reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem
  for W being Path of G st W is open holds for m, n being odd Element of
  NAT st m < n & n <= len W holds W.m <> W.n
proof
  let W be Path of G;
  assume
A1: W is open;
  let m, n be odd Element of NAT;
  assume that
A2: m < n and
A3: n <= len W;
  now
    assume
A4: W.m = W.n;
    then
A5: n = len W by A2,A3,Def28;
    m = 1 by A2,A3,A4,Def28;
    hence contradiction by A1,A4,A5;
  end;
  hence thesis;
end;
