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

theorem Th18:
  for f be with_the_same_dom Functional_Sequence of X,ExtREAL,
      F be SetSequence of S, r be Real st
    (for n be Nat holds F.n = dom(f.0) /\ great_eq_dom(f.n,r))
   holds for n be Nat holds
     (inferior_setsequence F).n = dom(f.0) /\
     great_eq_dom((inferior_realsequence f).n,r)
proof
  let f be with_the_same_dom Functional_Sequence of X,ExtREAL, F be
  SetSequence of S, r be Real;
  set E = dom(f.0);
  assume that
A1: for n be Nat holds F.n = E /\ great_eq_dom(f.n,r);
  let n be Nat;
  reconsider n9=n as Element of NAT by ORDINAL1:def 12;
  set f1=f^\n9;
  set F1=F^\n9;
A2: now
    let k be Nat;
    reconsider k9=k as Element of NAT by ORDINAL1:def 12;
    F1.k = F.(n+k9) by NAT_1:def 3;
    then F1.k = E /\ great_eq_dom(f.(n+k9),r) by A1;
    hence F1.k = E /\ great_eq_dom(f1.k,r) by NAT_1:def 3;
  end;
A3: meet rng(F^\n9) = (inferior_setsequence F).n by Th2;
  consider g be sequence of PFuncs(X,ExtREAL) such that
A4: f=g and
  f^\n9=g^\n9;
  f1.0 = g.(n+(0 qua Nat)) by A4,NAT_1:def 3;
  then dom(f1.0) = E by A4,Def2;
  then meet rng F1 = E /\ great_eq_dom(inf f1,r) by A2,Th16;
  hence thesis by A3,Th8;
end;
