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

theorem Th19:
  for f be with_the_same_dom Functional_Sequence of X,ExtREAL,
      E be Element of S st dom (f.0) = E &
  (for n be Nat holds f.n is E-measurable) holds
  for n holds (superior_realsequence f).n is E-measurable
proof
  let f be with_the_same_dom Functional_Sequence of X,ExtREAL, E be Element of
  S;
  assume that
A1: dom (f.0) = E and
A2: for n be Nat holds f.n is E-measurable;
  let n;
  reconsider n9=n as Element of NAT by ORDINAL1:def 12;
A3: now
    let r be Real;
    deffunc G(Element of NAT) = E /\ great_dom(f.$1,r);
    consider F being sequence of bool X such that
A4: for x being Element of NAT holds F.x = G(x) from FUNCT_2:sch 4;
    now
      let i be Nat;
A5:   f.i is E-measurable by A2;
      i in NAT by ORDINAL1:def 12;
      then
A6:   F.i=E /\ great_dom(f.i,r) by A4;
      dom (f.i) = E by A1,Def2;
      hence F.i in S by A6,A5,MESFUNC1:29;
    end;
    then
A7: rng F c= S by NAT_1:52;
A8: for x being Nat holds F.x = E /\ great_dom(f.x,r)
    proof
      let x be Nat;
      reconsider x9=x as Element of NAT by ORDINAL1:def 12;
      F.x9 = E /\ great_dom(f.x9,r) by A4;
      hence thesis;
    end;
    reconsider F as SetSequence of S by A7,RELAT_1:def 19;
    (superior_setsequence F).n9 in rng (superior_setsequence F) by NAT_1:51;
    then (superior_setsequence F).n9 in S;
    hence E /\ great_dom((superior_realsequence f).n,r) in S by A1,A8,Th17;
  end;
  dom((superior_realsequence f).n9) = E by A1,Def6;
  hence thesis by A3,MESFUNC1:29;
end;
