
theorem
  for M being PseudoMetricSpace, x,y being Element of M holds
  x-neighbour meets y-neighbour iff x tolerates y
proof
  let M be PseudoMetricSpace, x,y be Element of M;
  hereby
    assume x-neighbour meets y-neighbour;
    then consider p being object such that
A1: p in x-neighbour and
A2: p in y-neighbour by XBOOLE_0:3;
A3: ex q being Element of M st q=p & x tolerates q by A1;
    reconsider p as Element of M by A1;
    ex s being Element of M st s=p & y tolerates s by A2;
    hence x tolerates y by A3,Th1;
  end;
  assume x tolerates y; then
A4: x in y-neighbour;
  x in x-neighbour by Th4;
  hence thesis by A4,XBOOLE_0:3;
end;
