reserve n,m,k for Element of NAT,
  x,X for set,
  A1 for SetSequence of X,
  Si for SigmaField of X,
  XSeq for SetSequence of Si;
reserve Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th4:
  for S being non empty set holds S is SigmaField of X iff S c=
  bool X & (for A1 being SetSequence of X st rng A1 c= S holds Union A1 in S) &
  for A being Subset of X st A in S holds A` in S
proof
  let S be non empty set;
  thus S is SigmaField of X implies S c= bool X & (for A1 being SetSequence of
X st rng A1 c= S holds Union A1 in S) & for A being Subset of X st A in S holds
  A` in S
  proof
    assume S is SigmaField of X;
    then reconsider S as SigmaField of X;
    for A1 being SetSequence of X st rng A1 c= S holds Union A1 in S
    proof
      let A1 be SetSequence of X;
      assume rng A1 c= S;
      then reconsider A1 as SetSequence of S by RELAT_1:def 19;
      Union A1 in S by PROB_1:17;
      hence thesis;
    end;
    hence thesis by PROB_1:15;
  end;
  assume that
A1: S c= bool X and
A2: for A1 being SetSequence of X st rng A1 c= S holds Union A1 in S and
A3: for A being Subset of X st A in S holds A` in S;
  for A1 being SetSequence of X st rng A1 c= S holds Intersection A1 in S
  proof
    let A1 be SetSequence of X such that
A4: rng A1 c= S;
    for n being Nat holds (Complement A1).n in S
    proof
      let n be Nat;
      reconsider n1=n as Element of NAT by ORDINAL1:def 12;
      A1.n in rng A1 by NAT_1:51;
      then (A1.n1)` in S by A3,A4;
      hence thesis by PROB_1:def 2;
    end;
    then rng Complement A1 c= S by NAT_1:52;
    then (Union Complement A1)` in S by A2,A3;
    hence thesis by PROB_1:def 3;
  end;
  hence thesis by A1,A3,PROB_1:15;
end;
