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

theorem
  for k being Real holds REAL \ [.k,+infty.[ = ].-infty,k.[
 proof
 let k be Real;
A1: k in REAL by XREAL_0:def 1;
 for x being object holds x in REAL \ [.k,+infty.[ iff x in ].-infty,k.[
 proof
  let x be object;
  hereby assume A2: x in REAL \ [.k,+infty.[;
 A3: x in ].-infty,+infty.[ & not x in [.k,+infty.[
      by A2,XBOOLE_0:def 5,XXREAL_1:224;
     consider y being Element of REAL such that A4:x=y by A2;
     y in ].-infty,+infty.[ & not y>= k by A4,A3,XXREAL_1:236;
     hence x in ].-infty,k.[ by A4,XXREAL_1:233;
     end;
  assume A5: x in ].-infty,k.[; then
     k in REAL & x in ].-infty,k.[ &
     x in {a where a is Element of ExtREAL:
       -infty < a & a < k} by XREAL_0:def 1,XXREAL_1:def 4; then
     consider a being Element of ExtREAL such that
 A6: a=x & -infty < a & a < k;
     consider y being Element of ExtREAL such that A7: x=y by A6;
     reconsider y as Element of REAL by A1,A6,A7,XXREAL_0:47;
     y < k by A5,A7,XXREAL_1:233;
     then y in REAL & not y in [.k,+infty.[ by XXREAL_1:236;
     hence thesis by A7,XBOOLE_0:def 5;
  end;
 hence thesis by TARSKI:2;
 end;
