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

theorem Th20:
  for X being non empty set, f being PartFunc of [:X,X:],REAL,
      x,y being object
  st f is nonnegative Reflexive discerning holds [x,y] in low_toler(f,0)
  implies x = y
proof
  let X be non empty set, f be PartFunc of [:X,X:],REAL,
      x,y be object such that
A1: f is nonnegative Reflexive discerning;
  assume
A2: [x,y] in low_toler(f,0);
  then reconsider x1 = x, y1 = y as Element of X by ZFMISC_1:87;
  f.(x1,y1) <= 0 by A2,Def3;
  then f.(x1,y1) = 0 by A1;
  hence thesis by A1,METRIC_1:def 3;
end;
