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

theorem Th17:
  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_dom(f.n,r)) holds for n be Nat holds (
superior_setsequence F).n = dom(f.0) /\ great_dom((superior_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
A1: for n be Nat holds F.n = E /\ great_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_dom(f.(n+k9),r) by A1;
    hence F1.k = E /\ great_dom(f1.k,r) by NAT_1:def 3;
  end;
A3: union rng (F^\n9) = (superior_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 union rng F1 = E /\ great_dom(sup f1,r) by A2,Th15;
  hence thesis by A3,Th9;
end;
