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

theorem Th20:
  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 be Nat holds (inferior_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 be Nat;
  reconsider n9=n as Element of NAT by ORDINAL1:def 12;
A3: now
    let r be Real;
    deffunc G(Element of NAT) = E /\ great_eq_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_eq_dom(f.i,r) by A4;
      dom (f.i) = E by A1,Def2;
      hence F.i in S by A6,A5,MESFUNC1:27;
    end;
    then
A7: rng F c= S by NAT_1:52;
A8: for x being Nat holds F.x = E /\ great_eq_dom(f.x,r)
    proof
      let x be Nat;
      reconsider x9=x as Element of NAT by ORDINAL1:def 12;
      F.x9 = E /\ great_eq_dom(f.x9,r) by A4;
      hence thesis;
    end;
    reconsider F as SetSequence of S by A7,RELAT_1:def 19;
    (inferior_setsequence F).n9 in rng inferior_setsequence F by NAT_1:51;
    then (inferior_setsequence F).n9 in S;
    hence
    E /\ great_eq_dom((inferior_realsequence f).n,r ) in S by A1,A8,Th18;
  end;
  dom((inferior_realsequence f).n9) = E by A1,Def5;
  hence thesis by A3,MESFUNC1:27;
end;
