reserve V for set;

theorem Th17:
  for M being non empty MetrStruct holds V in M-neighbour iff V is
  equivalence_class of M
proof
  let M be non empty MetrStruct;
A1: V is equivalence_class of M implies V in M-neighbour
  proof
    assume V is equivalence_class of M;
    then ex x being Element of M st V=x-neighbour by Def3;
    hence thesis;
  end;
  V in M-neighbour implies V is equivalence_class of M
  proof
    assume V in M-neighbour;
    then ex x being Element of M st V=x-neighbour by Th15;
    hence thesis by Def3;
  end;
  hence thesis by A1;
end;
