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

theorem Th12:
  for f be Functional_Sequence of X,ExtREAL holds
  (for x be Element of X st x in dom lim_sup f holds
     (lim_sup f).x=inf superior_realsequence(f#x) &
     (lim_sup f).x=inf ((superior_realsequence f)#x) &
  (lim_sup f).x=inf (superior_realsequence f).x ) &
     lim_sup f = inf superior_realsequence f
proof
  let f be Functional_Sequence of X,ExtREAL;
A1: dom(inf superior_realsequence f) =dom((superior_realsequence f).0) by Def3
    .=dom(f.0) by Def6
    .=dom lim_sup f by Def8;
A2: now
    let x be Element of X;
    assume
A3: x in dom lim_sup f; then
A4: (lim_sup f).x=lim_sup (f#x) by Def8;
    hence (lim_sup f).x = inf superior_realsequence(f#x);
    dom lim_sup f = dom(f.0) by Def8;
    hence (lim_sup f).x = inf((superior_realsequence f)#x) by A3,A4,Th10;
    hence (lim_sup f).x = (inf superior_realsequence f).x by A1,A3,Def3;
  end;
  then for x be Element of X st x in dom lim_sup f holds (lim_sup f).x =(inf
  superior_realsequence f).x;
  hence thesis by A1,A2,PARTFUN1:5;
end;
