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

theorem Th4:
  for k1,k2 being Real holds
  [.k2,k1.[ is Element of Borel_Sets
proof
 let k1,k2 be Real;
 set R = ].-infty,k2.[;
 k1 in REAL & k2 in REAL by XREAL_0:def 1; then
A1: -infty<k2 & k1<+infty by XXREAL_0:9,12;
A2: REAL \ ].-infty,k2.[=[.k2,+infty.[ by Th2;
 R` is Element of Borel_Sets &
     ].-infty,k1.[ is Element of Borel_Sets &
     [.k1,+infty.[ is Element of Borel_Sets by Th3,A2; then
 ].-infty,k1.[ /\ R` is Element of Borel_Sets by FINSUB_1:def 2; then
 ].-infty,k1.[ /\ [.k2,+infty.[ is Element of Borel_Sets by Th2;
 hence thesis by A1,XXREAL_1:154;
end;
