 reserve Exx for Real;

theorem Th40:
  INT is Event of Borel_Sets
proof
  consider A1 being SetSequence of REAL such that
A1: for n being Nat holds A1.n={n} by Q00;
  consider A2 being SetSequence of REAL such that
A2: for n being Nat holds A2.n={-n} by H2;
A3: for n being Nat holds (Partial_Union A1).n is Event of Borel_Sets
 proof
  defpred J[Nat] means (Partial_Union A1).$1 in Borel_Sets;
B0: (Partial_Union A1).0=A1.0 by PROB_3:def 2;
  A1.0 = {0} by A1; then
J0: J[0] by Th1,B0;
J1: for n being Nat st J[n] holds J[n+1]
  proof
   let n be Nat;
   assume Q0: J[n];
   (Partial_Union A1).(n+1) in Borel_Sets
   proof
C0: (Partial_Union A1).(n+1) = (Partial_Union A1).n \/ A1.(n+1)
     by PROB_3:def 2;
C2: A1.(n+1) in Borel_Sets
    proof
     A1.(n+1)={n+1} by A1;
     hence thesis by Th1;
    end;
    (Partial_Union A1).n \/ A1.(n+1) is Event of Borel_Sets
     by Q0,C2,PROB_1:21;
    hence thesis by C0;
   end;
 hence thesis;
 end;
  for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
  hence thesis;
 end;
  defpred J[Nat] means (Partial_Union A2).$1 in Borel_Sets;
B0: (Partial_Union A2).0=A2.0 by PROB_3:def 2;
  A2.0 = {-0} by A2; then
J0: J[0] by Th1,B0;
A4: for n being Nat holds
 (Partial_Union A2).n is Event of Borel_Sets
 proof
  J1: for n being Nat st J[n] holds J[n+1]
  proof
   let n be Nat;
   assume Q0: J[n];
   (Partial_Union A2).(n+1) in Borel_Sets
   proof
C0: (Partial_Union A2).(n+1) = (Partial_Union A2).n \/ A2.(n+1)
     by PROB_3:def 2;
    A2.(n+1)={-(n+1)} by A2; then
    A2.(n+1) in Borel_Sets by Th1; then
    (Partial_Union A2).n \/ A2.(n+1) is Event of Borel_Sets
      by Q0,PROB_1:21;
    hence thesis by C0;
   end;
  hence thesis;
 end;
  for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
  hence thesis;
 end;
 consider B1 being SetSequence of REAL such that
A5: for n being Nat holds B1.n = (Partial_Union A1).n by ThA;
A6: for n being Nat holds B1.n is Event of Borel_Sets
 proof
  let n be Nat;
  (Partial_Union A1).n is Event of Borel_Sets by A3;
  hence thesis by A5;
 end;
 consider B2 being SetSequence of REAL such that
A7: for n being Nat holds B2.n = (Partial_Union A2).n by ThA;
 A8: for n being Nat holds B2.n is Event of Borel_Sets
 proof
  let n be Nat;
  (Partial_Union A2).n is Event of Borel_Sets by A4;
  hence thesis by A7;
 end;
 reconsider B1 as SetSequence of Borel_Sets by A6,PROB_1:25;
 reconsider B2 as SetSequence of Borel_Sets by A8,PROB_1:25;
A9: Union B1 is Event of Borel_Sets by PROB_1:26;
A10: Union B2 is Event of Borel_Sets by PROB_1:26;
 Union B1 \/ Union B2 = INT
 proof
  for x being object holds x in (Union B1 \/ Union B2) iff x in INT
  proof
   let x be object;
   thus x in Union B1 \/ Union B2 implies x in INT
   proof
    assume x in Union B1 \/ Union B2; then
    per cases by XBOOLE_0:def 3;
    suppose F0: x in Union B1;
      consider n being Nat such that F1: x in B1.n by F0,PROB_1:12;
  F2: x in (Partial_Union A1).n by F1,A5;
      consider k being Nat such that
  G1: k<=n & x in (Partial_Union A1).k by F2;
      consider m being Nat such that
  I0: m<=k & x in A1.m by G1,PROB_3:13;
     x in INT
     proof
      x in {m} by I0,A1; then
  I1: x=m by TARSKI:def 1;
      m+0 in NAT;
      hence thesis by I1,NUMBERS:17;
    end;
    hence thesis;
    end;
    suppose x in Union B2; then
      consider k being Nat such that H1: x in B2.k by PROB_1:12;
      x in (Partial_Union A2).k by H1,A7;
      then consider z being Nat such that
  G1: z<=k & x in A2.z by PROB_3:13;
      x in {-z} by G1,A2;
      then x=-z by TARSKI:def 1;
      hence thesis by INT_1:def 1;
     end;
   end;
   x in INT implies x in (Union B1 \/ Union B2)
   proof
    assume x in INT; then
    consider k being Nat such that E0: x=k or x=-k by INT_1:def 1;
    per cases by E0;
    suppose x=k; then
     reconsider p2 = x as Nat;
     x in Union B1 \/ Union B2
     proof
      ex k being Nat st x in B1.k
      proof
       ex q being Nat st x in (Partial_Union A1).q
       proof
        x in {p2} by TARSKI:def 1; then
        x in A1.p2 by A1; then
        x in (Partial_Union A1).p2 by PROB_3:13;
        hence thesis;
       end; then
       consider q being Nat such that I0: x in (Partial_Union A1).q;
       B1.q=(Partial_Union A1).q by A5;
       hence thesis by I0;
      end;
      then x in Union B1 by PROB_1:12;
      hence thesis by XBOOLE_0:def 3;
     end;
    hence thesis;
    end;
    suppose F1: x=-k;
     -k is Element of INT by INT_1:def 1;
     then consider z being Element of INT such that G0: z=-k & -k=x by F1;
     x in Union B1 \/ Union B2
     proof
       ex k being Nat st x in B2.k
       proof
       ex q being Nat st x in (Partial_Union A2).q
       proof
    K0: x in {-k} by G0,TARSKI:def 1;
        x in A2.k by A2,K0; then
        x in (Partial_Union A2).k by PROB_3:13;
        then consider q being Nat such that I0: x in (Partial_Union A2).q;
        thus thesis by I0;
       end; then
       consider q being Nat such that I0: x in (Partial_Union A2).q;
       B2.q=(Partial_Union A2).q by A7;
       hence thesis by I0;
      end; then
      x in Union B2 or x in Union B2 by PROB_1:12;
      hence thesis by XBOOLE_0:def 3;
     end;
     hence thesis;
    end;
   end;
   hence thesis;
  end;
 hence thesis;
 end;
hence thesis by A9,A10,PROB_1:3;
end;
