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

theorem Th33:
  for M being non empty MetrStruct, a being Real holds
  dist_toler(M,a) = low_toler(the distance of M,a)
proof
  let M be non empty MetrStruct, a be Real;
  now
    let z be object such that
A1: z in low_toler(the distance of M,a);
    consider x,y being object such that
A2: x in the carrier of M & y in the carrier of M and
A3: z = [x,y] by A1,ZFMISC_1:def 2;
    reconsider x1 = x, y1 = y as Element of M by A2;
    dist(x1,y1) = (the distance of M).(x1,y1) by METRIC_1:def 1;
    then dist(x1,y1) <= a by A1,A3,Def3;
    then x1,y1 are_in_tolerance_wrt a;
    hence z in dist_toler(M,a) by A3,Def7;
  end;
  then
A4: low_toler(the distance of M,a) c= dist_toler(M,a);
  now
    let z be object such that
A5: z in dist_toler(M,a);
    consider x,y being object such that
A6: x in the carrier of M & y in the carrier of M and
A7: z = [x,y] by A5,ZFMISC_1:def 2;
    reconsider x1 = x, y1 = y as Element of M by A6;
    (the distance of M).(x1,y1) = dist(x1,y1) & x1,y1 are_in_tolerance_wrt
    a by A5,A7,Def7,METRIC_1:def 1;
    then (the distance of M).(x1,y1) <= a;
    hence z in low_toler(the distance of M,a) by A7,Def3;
  end;
  then dist_toler(M,a) c= low_toler(the distance of M,a);
  hence thesis by A4,XBOOLE_0:def 10;
end;
