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

theorem Th16:
  f is Reflexive & a >= 0 implies low_toler(f,a) is_reflexive_in X
proof
  assume
A1: f is Reflexive & a >= 0;
  now
    let x be object;
    assume x in X;
    then reconsider x1 = x as Element of X;
    f.(x1,x1) <= a by A1,METRIC_1:def 2;
    hence [x,x] in low_toler(f,a) by Def3;
  end;
  hence thesis by RELAT_2:def 1;
end;
