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 Th131:
  W is non trivial implies ex lenW2 being odd Element of NAT st
  lenW2 = len W - 2 & W.cut(1,lenW2).addEdge(W.(lenW2+1)) = W
proof
  set lenW2 = len W - 2*1;
  assume W is non trivial;
  then len W >= 3 by Lm54;
  then reconsider lenW2 as odd Element of NAT by INT_1:5,XXREAL_0:2;
  set W1 = W.cut(1,lenW2), e = W.(lenW2+1);
  take lenW2;
  thus lenW2 = len W - 2;
  lenW2 < len W - 0 by XREAL_1:15;
  hence W1.addEdge(e) = W.cut(1,lenW2+2) by Th39,ABIAN:12,JORDAN12:2
    .= W by Lm18;
end;
