X) such that
  Seq fc = sc and
A2: Seq fvs = g and
A3: g1=g and
A4: g1 is_oriented_vertex_seq_of sc;
A5: len g1 = len sc + 1 by A4,GRAPH_4:def 5;
  reconsider dfvs=dom fvs as finite set;
A6: rng Sgm(dom fvs) c= dom fvs by FINSEQ_1:50;
A7: dom fvs c= dom f by RELAT_1:11;
  g=fvs*Sgm(dom fvs) by A2,FINSEQ_1:def 15;
  then dom g=dom (Sgm(dom fvs)) by A6,RELAT_1:27;
  then len g=len Sgm(dom fvs) by FINSEQ_3:29
    .=card dfvs by FINSEQ_3:39;
  hence thesis by A3,A5,A7,A1,NAT_1:12,43;
end;
