reserve V for set;

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