reserve i, k, m, n for Nat,
  r, s for Real,
  rn for Real,
  x, y , z, X for set,
  T, T1, T2 for non empty TopSpace,
  p, q for Point of T,
  A, B, C for Subset of T,
  A9 for non empty Subset of T,
  pq for Element of [:the carrier of T,the carrier of T:],
  pq9 for Point of [:T,T:],
  pmet,pmet1 for Function of [:the carrier of T,the carrier of T:],REAL,
  pmet9,pmet19 for RealMap of [:T,T:] ,
  f,f1 for RealMap of T,
  FS2 for Functional_Sequence of [:the carrier of T,the carrier of T:],REAL,
  seq for Real_Sequence;

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;
