 reserve Omega, Omega2 for non empty set;
 reserve Sigma, F for SigmaField of Omega;
 reserve Sigma2, F2 for SigmaField of Omega2;

theorem Th3:
  for k being Real holds
  [.k,+infty.[ is Element of Borel_Sets &
  ].-infty,k.[ is Element of Borel_Sets
proof
 let k be Real;
A1: k in REAL by XREAL_0:def 1;
 set R = ].-infty,k.[;
 A2: ].-infty,k.[ in Borel_Sets
 proof
  set L = halfline(k);
A3: L in Family_of_halflines & L=].-infty,k.[ by A1;
  Family_of_halflines c= sigma(Family_of_halflines) by PROB_1:def 9;
  hence thesis by A3;
 end; then
 R` in Borel_Sets by PROB_1:def 1;
 hence thesis by Th2,A2;
end;
