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;
reserve I for TheEvent of r;

theorem
  REAL is Element of Ext_Borel_Sets
proof
 reconsider Set1 = ExtREAL as Element of Ext_Borel_Sets by PROB_1:23;
 reconsider Set2 = {+infty} as Element of Ext_Borel_Sets by Th5000,Th6000;
 reconsider Set3 = {-infty} as Element of Ext_Borel_Sets by Th500,Th600;
 reconsider Set4 = Set1 \ Set2 as Element of Ext_Borel_Sets;
 Set4 = [.-infty,+infty.[ \/ ].+infty,+infty.] by XXREAL_1:388,209
     .= [.-infty,+infty.[; then
PP: Set4 \ Set3 = [.-infty,-infty.[ \/ ].-infty,+infty.[ by XXREAL_1:387
     .= ].-infty,+infty.[;
 reconsider Set5 = Set4 \ Set3 as Element of Ext_Borel_Sets;
 thus thesis by XXREAL_1:224,PP;
end;
