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

theorem Th26:
  for T,R being Tolerance of X st for Z holds Z is TolClass of T
  iff Z is TolClass of R holds T = R
proof
  let T,R be Tolerance of X;
  assume
A1: for Z holds Z is TolClass of T iff Z is TolClass of R;
  for x,y being object holds [x,y] in T iff [x,y] in R
  proof
    let x,y be object;
    thus [x,y] in T implies [x,y] in R
    proof
      assume [x,y] in T;
      then consider Z being TolClass of T such that
A2:   x in Z & y in Z by Th20;
      reconsider Z as TolClass of R by A1;
      Z is TolSet of R;
      hence thesis by A2,Def1;
    end;
    assume [x,y] in R;
    then consider Z being TolClass of R such that
A3: x in Z & y in Z by Th20;
    reconsider Z as TolClass of T by A1;
    Z is TolSet of T;
    hence thesis by A3,Def1;
  end;
  hence thesis;
end;
