reserve X for set;

theorem Th17:
  for S being SigmaField of X, M being sigma_Measure of S, F being
  sequence of bool X st (for n being Element of NAT holds F.n is thin of M)
holds ex G being sequence of S st for n being Element of NAT holds F.n c= G
  .n & M.(G.n) = 0.
proof
  let S be SigmaField of X, M be sigma_Measure of S, F be sequence of bool
  X;
  defpred P[Element of NAT, set] means for n being Element of NAT, y being set
  st n = $1 & y = $2 holds y in S & F.n c= y & M.y = 0.;
  assume
A1: for n being Element of NAT holds F.n is thin of M;
A2: for t being Element of NAT ex A being Element of S st P[t,A]
  proof
    let t be Element of NAT;
    F.t is thin of M by A1;
    then consider A being set such that
A3: A in S and
A4: F.t c= A & M.A = 0. by Def2;
    reconsider A as Element of S by A3;
    take A;
    thus thesis by A4;
  end;
  ex G being sequence of S st for t being Element of NAT holds P[t,G.t
  ] from FUNCT_2:sch 3(A2);
  then consider G being sequence of S such that
A5: for t being Element of NAT, n being Element of NAT, y being set st n
  = t & y = G.t holds y in S & F.n c= y & M.y = 0.;
  take G;
  thus thesis by A5;
end;
