reserve X for set;

theorem Th15:
  for S being SigmaField of X, M being sigma_Measure of S, F being
  sequence of COM(S,M) holds ex G being sequence of S st for n being
  Element of NAT holds G.n in MeasPart(F.n)
proof
  let S be SigmaField of X, M be sigma_Measure of S, F be sequence of COM(
  S,M);
  defpred P[Element of NAT, set] means for n being Element of NAT, y being set
  st n = $1 & y = $2 holds y in MeasPart(F.n);
A1: for t being Element of NAT ex A being Element of S st P[t,A]
  proof
    let t be Element of NAT;
    set A = the Element of MeasPart(F.t);
    reconsider A as Element of S by Def4;
    take A;
    thus thesis;
  end;
  ex G being sequence of S st for t being Element of NAT holds P[t,G.t
  ] from FUNCT_2:sch 3(A1);
  then consider G being sequence of S such that
A2: for t being Element of NAT, n being Element of NAT, y being set st n
  = t & y = G.t holds y in MeasPart(F.n);
  take G;
  thus thesis by A2;
end;
