reserve V for set;

theorem Th29:
  for M being PseudoMetricSpace,VQv being Element of [:M-neighbour
  ,M -neighbour,REAL:] holds VQv in set_in_rel M iff ex V,Q being Element of M
  -neighbour,v being Element of REAL st VQv = [V,Q,v] & V,Q is_dst v
proof
  let M be PseudoMetricSpace,VQv be Element of [:M-neighbour,M-neighbour ,REAL
  :];
  VQv in set_in_rel M implies ex V,Q being Element of M-neighbour,v being
  Element of REAL st VQv = [V,Q,v] & V,Q is_dst v
  proof
    assume VQv in set_in_rel M;
    then
    ex S being Element of [:M-neighbour,M-neighbour,REAL:] st VQv = S & ex
V,Q being Element of M-neighbour,v being Element of REAL st S = [V,Q,v] & V,Q
    is_dst v;
    hence thesis;
  end;
  hence thesis;
end;
