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

theorem Th18:
  a >= 0 & f is Reflexive symmetric implies low_toler(f,a) is Tolerance of X
proof
  set T = low_toler(f,a);
  assume that
A1: a >= 0 and
A2: f is Reflexive symmetric;
A3: low_toler(f,a) is_reflexive_in X by A1,A2,Th16;
A4: dom T = X by A1,A2,Th3,Th16;
  then
A5: field T = X \/ (rng low_toler(f,a))
    .= X by XBOOLE_1:12;
  then T is_symmetric_in field T by A2,Th17;
  hence thesis by A3,A4,A5,PARTFUN1:def 2,RELAT_2:def 9,def 11;
end;
