reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem Th36:
  for M being non empty MetrSpace for R being Equivalence_Relation of M st
  R = dist_toler(M,0)[*] holds Class R = SmallestPartition the carrier of M
proof
  let M be non empty MetrSpace;
  now
    let x,y be Element of M;
    dist(x,y) >= 0 by METRIC_1:5;
    hence (the distance of M).(x,y) >= 0 by METRIC_1:def 1;
  end;
  then
A1: the distance of M is nonnegative;
  let R be Equivalence_Relation of M;
  assume R = dist_toler(M,0)[*];
  then
  the distance of M is Reflexive discerning & low_toler(the distance of M,
  0) [*] = R by Th33,METRIC_1:def 6,def 7;
  hence thesis by A1,Th24;
end;
