
theorem
  for G being _Graph, W being Walk of G st
    W is Cycle-like & W is chordal & W.length()=4
  ex e being object st e Joins W.1,W.5,G or e Joins W.3,W.7,G
proof
  let G be _Graph, W be Walk of G such that
A1: W is Cycle-like and
A2: W is chordal and
A3: W.length() = 4;
A4: len W = 2*4+1 by A3,GLIB_001:112;
  consider m, n being odd Nat such that
A5: m+2 < n and
A6: n <= len W and
  W.m <> W.n and
A7: ex e being object st e Joins W.m,W.n,G and
A8: W is Cycle-like implies not (m=1 & n = len W) & not (m=1 & n = len
  W-2) & not (m=3 & n = len W) by A2,Th83;
  consider e being object such that
A9: e Joins W.m,W.n,G by A7;
  assume
A10: not(ex e being object st e Joins W.1,W.5,G or e Joins W.3,W.7,G);
A11: now
    assume
A12: m = 3;
    then n < len W by A1,A6,A8,XXREAL_0:1;
    then
A13: n <= 9 - 2 by A4,Th3;
    n <> 7 by A10,A9,A12;
    then n < 2*3+1 by A13,XXREAL_0:1;
    then n <= 7 - 2 by Th3;
    hence contradiction by A5,A12;
  end;
A14: now
    reconsider jj=2*3+1 as odd Nat;
    assume
A15: m = 1;
    then n < len W by A1,A6,A8,XXREAL_0:1;
    then n <= 9 - 2 by A4,Th3;
    then n < jj by A1,A4,A8,A15,XXREAL_0:1;
    then
A16: n <= jj-2 by Th3;
    n <> 5 by A10,A9,A15;
    then n < 2*2+1 by A16,XXREAL_0:1;
    then n <= 5-2 by Th3;
    hence contradiction by A5,A15;
  end;
A17: W.first() = W.last() by A1,GLIB_001:def 24;
A18: now
    assume
A19: m = 5;
    n <> 9 by A17,A4,A9,A19,GLIB_000:14,A10;
    then n < len W by A4,A6,XXREAL_0:1;
    then n <= len W - 2 by Th3;
    hence contradiction by A4,A5,A19;
  end;
  0+1 <= m by ABIAN:12;
  then 2*0+1 < m by A14,XXREAL_0:1;
  then 1+2 <= m by Th4;
  then 2*1+1 < m by A11,XXREAL_0:1;
  then 3+2 <= m by Th4;
  then 2*2+1 < m by A18,XXREAL_0:1;
  then 5+2 <= m by Th4;
  then 7+2 <= m + 2 by XREAL_1:7;
  hence contradiction by A4,A5,A6,XXREAL_0:2;
end;
