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
  W is non trivial implies for v being Vertex of G st v in W.vertices()
  holds not v is isolated
proof
  assume W is non trivial;
  then
A1: len W <> 1 by Lm55;
  let v be Vertex of G;
  assume v in W.vertices();
  then consider n being odd Element of NAT such that
A2: n <= len W and
A3: W.n = v by Lm45;
  now
    per cases;
    suppose
A4:   n = len W;
      1 <= len W by ABIAN:12;
      then 1 < len W by A1,XXREAL_0:1;
      then 1+1 <= len W by NAT_1:13;
      then reconsider lenW2 = len W - 2*1 as odd Element of NAT by INT_1:5;
      lenW2 < len W - 0 by XREAL_1:15;
      then W.(lenW2+1) Joins W.lenW2,W.(lenW2+2),G by Def3;
      then W.(lenW2+1) Joins v,W.lenW2,G by A3,A4,GLIB_000:14;
      hence ex e being set st e in v.edgesInOut() by GLIB_000:62;
    end;
    suppose
      n <> len W;
      then n < len W by A2,XXREAL_0:1;
      then W.(n+1) Joins v, W.(n+2), G by A3,Def3;
      hence ex e being set st e in v.edgesInOut() by GLIB_000:62;
    end;
  end;
  hence thesis by GLIB_000:def 49;
end;
