 reserve Exx for Real;

theorem
  Borel_Sets = sigma Family_of_halflines2
proof
A1: sigma Family_of_halflines2 c= sigma Family_of_halflines
 proof
  Family_of_halflines2 c= sigma(Family_of_halflines)
  proof
   for x being object holds x in Family_of_halflines2 implies
    x in sigma Family_of_halflines
   proof
    let x be object;
    assume x in Family_of_halflines2;
    then consider r being Element of REAL such that
B1: x=right_closed_halfline(r);
    x is Element of Borel_Sets by FINANCE1:3,B1;
    hence thesis;
   end;
   hence thesis;
  end;
  hence thesis by PROB_1:def 9;
  end;
 sigma Family_of_halflines c= sigma Family_of_halflines2
 proof
  Family_of_halflines c= sigma Family_of_halflines2
  proof
   for x being object holds x in Family_of_halflines implies
    x in sigma Family_of_halflines2
   proof
    let x be object;
    assume x in Family_of_halflines;
    then consider r being Element of REAL such that B1: x=halfline(r);
    right_closed_halfline(r) is Event of sigma Family_of_halflines2
    proof
      set L = right_closed_halfline(r);
  A3: L in Family_of_halflines2 & L=[.r,+infty.[;
      Family_of_halflines2 c= sigma(Family_of_halflines2) by PROB_1:def 9;
      hence thesis by A3;
    end; then
    (halfline(r))` is Event of sigma(Family_of_halflines2) by FINANCE1:2;
    then ((halfline(r))`)` is Event of sigma(Family_of_halflines2)
     by PROB_1:15;
    hence thesis by B1;
   end;
   hence thesis;
  end;
  hence thesis by PROB_1:def 9;
 end;
 hence thesis by A1;
end;
