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 Th26:
  seq is non-decreasing implies seq^\k is non-decreasing & sup seq
  = sup(seq^\k)
proof
  set seq0 = seq^\k;
  assume
A1: seq is non-decreasing;
  now
    let n,m be Nat;
    assume m <= n;
    then k+m <= k+n by XREAL_1:6;
    then seq.(k+m) <=seq.(k+n) by A1,Th7;
    then seq0.m <= seq.(k+n) by NAT_1:def 3;
    hence seq0.m <= seq0.n by NAT_1:def 3;
  end;
  hence seq^\k is non-decreasing;
  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: seq.(n+k) <= sup seq0 by Th23;
    n <=n+k by NAT_1:11;
    then seq.n <= seq.(n+k) by A1,Th7;
    hence y <= sup rng seq0 by A3,A4,XXREAL_0:2;
  end;
  then sup rng seq0 is UpperBound of rng seq by XXREAL_2:def 1;
  then
A5: sup rng seq <= sup rng seq0 by XXREAL_2:def 3;
  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 seq0.n <=sup seq by Th23;
    hence y <= sup rng seq by A7;
  end;
  then sup rng seq is UpperBound of rng seq0 by XXREAL_2:def 1;
  then sup rng seq0 <= sup rng seq by XXREAL_2:def 3;
  hence thesis by A5,XXREAL_0:1;
end;
