
theorem Th17:
  for G being WGraph, W being Walk of G, e being set st e in W
  .last().edgesInOut() holds W.addEdge(e).weightSeq() = W.weightSeq() ^ <* (
  the_Weight_of G).e *>
proof
  let G be WGraph, W be Walk of G, e be set;
  set W2 = W.addEdge(e), WA = G.walkOf(W.last(),e,W.last().adj(e));
  assume e in W.last().edgesInOut();
  then
A1: e Joins W.last(), W.last().adj(e), G by GLIB_000:67;
  then W2 = W.append(WA) & W.last() = WA.first() by GLIB_001:15,def 13;
  hence W2.weightSeq() = W.weightSeq() ^ WA.weightSeq() by Th16
    .= W.weightSeq() ^ <* (the_Weight_of G).e *> by A1,Th14;
end;
