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;
reserve T,L1 for Sequence,
  O,O1,O2,O3,C for Ordinal;

theorem Th25:
  for d being BiFunction of A,L, q being QuadrSeq of d holds O in
  DistEsti(d) iff O in dom q
proof
  let d be BiFunction of A,L;
  let q be QuadrSeq of d;
  reconsider N = dom q as Cardinal by Def13;
  reconsider M = DistEsti(d) as Cardinal;
  q is one-to-one by Def13;
  then
A1: dom q,rng q are_equipotent by WELLORD2:def 4;
  DistEsti(d),{[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} are_equipotent by Def11;
  then DistEsti(d),rng q are_equipotent by Def13;
  then DistEsti(d),dom q are_equipotent by A1,WELLORD2:15;
  then
A2: M = N by CARD_1:2;
  hence O in DistEsti(d) implies O in dom q;
  assume O in dom q;
  hence thesis by A2;
end;
