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

theorem Th5:
  for a,b being Real holds
   Intersection half_open_sets(a,b) is Element of Borel_Sets
 proof
  let a,b be Real;
  for n being Nat holds
    (Complement half_open_sets(a,b)).n is Element of Borel_Sets
  proof
   let n be Nat;
    reconsider nn=n as Element of NAT by ORDINAL1:def 12;
   (half_open_sets(a,b).nn)` is Element of Borel_Sets
   proof
     half_open_sets(a,b).n is Element of Borel_Sets
     proof
      per cases by NAT_1:6;
      suppose A1: n=0;
        half_open_sets(a,b).0=halfline_fin(a,b+1) by Def1;
        hence thesis by A1,Th4;
      end;
      suppose ex k being Nat st n = k + 1; then
        consider k being Nat such that A2: n=k+1;
        reconsider k as Element of NAT by ORDINAL1:def 12;
        half_open_sets(a,b).(k+1) =
          halfline_fin(a,b+(1/(k+1))) by Def1;
        hence thesis by A2,Th4;
      end;
      end;
     hence thesis by PROB_1:def 1;
     end;
     hence thesis by PROB_1:def 2;
  end; then
  Complement half_open_sets(a,b)
    is SetSequence of Borel_Sets by PROB_1:25; then
  Union Complement half_open_sets(a,b)
    is Element of Borel_Sets by PROB_1:26;
  hence thesis by PROB_1:def 1;
end;
