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 Th500:
  for b being Real holds
    Intersection ext_right_closed_sets(b) is Element of Ext_Borel_Sets
proof
  let b be Real;
  for n being Nat holds
    (Complement ext_right_closed_sets(b)).n is Element of Ext_Borel_Sets
  proof
   let n be Nat;
    reconsider nn=n as Element of NAT by ORDINAL1:def 12;
   ((ext_right_closed_sets(b)).nn)` is Element of Ext_Borel_Sets
   proof
     (ext_right_closed_sets(b)).n is Element of Ext_Borel_Sets
     proof
      (ext_right_closed_sets(b)).n = [.-infty,b-n.] by Def3000;
      hence thesis by Th3;
     end;
     hence thesis by PROB_1:def 1;
     end;
     hence thesis by PROB_1:def 2;
  end; then
  Complement ext_right_closed_sets(b)
    is SetSequence of Ext_Borel_Sets by PROB_1:25; then
  Union Complement ext_right_closed_sets(b)
    is Element of Ext_Borel_Sets by PROB_1:26;
  hence thesis by PROB_1:def 1;
end;
