
theorem Th45: :: PathLike05
  for G being _Graph, W being Walk of G st W.first() <> W.last() &
  not W.first(),W.last() are_adjacent holds W.length() >= 2
proof
  let G be _Graph, W be Walk of G such that
A1: W.first() <> W.last() and
A2: not W.first(),W.last() are_adjacent;
  set l = W.length();
  assume l < 2;
  then l < 1+1;
  then
A3: l <= 1 by NAT_1:13;
  per cases by A3,NAT_1:25;
  suppose
    l = 0;
    then W is trivial;
    hence contradiction by A1,GLIB_001:127;
  end;
  suppose
    l = 1;
    then
A4: len W = 2*1+1 by GLIB_001:112
      .= 3;
    1 = 2*0+1;
    then W.(1+1) Joins W.1,W.(1+2),G by A4,GLIB_001:def 3;
    hence contradiction by A2,A4;
  end;
end;
