reserve V for set;

theorem Th34:
  for M being PseudoMetricSpace, V,Q being Element of M-neighbour
  holds (nbourdist M).(V,Q) = 0 iff V = Q
proof
  let M be PseudoMetricSpace, V,Q be Element of M-neighbour;
A1: V=Q implies (nbourdist M).(V,Q) = 0
  proof
    consider p being Element of M such that
A2: V=p-neighbour by Th15;
A3: p in V by A2,Th4;
    consider q being Element of M such that
A4: Q=q-neighbour by Th15;
    assume V = Q;
    then
A5: p tolerates q by A3,A4,Th2;
    q in Q by A4,Th4;
    then (nbourdist M).(V,Q) = dist(p,q) by A3,Def13
      .= 0 by A5;
    hence thesis;
  end;
  (nbourdist M).(V,Q) = 0 implies V = Q
  proof
    assume
A6: (nbourdist M).(V,Q) = 0;
    consider p being Element of M such that
A7: V=p-neighbour by Th15;
    consider q being Element of M such that
A8: Q=q-neighbour by Th15;
A9: q in Q by A8,Th4;
    p in V by A7,Th4;
    then dist(p,q) = 0 by A6,A9,Def13;
    then p tolerates q;
    hence thesis by A7,A8,Th8;
  end;
  hence thesis by A1;
end;
