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

theorem
  for M being non empty Reflexive symmetric MetrStruct, a being Real,
  T being Relation of the carrier of M,the carrier of M st
  T = dist_toler(M,a) & a >= 0 holds T is Tolerance of the carrier of M
proof
  let M be non empty Reflexive symmetric MetrStruct, a be Real, T be
  Relation of the carrier of M,the carrier of M such that
A1: T = dist_toler(M,a) and
A2: a >= 0;
A3: the distance of M is symmetric & the distance of M is Reflexive by
METRIC_1:def 6,def 8;
  T = low_toler(the distance of M,a) by A1,Th33;
  hence thesis by A2,A3,Th18;
end;
