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 Th39:
  for m,n being odd Element of NAT st m <= n & n < len W holds W
  .cut(m,n).addEdge(W.(n+1)) = W.cut(m,n+2)
proof
  let m,n be odd Element of NAT;
  set W1 = W.cut(m,n);
  set e = W.(n+1);
  assume that
A1: m <= n and
A2: n < len W;
A3: n+2 <= len W by A2,Th1;
A4: W1.last() = W.n by A1,A2,Lm16;
  then e Joins W1.last(), W.(n+2), G by A2,Def3;
  then e Joins W1.last(), W.vertexAt(n+2), G by A3,Def8;
  then W1.last().adj(e) = W.vertexAt(n+2) by GLIB_000:66;
  then W1.last().adj(e) = W.(n+2) by A3,Def8;
  then
A5: G.walkOf(W1.last(),e,W1.last().adj(e)) = W.cut(n,n+2) by A2,A4,Th38;
  n <= n+2 by Th1;
  hence thesis by A1,A3,A5,Lm17;
end;
