reserve X,Y,Z,x,y,z for set;
reserve T,R for Tolerance of X;

theorem
  for T being Tolerance of X holds rng T = X
proof
  let T be Tolerance of X;
  for x being object holds x in rng T iff x in X
  proof
    let x be object;
    x in X implies x in rng T
    proof
      field T = X by ORDERS_1:12;
      then
A1:   T is_reflexive_in X by RELAT_2:def 9;
      assume x in X;
      then [x,x] in T by A1;
      hence thesis by XTUPLE_0:def 13;
    end;
    hence thesis;
  end;
  hence thesis by TARSKI:2;
end;
