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

theorem Th2:
  for k being Real holds REAL \ ].-infty,k.[ = [.k,+infty.[
 proof
 let k be Real;
A1: k in REAL by XREAL_0:def 1;
 for x being object holds x in REAL \ ].-infty,k.[ iff x in [.k,+infty.[
 proof
  let x be object;
  hereby assume A2: x in REAL \ ].-infty,k.[;
 A3: x in ].-infty,+infty.[ & not x in ].-infty,k.[
      by A2,XBOOLE_0:def 5,XXREAL_1:224;
     consider y being Element of REAL such that A4:x=y by A2;
 A5: y in ].-infty,+infty.[ & not y< k by A4,A3,XXREAL_1:233;
     thus x in [.k,+infty.[ by A5,A4,XXREAL_1:236;
     end;
  assume A6: x in [.k,+infty.[;
     then k in REAL & x in [.k,+infty.[ &
       x in {a where a is Element of ExtREAL:
         k <= a & a < +infty} by XREAL_0:def 1,XXREAL_1:def 2; then
     consider a being Element of ExtREAL such that
 A7: a=x & k <= a & a < +infty;
     consider y being Element of ExtREAL such that A8: x=y by A7;
     reconsider y as Element of REAL by A7,A8,A1,XXREAL_0:46;
     y >= k by A6,A8,XXREAL_1:236; then
     y in REAL & not y in ].-infty,k.[ by XXREAL_1:233;
     hence thesis by A8,XBOOLE_0:def 5;
  end;
 hence thesis by TARSKI:2;
 end;
