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

theorem
  for a,b being Real, n being Nat holds
  (Partial_Intersection half_open_sets(a,b)).n is Element of Borel_Sets
proof
 let a,b be Real;
 defpred J[Nat] means
  (Partial_Intersection half_open_sets(a,b)).$1 is Element of Borel_Sets;
A1: J[0]
proof
  (Partial_Intersection half_open_sets(a,b)).0 =
    half_open_sets(a,b).0 by PROB_3:def 1; then
  (Partial_Intersection half_open_sets(a,b)).0 =
    halfline_fin(a,(b+1)) by Def1;
  hence thesis by Th4;
end;
A2: for k being Nat st J[k] holds J[k+1]
proof
 let k be Nat;
 assume A3: (Partial_Intersection half_open_sets(a,b)).k
   is Element of Borel_Sets;
 reconsider k as Element of NAT by ORDINAL1:def 12;
(Partial_Intersection half_open_sets(a,b)).(k+1) =
(Partial_Intersection half_open_sets(a,b)).k /\
half_open_sets(a,b).(k+1) by PROB_3:def 1; then
A4: (Partial_Intersection half_open_sets(a,b)).(k+1) =
(Partial_Intersection half_open_sets(a,b)).k /\
halfline_fin(a,(b+1/(k+1))) by Def1;
[.a,b+1/(k+1).[ is Element of Borel_Sets &
(Partial_Intersection half_open_sets(a,b)).k
  is Element of Borel_Sets by Th4,A3;
hence thesis by A4,FINSUB_1:def 2;
end;
for k being Nat holds J[k] from NAT_1:sch 2(A1,A2);
hence thesis;
end;
