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 Th25:
  seq is non-increasing implies seq^\k is non-increasing & inf seq
  = inf(seq^\k)
proof
  set seq0 = seq^\k;
  assume
A1: seq is non-increasing;
  now
    let n,m be Nat;
    assume m<=n;
    then k+m <= k+n by XREAL_1:6;
    then seq.(k+n) <=seq.(k+m) by A1,Th7;
    then seq0.n <= seq.(k+m) by NAT_1:def 3;
    hence seq0.n <= seq0.m by NAT_1:def 3;
  end;
  hence seq^\k is non-increasing;
  now
    let y be ExtReal;
    assume y in rng seq;
    then consider n be object such that
A2: n in dom seq and
A3: y=seq.n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A2;
    seq0.n = seq.(n+k) by NAT_1:def 3;
    then
A4: inf seq0 <= seq.(n+k) by Th23;
    n <= n+k by NAT_1:11;
    then seq.(n+k) <= seq.n by A1,Th7;
    hence inf rng seq0 <= y by A3,A4,XXREAL_0:2;
  end;
  then inf rng seq0 is LowerBound of rng seq by XXREAL_2:def 2;
  then
A5: inf rng seq0 <= inf rng seq by XXREAL_2:def 4;
  now
    let y be ExtReal;
    assume y in rng seq0;
    then consider n be object such that
A6: n in dom seq0 and
A7: y=seq0.n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A6;
    seq0.n= seq.(n+k) by NAT_1:def 3;
    then inf seq <= seq0.n by Th23;
    hence inf rng seq <= y by A7;
  end;
  then inf rng seq is LowerBound of rng seq0 by XXREAL_2:def 2;
  then inf rng seq <= inf rng seq0 by XXREAL_2:def 4;
  hence thesis by A5,XXREAL_0:1;
end;
