
theorem
  for G being nonnegative-weighted WGraph, e be set holds e in
  the_Edges_of G implies 0 <= (the_Weight_of G).e
proof
  let G be nonnegative-weighted WGraph, e be set;
  assume e in the_Edges_of G;
  then e in dom the_Weight_of G by PARTFUN1:def 2;
  then
A1: (the_Weight_of G).e in rng (the_Weight_of G) by FUNCT_1:3;
  rng (the_Weight_of G) c= Real>=0 by Def14;
  then (the_Weight_of G).e in Real>=0 by A1;
  then ex r being Real st (the_Weight_of G).e = r & r >= 0 by GRAPH_5:def 12;
  hence thesis;
end;
