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 Th7:
  for pmet st pmet is_a_pseudometric_of the carrier of T for x,y be
Point of T,A be non empty Subset of T
 holds |.lower_bound(pmet,A).x-lower_bound(pmet,A).y.|<=
  pmet.(x,y)
proof
  let pmet such that
A1: pmet is_a_pseudometric_of the carrier of T;
A2: pmet is symmetric by A1,NAGATA_1:def 10;
  let x,y be Point of T,A be non empty Subset of T;
A3: for x1,y1 be Point of T
 holds lower_bound(pmet,A).y1-lower_bound(pmet,A).x1>=-pmet.(x1,
  y1)
  proof
    let x1,y1 be Point of T;
A4: dist(pmet,x1).:A is non empty bounded_below by A1,Lm1;
A5: for rn st rn in dist(pmet,y1).:A
 holds rn>=lower_bound(dist(pmet,x1).:A)-pmet.
    (x1,y1)
    proof
      let rn;
      assume rn in dist(pmet,y1).:A;
      then consider z being object such that
A6:   z in dom dist(pmet,y1) and
A7:   z in A and
A8:   dist(pmet,y1).z=rn by FUNCT_1:def 6;
      reconsider z as Point of T by A6;
A9:   dist(pmet,x1).z= pmet.(x1,z) by Def2;
      pmet is triangle by A1,NAGATA_1:def 10;
      then
A10:  pmet.(x1,y1)+pmet.(y1,z)>=pmet.(x1,z) by METRIC_1:def 5;
      dom dist(pmet,x1)= the carrier of T by FUNCT_2:def 1;
      then dist(pmet,x1).z in dist(pmet,x1).:A by A7,FUNCT_1:def 6;
      then pmet.(x1,z)>=lower_bound(dist(pmet,x1).:A) by A4,A9,SEQ_4:def 2;
      then pmet.(x1,y1)+pmet.(y1,z)>=lower_bound(dist(pmet,x1).:A)+0
      by A10,XXREAL_0:2;
      then pmet.(y1,z)-0>=lower_bound(dist(pmet,x1).:A)-pmet.(x1,y1)
      by XREAL_1:21;
      hence thesis by A8,Def2;
    end;
    dist(pmet,y1).:A is non empty bounded_below by A1,Lm1;
    then lower_bound(dist(pmet,y1).:A)-0>=
    lower_bound(dist(pmet,x1).:A)-pmet.(x1,y1) by A5,SEQ_4:43;
    then
A11: lower_bound(dist(pmet,y1).:A)-lower_bound(dist(pmet,x1).:A)
>=0-pmet.(x1,y1) by XREAL_1:17;
    lower_bound(dist(pmet,y1).:A)=lower_bound(pmet,A).y1 by Def3;
    hence thesis by A11,Def3;
  end;
  then lower_bound(pmet,A).y-lower_bound(pmet,A).x>=-pmet.(x,y);
  then -(lower_bound(pmet,A).x-lower_bound(pmet,A).y)>=-pmet.(x,y);
  then
A12: (lower_bound(pmet,A).x-lower_bound(pmet,A).y)<= pmet.(x,y) by XREAL_1:24;
  lower_bound(pmet,A).x-lower_bound(pmet,A).y>=-pmet.(y,x) by A3;
  then (lower_bound(pmet,A).x-lower_bound(pmet,A).y)>=-pmet.(x,y)
  by A2,METRIC_1:def 4;
  hence thesis by A12,ABSVALUE:5;
end;
