
theorem Th28:
  for G being nonnegative-weighted WGraph, W being Walk of G, n
  being Element of NAT st n in dom W.weightSeq() holds 0 <= (W.weightSeq()).n
proof
  let G be nonnegative-weighted WGraph, W be Walk of G, n be Element of NAT;
  set WS = W.weightSeq();
  assume
A1: n in dom W.weightSeq();
  then
A2: 1 <= n by FINSEQ_3:25;
A3: n <= len WS by A1,FINSEQ_3:25;
  then n <= len W.edgeSeq() by Def18;
  then dom the_Weight_of G = the_Edges_of G & n in dom W.edgeSeq() by A2,
FINSEQ_3:25,PARTFUN1:def 2;
  then
A4: (W.edgeSeq().n) in dom the_Weight_of G by GLIB_001:79;
  WS.n = (the_Weight_of G).(W.edgeSeq().n) by A2,A3,Def18;
  then
A5: WS.n in rng (the_Weight_of G) by A4,FUNCT_1:def 3;
  rng (the_Weight_of G) c= Real>=0 by Def14;
  then WS.n in Real>=0 by A5;
  then ex r being Real st r = WS.n & r >= 0 by GRAPH_5:def 12;
  hence thesis;
end;
