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;
