reserve V for set;

theorem Th21:
  for M being PseudoMetricSpace, V,Q being Element of M-neighbour,
  v being Real holds V,Q is_dst v iff ex p,q being Element of M st p
  in V & q in Q & dist(p,q)=v
proof
  let M be PseudoMetricSpace, V,Q be Element of M-neighbour,
      v be Real;
A1: V,Q is_dst v implies ex p,q being Element of M st p in V & q in Q & dist
  (p,q)=v
  proof
    consider q being Element of M such that
A2: Q=q-neighbour by Th15;
A3: q in Q by A2,Th4;
    assume
A4: V,Q is_dst v;
    consider p being Element of M such that
A5: V=p-neighbour by Th15;
    p in V by A5,Th4;
    then dist(p,q)=v by A4,A3;
    hence thesis by A5,A3,Th4;
  end;
  (ex p,q being Element of M st (p in V & q in Q & dist(p,q)=v)) implies V
  ,Q is_dst v
  by Th20;
  hence thesis by A1;
end;
