reserve V for set;

theorem
  for M being PseudoMetricSpace, v being Element of REAL holds v in
  real_in_rel M iff ex V,Q being Element of M-neighbour st V,Q is_dst v
proof
  let M be PseudoMetricSpace, v be Element of REAL;
  v in real_in_rel M implies ex V,Q being Element of M-neighbour st V,Q
  is_dst v
  proof
    assume v in real_in_rel M;
    then ex r be Element of REAL st v=r & ex V,Q being Element of M-neighbour
    st V,Q is_dst r;
    hence thesis;
  end;
  hence thesis;
end;
