theorem Th55:
  Degree(v, X) <> 0 & (for v holds Degree(v, X) is even) implies
for c being Element of X-CycleSet v holds c is non empty & rng c c= X & v in G
  -VSet rng c
proof
  assume Degree(v, X) <> 0 & for v holds Degree(v, X) is even;
  then
A1: X-CycleSet v = { c9 where c9 is Element of G-CycleSet : rng c9 c= X & c9
  is non empty & ex vs being FinSequence of the carrier of G st vs
  is_vertex_seq_of c9 & vs.1 = v} by Def12;
  let c be Element of X-CycleSet v;
  c in X-CycleSet v;
  then consider c9 being Element of G-CycleSet such that
A2: c = c9 and
A3: rng c9 c= X and
A4: c9 is non empty and
A5: ex vs being FinSequence of the carrier of G st vs is_vertex_seq_of
  c9 & vs.1 = v by A1;
  thus c is non empty by A2,A4;
  thus rng c c= X by A2,A3;
  reconsider c9 as Path of G by Def8;
  consider vs being FinSequence of the carrier of G such that
A6: vs is_vertex_seq_of c9 and
A7: vs.1 = v by A5;
  len vs = len c9 +1 by A6;
  then 1 <= len vs by NAT_1:11;
  then
A8: 1 in dom vs by FINSEQ_3:25;
  G-VSet rng c9 = rng vs by A4,A6,GRAPH_2:31;
  hence thesis by A2,A7,A8,FUNCT_1:def 3;
end;
