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

theorem
  union TolSets T = X
proof
  for x being object holds x in union TolSets T iff x in X
  proof
    let x be object;
    thus x in union TolSets T implies x in X
    proof
      assume x in union TolSets T;
      then consider Z such that
A1:   x in Z and
A2:   Z in TolSets T by TARSKI:def 4;
      Z is TolSet of T by A2,Def3;
      then Z c= X by Th18;
      hence thesis by A1;
    end;
    assume x in X;
    then consider Z being TolClass of T such that
A3: x in Z by Th21;
    Z in TolSets T by Def3;
    hence thesis by A3,TARSKI:def 4;
  end;
  hence thesis by TARSKI:2;
end;
