reserve Omega, I for non empty set;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve D, E, F for Subset-Family of Omega;
reserve  B, sB for non empty Subset of Sigma;
reserve b for Element of B;
reserve a for Element of Sigma;
reserve p, q, u, v for Event of Sigma;
reserve n, m for Element of NAT;
reserve S, S9, X, x, y, z, i, j for set;

theorem Th15:
  for F being ManySortedSigmaField of I, Sigma holds
  tailSigmaField(F,I) is SigmaField of Omega
proof
  let F be ManySortedSigmaField of I, Sigma;
A1: for A1 being SetSequence of Omega st rng A1 c= tailSigmaField(F,I) holds
  Intersection A1 in tailSigmaField(F,I)
  proof
    let A1 be SetSequence of Omega;
    assume
A2: rng A1 c= tailSigmaField(F,I);
A3: for n holds for S holds S in futSigmaFields(F,I) implies A1.n in S
    proof
      let n, S;
A4:   A1.n in rng A1 by NAT_1:51;
      assume S in futSigmaFields(F,I);
      hence thesis by A2,A4,SETFAM_1:def 1;
    end;
    for S st S in futSigmaFields(F,I) holds (Union Complement A1)` in S
    proof
      let S;
      assume
A5:   S in futSigmaFields(F,I);
      then ex E being finite Subset of I st S = sigUn(F,I\E) by Def7;
      then reconsider S as SigmaField of Omega;
      for n being Nat holds (Complement A1).n in S
      proof
        let n be Nat;
        reconsider n as Element of NAT by ORDINAL1:def 12;
        A1.n in S by A3,A5;
        then (A1.n)` is Event of S by PROB_1:20;
        then (A1.n)` in S;
        hence thesis by PROB_1:def 2;
      end;
      then rng Complement A1 c= S by NAT_1:52;
      then reconsider CA1= Complement A1 as SetSequence of S by RELAT_1:def 19;
      Union CA1 in S by PROB_1:17;
      then (Union Complement A1)` is Event of S by PROB_1:20;
      hence thesis;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
  for A being Subset of Omega st A in tailSigmaField(F,I) holds A` in
  tailSigmaField(F,I)
  proof
    let A be Subset of Omega;
    assume
A6: A in tailSigmaField(F,I);
    for S holds S in futSigmaFields(F,I) implies A` in S
    proof
      let S;
      assume
A7:   S in futSigmaFields(F,I);
      then consider E being finite Subset of I such that
A8:   S=sigUn(F,I\E) by Def7;
      A in S by A6,A7,SETFAM_1:def 1;
      then A` is Event of sigma (Union (F|(I\E))) by A8,PROB_1:20;
      hence thesis by A8;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
  hence thesis by A1,PROB_1:15;
end;
