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 Th5:
  for X be non empty set,f be Function of [:X,X:],REAL st f
  is_a_pseudometric_of X for A be non empty Subset of X,x be Element of X holds
  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 non empty Subset of X,x be Element of X;
A2: now
    let rn;
    assume rn in (dist(f,x)).:A;
    then consider y being object such that
A3: y in dom dist(f,x) and
    y in A and
A4: rn = dist(f,x).y by FUNCT_1:def 6;
    dist(f,x).y=f.(x,y) by A3,Def2;
    hence rn>=0 by A1,A3,A4,NAGATA_1:29;
  end;
  X=dom dist(f,x) by FUNCT_2:def 1;
  then lower_bound ((dist(f,x)).:A)>=0 by A2,SEQ_4:43;
  hence thesis by Def3;
end;
