
theorem Th13:
  for M being non empty MetrSpace,x being Element of M holds x -neighbour = {x}
proof
  let M be non empty MetrSpace,x be Element of M;
  for p being Element of M holds p in {x} implies p in x -neighbour
  proof
    let p be Element of M;
    assume p in {x};
    then p = x by TARSKI:def 1;
    hence thesis by Th12;
  end;
  then
A1: {x} c= x-neighbour by SUBSET_1:2;
  for p being Element of M holds p in x-neighbour implies p in {x}
  proof
    let p be Element of M;
    assume p in x-neighbour;
    then p = x by Th12;
    hence thesis by TARSKI:def 1;
  end;
  then x-neighbour c= {x} by SUBSET_1:2;
  hence thesis by A1,XBOOLE_0:def 10;
end;
