 reserve Exx for Real;

theorem ZV5:
  NAT is Event of Borel_Sets
proof
 consider A being SetSequence of Borel_Sets such that
Q10: for n being Nat holds A.n={n} by Q0;
 reconsider A as SetSequence of REAL;
 Union A = NAT
 proof
  for x being object holds x in Union A iff x in NAT
  proof
   let x be object;
H0: for n being Element of NAT holds
        for x being object holds x in {n} iff x=n by TARSKI:def 1;
H1: (ex n being Nat st x in A.n) implies
     ex n being Element of NAT st x in A.n
   proof
    given n being Nat such that C0: x in A.n;
    n+0 in NAT;
    then consider n being Element of NAT such that C1: x in A.n by C0;
    thus thesis by C1;
   end;
   thus x in Union A implies x in NAT
   proof
    assume C0: x in Union A;
    ex n being Element of NAT st x in A.n & A.n={n}
    proof
     consider n being Element of NAT such that D0: x in A.n by C0,PROB_1:12,H1;
     A.n = {n} by Q10;
     hence thesis by D0;
    end;
    then ex n being Element of NAT st x=n by H0;
    hence thesis;
   end;
   assume C000: x in NAT;
   ex k being Nat st x in {k} & A.k={k}
   proof
    reconsider k = x as Nat by C000;
A:  x in {k} by TARSKI:def 1;
    A.k = {k} by Q10;
    hence thesis by A;
   end;
   hence thesis by PROB_1:12;
  end;
 hence thesis;
 end;
hence thesis by PROB_1:17;
end;
