theorem Th9:
  for f be Function of [:X,X:],REAL st f is_metric_of X holds f
  is_a_pseudometric_of X
proof
  let f be Function of [:X,X:],REAL;
  assume f is_metric_of X;
  then
  for a,b,c be Element of X holds f.(a,a)=0 & f.(a,b)=f.(b,a) & f.(a,c)<=f
  .(a,b)+f.(b,c) by PCOMPS_1:def 6;
  then f is Reflexive symmetric triangle by METRIC_1:def 2,def 4,def 5;
  hence thesis by NAGATA_1:def 10;
end;
