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 Th32:
  for F being sequence of COM(Sigma,P), BSeq being SetSequence
of Sigma holds ex CSeq being SetSequence of Omega st for n holds CSeq.n = F.n \
  BSeq.n
proof
  let F be sequence of COM(Sigma,P), G be SetSequence of Sigma;
  defpred P[Element of NAT, set] means for n being Element of NAT for y being
  set holds (n = $1 & y = $2 implies y = F.n \ G.n);
A1: for t being Element of NAT ex A being Element of bool Omega st P[t,A]
  proof
    let t be Element of NAT;
    F.t is Element of COM(Sigma,P);
    then reconsider A = F.t \ G.t as Element of bool Omega by XBOOLE_1:1;
    take A;
    thus thesis;
  end;
  ex H being sequence of bool Omega st for t being Element of NAT
  holds P[t,H.t] from FUNCT_2:sch 3(A1);
  then consider H being sequence of bool Omega such that
A2: for t being Element of NAT holds for n being Element of NAT for y
  being set holds n = t & y = H.t implies y = F.n \ G.n;
  take H;
  thus thesis by A2;
end;
