reserve X for set;

theorem Th16:
  for S being SigmaField of X, M being sigma_Measure of S, F being
sequence of COM(S,M), G being sequence of S ex H being sequence of
  bool X st for n being Element of NAT holds H.n = F.n \ G.n
proof
  let S be SigmaField of X, M be sigma_Measure of S, F be sequence of COM(
  S,M), G be sequence of S;
  defpred P[Element of NAT, set] means for n being Element of NAT, y being set
  st n = $1 & y = $2 holds y = F.n \ G.n;
A1: for t being Element of NAT ex A being Subset of X st P[t,A]
  proof
    let t be Element of NAT;
    F.t is Element of COM(S,M);
    then reconsider A = F.t \ G.t as Subset of X by XBOOLE_1:1;
    take A;
    thus thesis;
  end;
  ex H being sequence of bool X 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 X 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;
