theorem Th50: :: CycVerDeg2
  for c being Path of G st c is non cyclic & vs is_vertex_seq_of c
  holds Degree(v, rng c) is even iff v <> vs.1 & v <> vs.len vs
proof
  let c be Path of G such that
A1: c is non cyclic and
A2: vs is_vertex_seq_of c;
  len vs = len c +1 by A2;
  then
A3: 1 <= len vs by NAT_1:11;
  then 1 in dom vs & len vs in dom vs by FINSEQ_3:25;
  then reconsider v1 = vs.1, v2 = vs.len vs as Vertex of G by FINSEQ_2:11;
A4: v1 <> v2 by A1,A2;
  set G9 = AddNewEdge(v1, v2);
  reconsider vs9 = vs as FinSequence of the carrier of G9 by Def7;
  reconsider c9 = c as Path of G9 by Th37;
A5: vs9 is_vertex_seq_of c9 by A2,Th36;
  reconsider v9 = v, v19 = v1, v29 = v2 as Vertex of G9 by Def7;
  set v219 = <*v29, v19*>;
  set vs9e = vs9^'<*v29, v19*>;
  len v219 = 2 by FINSEQ_1:44;
  then
A6: vs9e.len vs9e = v219.2 by FINSEQ_6:142;
  set E = the carrier' of G;
  set e = E;
A7: e in {E} by TARSKI:def 1;
  the carrier' of G9 = (the carrier' of G) \/ {E} by Def7;
  then e in the carrier' of G9 by A7,XBOOLE_0:def 3;
  then reconsider ep = <*e*> as Path of G9 by Th4;
A8: rng ep = {e} by FINSEQ_1:39;
A9: not e in e;
  then rng ep misses E by A8,ZFMISC_1:50;
  then
A10: rng ep /\ E = {};
  (the Source of G9).e = v19 & (the Target of G9).e = v29 by Th34;
  then
A11: vs9.len vs9 = <*v29,v19*>.1 & <*v29,v19*> is_vertex_seq_of ep by Th11;
A12: rng c c= the carrier' of G by FINSEQ_1:def 4;
  then
A13: not e in rng c by A9;
  rng c9 misses rng ep
  proof
    assume not thesis;
    then ex x being object st x in rng c9 & x in rng ep by XBOOLE_0:3;
    hence contradiction by A13,A8,TARSKI:def 1;
  end;
  then reconsider c9e = c9^ep as Path of G9 by A5,A11,Th6;
A14: vs9e is_vertex_seq_of c9e by A5,A11,GRAPH_2:44;
  vs9e.1 = vs.1 by A3,FINSEQ_6:140;
  then vs9e.1 = vs9e.len vs9e by A6;
  then reconsider c9e as cyclic Path of G9 by A14,MSSCYC_1:def 2;
  rng c9e = rng c \/ rng ep by FINSEQ_1:31;
  then
A15: e in rng c9e by A7,A8,XBOOLE_0:def 3;
A16: rng c9e /\ E = (rng c \/ rng ep) /\ E by FINSEQ_1:31
    .= (rng c /\ E) \/ {} by A10,XBOOLE_1:23
    .= rng c by A12,XBOOLE_1:28;
  then
A17: Degree(v, rng c9e) = Degree(v, rng c) by Th31;
  reconsider dv9 = Degree(v9, rng c9e) as even Element of NAT by Th49;
A18: Degree(v9, rng c9e) is even by Th49;
  per cases;
  suppose
    v <> v1 & v <> v2;
    hence thesis by A18,A17,Th48;
  end;
  suppose
A19: v = v1 or v = v2;
    then Degree(v9, rng c9e) = Degree(v, rng c9e) +1 by A4,A15,Th47;
    then Degree(v, rng c9e) = dv9 -1;
    hence thesis by A16,A19,Th31;
  end;
end;
