reserve n,k for Nat,
  X for non empty set,
  S for SigmaField of X;

theorem Th1:
  for M be sigma_Measure of S, F be sequence of S, n holds
  {x where x is Element of X : for k st n <= k holds x in F.k}
    is Element of S
proof
  let M be sigma_Measure of S, F be sequence of S, n;
  set G = {x where x is Element of X : for k st n <= k holds x in F.k};
  deffunc Fn(Element of NAT) = F.(n+$1);
  consider E be sequence of S such that
A1: for k be Element of NAT holds E.k=Fn(k) from FUNCT_2:sch 4;
  now
    let z be object;
    assume z in G; then
A2: ex x be Element of X st z=x & for k be Nat st n <= k holds
    x in F.k;
    for Y be set st Y in rng E holds z in Y
    proof
      let Y be set;
      assume Y in rng E;
      then consider l be object such that
A3:   l in NAT and
A4:   Y=E.l by FUNCT_2:11;
      reconsider l as Element of NAT by A3;
      z in F.(n+l) by A2,NAT_1:12;
      hence thesis by A1,A4;
    end;
    hence z in meet rng E by SETFAM_1:def 1;
  end; then
A5: G c= meet rng E;
  rng E is N_Sub_set_fam of X by MEASURE1:23;
  then rng E is N_Measure_fam of S by MEASURE2:def 1; then
A6: meet rng E is Element of S by MEASURE2:2;
  now
    let z be object;
    assume
A7: z in meet rng E;
    now
      let k be Nat;
      assume n <= k;
      then reconsider l=k-n as Element of NAT by NAT_1:21;
      E.l in rng E by FUNCT_2:4;
      then z in E.l by A7,SETFAM_1:def 1;
      then z in F.(n+l) by A1;
      hence z in F.k;
    end;
    hence z in G by A6,A7;
  end;
  then meet rng E c= G;
  hence thesis by A6,A5,XBOOLE_0:def 10;
end;
