
theorem Th1:
  for M being PseudoMetricSpace, x,y,z being Element of M holds x
  tolerates y & y tolerates z implies x tolerates z
proof
  let M be PseudoMetricSpace, x,y,z be Element of M;
  assume x tolerates y & y tolerates z;
  then dist(x,y)=0 & dist(y,z)=0;
  then dist(x,z) <= 0 + 0 by METRIC_1:4;
  then dist(x,z) = 0 by METRIC_1:5;
  hence thesis;
end;
