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 Th31:
  for F being sequence of COM(Sigma,P) holds ex BSeq being
  SetSequence of Sigma st for n holds BSeq.n in ProbPart(F.n)
proof
  let F be sequence of COM(Sigma,P);
  defpred P[Element of NAT, set] means for n being Element of NAT for y being
  set holds (n = $1 & y = $2 implies y in ProbPart(F.n));
A1: for t being Element of NAT ex A being Element of Sigma st P[t,A]
  proof
    let t be Element of NAT;
    set A = the Element of ProbPart(F.t);
    reconsider A as Element of Sigma by Def7;
    take A;
    thus thesis;
  end;
  ex G being sequence of Sigma st for t being Element of NAT holds P[t
  ,G.t] from FUNCT_2:sch 3(A1);
  then consider G being sequence of Sigma such that
A2: for t being Element of NAT holds for n being Element of NAT for y
  being set holds (n = t & y = G.t implies y in ProbPart(F.n));
  reconsider BSeq = G as SetSequence of Omega by FUNCT_2:7;
  reconsider BSeq as SetSequence of Sigma;
  take BSeq;
  thus thesis by A2;
end;
