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 Th6:
  for X be non empty set,f be Function of [:X,X:],REAL st f
  is_a_pseudometric_of X for A be Subset of X,x be Element of X holds x in A
  implies lower_bound(f,A).x=0
proof
  let X be non empty set,f be Function of [:X,X:],REAL such that
A1: f is_a_pseudometric_of X;
  let A be Subset of X,x be Element of X;
  assume
A2: x in A;
  then reconsider A as non empty Subset of X;
A3: dist(f,x).:A is non empty bounded_below by A1,Lm1;
  f is Reflexive by A1,NAGATA_1:def 10;
  then f.(x,x)=0 by METRIC_1:def 2;
  then X=dom dist(f,x) & dist(f,x).x=0 by Def2,FUNCT_2:def 1;
  then 0 in dist(f,x).:A by A2,FUNCT_1:def 6;
  then lower_bound(dist(f,x).:A)<=0 by A3,SEQ_4:def 2;
  then lower_bound(f,A).x <=0 by Def3;
  hence thesis by A1,Th5;
end;
