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 Th31:
  for seq be ExtREAL_sequence, k be Element of NAT st seq is
non-decreasing & -infty < seq.k & seq.k < +infty holds seq^\k is bounded_below
  & inf (seq^\k) = seq.k
proof
  let seq be ExtREAL_sequence, k be Element of NAT;
  assume that
A1: seq is non-decreasing and
A2: -infty < seq.k and
A3: seq.k < +infty;
  set seq0=seq^\k;
  now
    let y be ExtReal;
    assume y in rng seq0;
    then consider n be object such that
A4: n in dom seq0 and
A5: y=seq0.n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A4;
    k <= n+k by NAT_1:11;
    then seq.k <= seq.(n+k) by A1,Th7;
    hence seq.k <= y by A5,NAT_1:def 3;
  end;
  then
A6: seq.k is LowerBound of rng seq0 by XXREAL_2:def 2;
  seq0.0 = seq.(0+k) by NAT_1:def 3;
  then
A7: seq.k in rng seq0 by FUNCT_2:4;
  seq.k in REAL by A2,A3,XXREAL_0:14;
  then rng seq0 is bounded_below by A6,XXREAL_2:def 9;
  hence thesis by A6,A7,XXREAL_2:56;
end;
