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
  (superior_realsequence seq).n = sup(seq^\n) & (inferior_realsequence
  seq).n = inf(seq^\n)
proof
  set rseq=seq^\n;
A1: ex Z being non empty Subset of ExtREAL
st Z = {seq.k : n <= k} & (inferior_realsequence seq).n = inf Z by Def6;
  now
    let x be object;
    assume x in {seq.k: n <= k};
    then consider k be Nat such that
A2: x=seq.k and
A3: n <= k;
    reconsider k1=k-n as Element of NAT by A3,INT_1:5;
    x = seq.(n+k1) by A2;
    then x = rseq.k1 by NAT_1:def 3;
    hence x in rng rseq by FUNCT_2:4;
  end;
  then
A4: {seq.k: n <= k} c= rng rseq;
  now
    let x be object;
    assume x in rng rseq;
    then consider z be object such that
A5: z in dom rseq and
A6: x=rseq.z by FUNCT_1:def 3;
    reconsider k0=z as Element of NAT by A5;
A7: n <=n+k0 by NAT_1:11;
    x = seq.(n+k0) by A6,NAT_1:def 3;
    hence x in {seq.k: n <= k} by A7;
  end;
  then
A8: rng rseq c= { seq.k: n <= k};
  ex Y being non empty Subset of ExtREAL
  st Y = {seq.k : n <= k} & (superior_realsequence seq).n = sup Y by Def7;
  hence thesis by A1,A4,A8,XBOOLE_0:def 10;
end;
