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

theorem
  TolSets R c= TolSets T iff R c= T
proof
  thus TolSets R c= TolSets T implies R c= T
  proof
    assume
A1: TolSets R c= TolSets T;
    let x,y be object;
    assume [x,y] in R;
    then consider Z being TolClass of R such that
A2: x in Z & y in Z by Th20;
    Z in TolSets R by Def3;
    then Z is TolSet of T by A1,Def3;
    hence thesis by A2,Def1;
  end;
  assume
A3: R c= T;
  let x be object;
  assume x in TolSets R;
  then reconsider Z = x as TolSet of R by Def3;
  for x,y st x in Z & y in Z holds [x,y] in T
  by Def1,A3;
  then Z is TolSet of T by Def1;
  hence thesis by Def3;
end;
