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;

theorem Th8:
  (inferior_realsequence seq).n <= seq.n & seq.n <= (
  superior_realsequence seq).n
proof
  consider Y being non empty Subset of ExtREAL such that
A1: Y = {seq.k: n <= k} and
A2: (inferior_realsequence seq).n = inf Y by Def6;
A3: seq.n in Y by A1;
  hence (inferior_realsequence seq).n <= seq.n by A2,XXREAL_2:3;
  ex Z being non empty Subset of ExtREAL
  st Z = {seq.k: n <= k} & (superior_realsequence seq).n = sup Z by Def7;
  hence thesis by A1,A3,XXREAL_2:4;
end;
