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

theorem Th21:
  for f be 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((superior_realsequence f).n,r)) holds
  meet F = dom(f.0) /\ great_eq_dom(lim_sup f,r)
proof
  let f be Functional_Sequence of X,ExtREAL, F be SetSequence of S, r be Real;
  set E = dom(f.0);
  set g=superior_realsequence f;
  assume
A1: for n be Nat holds F.n = E /\ great_eq_dom(g.n,r);
  dom(g.0) = dom(f.0) by Def6;
  then meet rng F = E /\ great_eq_dom(inf g,r) by A1,Th16;
  hence thesis by Th12;
end;
