reserve V for set;

theorem Th20:
  for M being PseudoMetricSpace, V,Q being Element of M-neighbour
for p1,p2,q1,q2 being Element of M holds ( p1 in V & q1 in Q & p2 in V & q2 in
  Q implies dist(p1,q1)=dist(p2,q2))
proof
  let M be PseudoMetricSpace, V,Q be Element of M-neighbour;
  let p1,p2,q1,q2 be Element of M;
  assume that
A1: p1 in V and
A2: q1 in Q and
A3: p2 in V and
A4: q2 in Q;
  V is equivalence_class of M by Th17;
  then ex x being Element of M st V=x-neighbour by Def3;
  then
A5: dist(p1,p2)=0 by A1,A3,Th10;
  Q is equivalence_class of M by Th17;
  then ex y being Element of M st Q=y-neighbour by Def3;
  then
A6: dist(q1,q2)=0 by A2,A4,Th10;
  dist(p2,q2) <= dist(p2,p1) + dist(p1,q2) & dist(p1,q2) <= dist(p1,q1) +
  dist (q1,q2) by METRIC_1:4;
  then
A7: dist(p2,q2) <= dist(p1,q1) by A5,A6,XXREAL_0:2;
  dist(p1,q1) <= dist(p1,p2) + dist(p2,q1) & dist(p2,q1) <= dist(p2,q2) +
  dist (q2,q1) by METRIC_1:4;
  then dist(p1,q1) <= dist(p2,q2) by A5,A6,XXREAL_0:2;
  hence thesis by A7,XXREAL_0:1;
end;
