theorem Th57:
  for G being finite connected Graph, c being Element of G
  -CycleSet st rng c <> the carrier' of G & c is non empty & for v being Vertex
  of G holds Degree v is even holds ExtendCycle c is non empty & card rng c <
  card rng ExtendCycle c
proof
  let G be finite connected Graph, c be Element of G-CycleSet;
  set E = the carrier' of G;
  reconsider E9 = E as finite set by GRAPH_1:def 11;
  reconsider ccp = c as cyclic Path of G by Def8;
  assume that
A1: rng c <> the carrier' of G and
A2: c is non empty and
A3: for v being Vertex of G holds Degree v is even;
A4: rng c c= E by FINSEQ_1:def 4;
  then rng c c< the carrier' of G by A1;
  then ex xx being object st xx in E & not xx in rng c by XBOOLE_0:6;
  then reconsider Erc = E9 \ rng c as finite non empty set by XBOOLE_0:def 5;
  reconsider X = {v9 where v9 is Vertex of G : v9 in G-VSet rng c & Degree v9
  <> Degree(v9, rng c)} as non empty set by A1,A2,Th56;
  consider c9 being Element of G-CycleSet, v being Vertex of G such that
A5: v = the Element of X and
A6: c9 = the Element of Erc-CycleSet v and
A7: ExtendCycle c = CatCycles(c, c9) by A1,A2,Def13;
  v in X by A5;
  then
A8: ex v9 being Vertex of G st v = v9 & v9 in G-VSet rng c & Degree v9 <>
  Degree(v9, rng c);
A9: now
    let v be Vertex of G;
A10: Degree v = Degree(v, E) & Degree v is even by A3,Th24;
    Degree(v, Erc) = Degree(v, E9) - Degree(v, rng c) & Degree(v, rng ccp
    ) is even by A4,Th29,Th49;
    hence Degree(v, Erc) is even by A10;
  end;
  rng c misses (E\rng c) by XBOOLE_1:79;
  then
A11: rng c /\ (E\rng c) = {};
  Degree(v, Erc) = Degree(v, E9) - Degree(v, rng c) by A4,Th29;
  then
A12: Degree(v, Erc) <> 0 by A8,Th24;
  then rng c9 c= E \ rng c by A6,A9,Th55;
  then rng c /\ rng c9 = rng c /\ ((E\rng c) /\ rng c9) by XBOOLE_1:28
    .= {} /\ rng c9 by A11,XBOOLE_1:16
    .= {};
  then
A13: rng c misses rng c9;
  v in G-VSet rng c9 by A6,A9,A12,Th55;
  then
A14: G-VSet rng c meets G-VSet rng c9 by A8,XBOOLE_0:3;
  hence ExtendCycle c is non empty by A2,A7,A13,Th53;
  consider vr being Vertex of G such that
A15: vr = the Element of (G-VSet rng c) /\ (G-VSet rng c9) and
A16: CatCycles(c, c9) = Rotate(c, vr)^Rotate(c9, vr) by A14,A13,Def10;
A17: (G-VSet rng c) /\ (G-VSet rng c9) <> {} by A14;
  then vr in G-VSet rng c9 by A15,XBOOLE_0:def 4;
  then
A18: rng Rotate(c9, vr) = rng c9 by Lm5;
  vr in G-VSet rng c by A17,A15,XBOOLE_0:def 4;
  then rng Rotate(c, vr) = rng c by Lm5;
  then rng ExtendCycle c = rng c \/ rng c9 by A7,A16,A18,FINSEQ_1:31;
  then
A19: card rng ExtendCycle c = card rng c + card rng c9 by A13,CARD_2:40;
  c9 is non empty by A6,A9,A12,Th55;
  hence thesis by A19,XREAL_1:29;
end;
