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

theorem Th9:
  for f be with_the_same_dom Functional_Sequence of X,ExtREAL,
      n be Nat holds (superior_realsequence f).n = sup(f^\n)
proof
  let f be with_the_same_dom Functional_Sequence of X,ExtREAL, n be Nat;
  reconsider g=f as sequence of PFuncs(X,ExtREAL);
  dom sup(f^\n) = dom((f^\n).0) by Def4;
  then dom sup(f^\n) = dom(g.(n+(0 qua Nat))) by NAT_1:def 3;
  then
A1: dom sup(f^\n) = dom(f.0) by Def2;
A2: dom ((superior_realsequence f).n) = dom(f.0) by Def6;
  now
    let x be Element of X;
    assume
A3: x in dom sup(f^\n);
    now
      let k be Element of NAT;
      ((f^\n)#x).k =((f^\n).k).x by MESFUNC5:def 13;
      then ((f^\n)#x).k =(g.(n+k)).x by NAT_1:def 3;
      then ((f^\n)#x).k =(f#x).(n+k) by MESFUNC5:def 13;
      hence ((f^\n)#x).k = ((f#x)^\n).k by NAT_1:def 3;
    end;
    then (f^\n)#x = (f#x)^\n by FUNCT_2:63;
    then
A4: (sup(f^\n)).x =sup ((f#x)^\n) by A3,Def4;
    ((superior_realsequence f).n ).x = (superior_realsequence (f#x)).n by A2,A1
,A3,Def6;
    hence (sup (f^\n)).x = ((superior_realsequence f).n ).x by A4,RINFSUP2:27;
  end;
  hence thesis by A2,A1,PARTFUN1:5;
end;
