reserve V for set;

theorem Th36:
  for M being PseudoMetricSpace, V,Q,W being Element of M
-neighbour holds (nbourdist M).(V,W) <= (nbourdist M).(V,Q) + (nbourdist M).(Q,
  W)
proof
  let M be PseudoMetricSpace,V,Q,W be Element of M-neighbour;
  consider p being Element of M such that
A1: V=p-neighbour by Th15;
  consider w being Element of M such that
A2: W=w-neighbour by Th15;
A3: w in W by A2,Th4;
  consider q being Element of M such that
A4: Q=q-neighbour by Th15;
A5: q in Q by A4,Th4; then
A6: (nbourdist M).(Q,W) = dist(q,w) by A3,Def13;
A7: p in V by A1,Th4; then
A8: (nbourdist M).(V,W) = dist(p,w) by A3,Def13;
  (nbourdist M).(V,Q) = dist(p,q) by A7,A5,Def13;
  hence thesis by A8,A6,METRIC_1:4;
end;
