
theorem
  for G being real-weighted WGraph, v1,v2 being Vertex of G, e being set
  st e Joins v1,v2,G holds (G.walkOf(v1,e,v2)).cost() = (the_Weight_of G).e
proof
  let G be real-weighted WGraph, v1,v2 be Vertex of G, e be set;
  set W = G.walkOf(v1,e,v2);
   reconsider We = (the_Weight_of G).e as Element of REAL by XREAL_0:def 1;
  assume e Joins v1,v2,G;
  then W.weightSeq() = <* We *> by Th14;
  hence thesis by FINSOP_1:11;
end;
