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 Th16:
  for pmet st pmet is_a_pseudometric_of the carrier of T & for
pmet9 st pmet=pmet9 holds pmet9 is continuous for A be non empty Subset of T,p
  holds p in Cl A implies lower_bound(pmet,A).p=0
proof
  set rn=In(0,REAL);
  let pmet such that
A1: pmet is_a_pseudometric_of the carrier of T and
A2: for pmet9 st pmet=pmet9 holds pmet9 is continuous;
  let A be non empty Subset of T,p;
A3: dom lower_bound(pmet,A)= the carrier of T by FUNCT_2:def 1;
  reconsider Z={rn},infA=lower_bound(pmet,A).:A as Subset of R^1 by TOPMETR:17;
  infA c= Z
  proof
    let infa be object;
    assume infa in infA;
    then ex a be object st a in dom lower_bound(pmet,A) & a in A &
    infa=lower_bound (pmet,A).a by FUNCT_1:def 6;
    then infa=0 by A1,Th6;
    hence thesis by TARSKI:def 1;
  end;
  then
A4: Cl (infA) c= Cl Z by PRE_TOPC:19;
  reconsider InfA=lower_bound(pmet,A) as Function of T,R^1 by TOPMETR:17;
  for p holds dist(pmet,p) is continuous by A2,Th4;
  then lower_bound(pmet,A) is continuous by A1,Th8;
  then InfA is continuous by JORDAN5A:27;
  then
A5: InfA.:(Cl A) c= Cl(InfA.:A) by TOPS_2:45;
  assume p in Cl A;
  then
A6: lower_bound(pmet,A).p in lower_bound(pmet,A).:(Cl A) by A3,FUNCT_1:def 6;
  Z is closed by PCOMPS_1:7,TOPMETR:17;
  then Z=Cl Z by PRE_TOPC:22;
  then InfA.:(Cl A)c=Z by A4,A5;
  hence thesis by A6,TARSKI:def 1;
end;
