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

theorem Th37:
  for M being Reflexive symmetric bounded non empty MetrStruct
  st a >= diameter [#]M holds dist_toler(M,a) = nabla the carrier of M
proof
  let M be Reflexive symmetric bounded non empty MetrStruct such that
A1: a >= diameter [#]M;
  now
    let z be object;
    assume z in nabla the carrier of M;
    then 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 ZFMISC_1:def 2;
    reconsider x1=x, y1=y as Element of M by A2;
    dist(x1,y1) <= diameter [#]M by TBSP_1:def 8;
    then dist(x1,y1) <= a by A1,XXREAL_0:2;
    then x1, y1 are_in_tolerance_wrt a;
    hence z in dist_toler(M,a) by A3,Def7;
  end;
  then nabla the carrier of M c= dist_toler(M,a);
  hence thesis by XBOOLE_0:def 10;
end;
