reserve V for set;

theorem
  for M being non empty MetrSpace holds V in M-neighbour iff ex x being
  Element of M st V={x}
proof
  let M be non empty MetrSpace;
A1: V in M-neighbour implies ex x being Element of M st V={x}
  proof
    assume V in M-neighbour;
    then consider x being Element of M such that
A2: V=x -neighbour by Th15;
    x-neighbour = {x} by Th13;
    hence thesis by A2;
  end;
  (ex x being Element of M st V={x}) implies V in M-neighbour
  proof
    given x being Element of M such that
A3: V={x};
    x-neighbour = {x} by Th13;
    hence thesis by A3;
  end;
  hence thesis by A1;
end;
