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;
