reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve S for non empty Subset of REAL;
reserve r for Real;
reserve T for Nat;

theorem Th2:
  for k being ExtReal holds ExtREAL \ [.-infty,k.] = ].k,+infty.]
 proof
 let k be ExtReal;
 for x being object holds x in ExtREAL \ [.-infty,k.] iff x in ].k,+infty.]
 proof
  let x be object;
  hereby assume A2: x in ExtREAL \ [.-infty,k.];
 A3: x in [.-infty,+infty.] & not x in [.-infty,k.]
      by A2,XBOOLE_0:def 5,XXREAL_1:209;
     consider y being Element of ExtREAL such that A4:x=y by A2;
     y in [.-infty,+infty.] & not (-infty <= y & y <= k) by A4,A3,XXREAL_1:1;
     hence x in ].k,+infty.] by XXREAL_1:2,A4,XXREAL_0:3,5;
     end;
     assume A6: x in ].k,+infty.];
     then k in ExtREAL & x in ].k,+infty.] &
       x in {a where a is Element of ExtREAL:
         k < a & a <= +infty} by XXREAL_0:def 1,XXREAL_1:def 3; 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 ExtREAL;
     y>k by A6,A8,XXREAL_1:2;
     then y in ExtREAL & not y in [.-infty,k.] by XXREAL_1:1;
    hence thesis by A8,XBOOLE_0:def 5;
  end;
 hence thesis;
 end;
