
theorem
  for G being real-weighted WGraph, W be Walk of G, e be set st e in W
.last().edgesInOut() holds W.addEdge(e).cost() = W.cost() + (the_Weight_of G).e
proof
  let G be real-weighted WGraph, W be Walk of G, e be set;
  set W2 = W.addEdge(e);
   reconsider We = (the_Weight_of G). e as Element of REAL by XREAL_0:def 1;
  assume e in W.last().edgesInOut();
  then W2.weightSeq()=W.weightSeq() ^ <*(the_Weight_of G).e*> by Th17;
  then
  Sum (W2.weightSeq()) = Sum (W.weightSeq()) + Sum <*We*> by RVSUM_1:75;
  hence thesis by FINSOP_1:11;
end;
