reserve V for set;

theorem
  for M being PseudoMetricSpace holds set_in_rel M c= [:elem_in_rel_1 M,
  elem_in_rel_2 M,real_in_rel M:]
proof
  let M be PseudoMetricSpace;
  for VQv being Element of [:M-neighbour,M-neighbour ,REAL:] holds (VQv in
set_in_rel M implies VQv in [:elem_in_rel_1 M,elem_in_rel_2 M,real_in_rel M:])
  proof
    let VQv be Element of [:M-neighbour,M-neighbour,REAL:];
    assume VQv in set_in_rel M;
    then consider
    V,Q being Element of M-neighbour ,v being Element of REAL such that
A1: VQv = [V,Q,v] and
A2: V,Q is_dst v by Th29;
A3: v in real_in_rel M by A2;
    V in elem_in_rel_1 M & Q in elem_in_rel_2 M by A2;
    hence thesis by A1,A3,MCART_1:69;
  end;
  hence thesis by SUBSET_1:2;
end;
