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;
reserve G for finite Graph,
  v for Vertex of G,
  c for Chain of G,
  vs for FinSequence of the carrier of G,
  X1, X2 for set;
reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p for Path of G,
  vs for FinSequence of the carrier of G,
  v9 for Vertex of AddNewEdge(v1, v2),
  p9 for Path of AddNewEdge(v1, v2),
  vs9 for FinSequence of the carrier of AddNewEdge(v1, v2);
reserve G for finite Graph,
  v, v1, v2 for Vertex of G,
  vs for FinSequence of the carrier of G,
  v9 for Vertex of AddNewEdge(v1, v2);
reserve G for Graph,
  v for Vertex of G,
  vs for FinSequence of the carrier of G;
reserve G for finite Graph,
  v for Vertex of G,
  vs for FinSequence of the carrier of G;

theorem Th54:
  for p being Element of X-PathSet v, Y being finite set st Y =
  the carrier' of G & Degree(v, X) <> 0 holds len p <= card Y
proof
  let p be Element of X-PathSet v, Y be finite set;
  assume that
A1: Y = the carrier' of G and
A2: Degree(v, X) <> 0;
A3: p in X-PathSet v;
  X-PathSet v = { c where c is Element of X* : c is Path of G & c is non
empty & ex vs being FinSequence of the carrier of G st vs is_vertex_seq_of c &
  vs.1 = v } by A2,Def11;
  then
  ex c being Element of X* st p = c & c is Path of G & c is non empty & ex
vs being FinSequence of the carrier of G st vs is_vertex_seq_of c & vs.1 = v
by A3;
  then
A4: card (rng p) = len p by FINSEQ_4:62;
  rng p c= Y by A1,FINSEQ_1:def 4;
  hence thesis by A4,NAT_1:43;
end;
