reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;
reserve seq for ExtREAL_sequence;
reserve e1,e2 for ExtReal;
reserve rseq for Real_Sequence;

theorem Th28:
  for seq be ExtREAL_sequence,j be Element of NAT holds
superior_realsequence (seq^\j) = (superior_realsequence seq)^\j & lim_sup (seq
  ^\j) = lim_sup seq
proof
  let seq be ExtREAL_sequence,j be Element of NAT;
  set rseq=seq^\j;
  now
    let n be Element of NAT;
A1: ex Y2 being non empty Subset of ExtREAL
    st Y2 = {rseq.k : n <= k} & (superior_realsequence rseq).n = sup Y2
          by Def7;
A2: ex Y3 being non empty Subset of ExtREAL
   st Y3 = { seq.k: (j+n) <= k} & (superior_realsequence seq).(j+n) = sup Y3
by Def7;
    now
      let x be object;
      assume x in {seq.k where k: j+n <= k};
      then consider k be Nat such that
A3:   x=seq.k and
A4:   j+n <= k;
      j <= j+n by NAT_1:11;
      then reconsider k1=k-j as Element of NAT by A4,INT_1:5,XXREAL_0:2;
A5:   x = seq.(j+k1) by A3;
      j+n - j <= k - j by A4,XREAL_1:9;
      hence x in {seq.(j+k2) where k2: n<= k2} by A5;
    end;
    then
A6: {seq.k where k: j+n <= k} c= {seq.(j+k) where k: n <= k};
    now
      let x be object;
      assume x in {seq.(j+k) where k: n <= k};
      then consider k be Nat such that
A7:   x=seq.(j+k) and
A8:   n <= k;
      j+n <= j+k by A8,XREAL_1:6;
      hence x in { seq.k1 where k1: j+n <= k1} by A7;
    end;
    then {seq.(j+k) where k: n <= k} c= {seq.k where k: j+n <= k};
    then
A9: {seq.(j+k) where k: n <= k} = { seq.k where k: (j+n) <= k}
    by A6,XBOOLE_0:def 10;
    now
      let x be object;
      assume x in {seq.(j+k) where k: n <= k};
      then consider k be Nat such that
A10:  x=seq.(j+k) and
A11:  n <= k;
      x = rseq.k by A10,NAT_1:def 3;
      hence x in { rseq.k1 where k1: n<= k1} by A11;
    end;
    then
A12: {seq.(j+k) where k: n <= k} c= {rseq.k where k: n <= k};
    now
      let x be object;
      assume x in {rseq.k where k: n <= k};
      then consider k be Nat such that
A13:  x=rseq.k and
A14:  n <= k;
      x = seq.(j+k) by A13,NAT_1:def 3;
      hence x in { seq.(j+k1) where k1: n <= k1} by A14;
    end;
    then {rseq.k where k: n <= k} c= { seq.(j+k) where k: n <= k};
    then {rseq.k where k: n <= k} = {seq.(j+k) where k: n <= k}
     by A12,XBOOLE_0:def 10;
    hence (superior_realsequence rseq).n = ((superior_realsequence seq)^\j).n
    by A1,A2,A9,NAT_1:def 3;
  end;
  then (superior_realsequence seq)^\j = superior_realsequence rseq by
FUNCT_2:63;
  hence thesis by Th25;
end;
