reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;

theorem Th20:
  for d be distance_function of A,L holds DistEsti(d) <> {}
proof
  let d be distance_function of A,L;
  set X = { [x,y,a,b] where x is Element of A, y is Element of A, a is Element
  of L, b is Element of L: d.(x,y) <= a"\/"b};
  set x9 = the Element of A;
  consider z being set such that
A1: z = [x9,x9,Bottom L,Bottom L];
A2: DistEsti(d),X are_equipotent by Def11;
  d.(x9,x9) = Bottom L by Def6
    .= Bottom L "\/" Bottom L by YELLOW_5:1;
  then z in X by A1;
  hence thesis by A2,CARD_1:26;
end;
