 reserve Exx for Real;

theorem
  for A being SetSequence of NAT st
    (for n being Nat holds A.n = {n}) holds
    for n being Nat holds (Partial_Union A).n in Borel_Sets
proof
  let A be SetSequence of NAT;
  assume A0: for n being Nat holds A.n={n};
  defpred J[Nat] means (Partial_Union A).$1 in Borel_Sets;
S0: (Partial_Union A).0=A.0 by PROB_3:def 2;
   (Partial_Union A).0 in Borel_Sets
   proof
    A.0={0} by A0;
    hence thesis by Th1,S0;
   end; then
J0: J[0];
J1: for n being Nat st J[n] holds J[n+1]
  proof
   let n be Nat;
   assume Q0: J[n];
   (Partial_Union A).(n+1) in Borel_Sets
   proof
Q00: (Partial_Union A).(n+1) = (Partial_Union A).n \/ A.(n+1)
       by PROB_3:def 2;
     {n+1} in Borel_Sets by Th1; then
     A.(n+1) in Borel_Sets by A0;
     hence thesis by Q0,PROB_1:3,Q00;
   end;
   hence thesis;
  end;
  for n being Nat holds (Partial_Union A).n in Borel_Sets
  proof
    for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
    hence thesis;
  end;
  hence thesis;
end;
