
theorem Th7:
  for M being PseudoMetricSpace, x,y being Element of M holds y in
  x-neighbour implies x-neighbour = y-neighbour
proof
  let M be PseudoMetricSpace, x,y be Element of M;
  assume
A1: y in x-neighbour;
  for p being Element of M holds p in y-neighbour implies p in x -neighbour
  proof
    let p be Element of M;
    assume p in y-neighbour; then
A2: p tolerates y by Th2;
    y tolerates x by A1,Th2;
    then p tolerates x by A2,Th1;
    hence thesis;
  end;
  then
A3: y-neighbour c= x-neighbour by SUBSET_1:2;
  for p being Element of M holds p in x-neighbour implies p in y-neighbour
  proof
    let p be Element of M;
    assume p in x-neighbour; then
A4: p tolerates x by Th2;
    x tolerates y by A1,Th2;
    then p tolerates y by A4,Th1;
    hence thesis;
  end;
  then x-neighbour c= y-neighbour by SUBSET_1:2;
  hence thesis by A3,XBOOLE_0:def 10;
end;
