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

theorem Th11:
  for f be Functional_Sequence of X,ExtREAL holds
  (for x be Element of X st x in dom lim_inf f holds
  (lim_inf f).x=sup inferior_realsequence(f#x) &
  (lim_inf f).x=sup ((inferior_realsequence f)#x) &
  (lim_inf f).x=sup (inferior_realsequence f).x) &
  lim_inf f = sup inferior_realsequence f
proof
  let f be Functional_Sequence of X,ExtREAL;
  dom(sup inferior_realsequence f) =dom((inferior_realsequence f).0) by Def4;
  then dom(sup inferior_realsequence f) = dom(f.0) by Def5;
  then
A1: dom(sup inferior_realsequence f) = dom(lim_inf f) by Def7;
A2: now
    let x be Element of X;
    assume
A3: x in dom lim_inf f; then
A4: (lim_inf f).x=lim_inf(f#x) by Def7;
    hence (lim_inf f).x= sup inferior_realsequence(f#x);
    dom lim_inf f = dom(f.0) by Def7;
    hence (lim_inf f).x = sup((inferior_realsequence f)#x) by A3,A4,Th7;
    hence (lim_inf f).x =(sup inferior_realsequence f).x by A1,A3,Def4;
  end;
  then for x be Element of X st x in dom lim_inf f holds (lim_inf f).x =(sup
  inferior_realsequence f).x;
  hence thesis by A1,A2,PARTFUN1:5;
end;
