
theorem
  for G being _Graph, W being Walk of G st
    len W = 5 & not W.first(),W.last() are_adjacent holds W is chordless
proof
  let G be _Graph, W be Walk of G such that
A1: len W = 5 and
A2: not W.first(),W.last() are_adjacent;
  assume W is chordal;
  then consider m,n being odd Nat such that
A3: m+2 < n and
A4: n <= len W and
  W.m <> W.n and
A5: ex e being object st e Joins W.m,W.n,G and
  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 Th83;
A6: now
    assume
A7: m <> 1;
    1 <= m by Th2;
    then 2*0+1 < m by A7,XXREAL_0:1;
    then 1+2 <= m by Th4;
    then 3+2 <= m+2 by XREAL_1:7;
    hence contradiction by A1,A3,A4,XXREAL_0:2;
  end;
  then 3+2 <= n by A3,Th4;
  then W.n = W.last() by A1,A4,XXREAL_0:1;
  hence contradiction by A2,A5,A6;
end;
