reserve V for set;

theorem
  for M being PseudoMetricSpace holds elem_in_rel_1 M = elem_in_rel_2 M
proof
  let M be PseudoMetricSpace;
  for V being Element of M-neighbour holds (V in elem_in_rel_2 M implies V
  in elem_in_rel_1 M)
  proof
    let V be Element of M-neighbour;
    assume V in elem_in_rel_2 M;
    then consider Q being Element of M-neighbour,
        v being Element of REAL such that
A1: Q,V is_dst v by Th27;
    V,Q is_dst v by A1,Th22;
    hence thesis;
  end;
  then
A2: elem_in_rel_2 M c= elem_in_rel_1 M by SUBSET_1:2;
  for V being Element of M-neighbour holds (V in elem_in_rel_1 M implies V
  in elem_in_rel_2 M)
  proof
    let V be Element of M-neighbour;
    assume V in elem_in_rel_1 M;
    then consider
    Q being Element of M-neighbour, v being Element of REAL such that
A3: V,Q is_dst v by Th26;
    Q,V is_dst v by A3,Th22;
    hence thesis;
  end;
  then elem_in_rel_1 M c= elem_in_rel_2 M by SUBSET_1:2;
  hence thesis by A2,XBOOLE_0:def 10;
end;
