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
  seq is convergent implies seq^\k is convergent & lim seq = lim(seq^\k)
proof
  set seq0 = seq^\k;
  assume
A1: seq is convergent;
  per cases by A1,MESFUNC5:def 11;
  suppose
    seq is convergent_to_finite_number;
    hence thesis by Th20;
  end;
  suppose
A2: seq is convergent_to_+infty;
    for g be Real st 0 < g ex n be Nat st for m be Nat st n<=m
    holds g <= seq0.m
    proof
      let g be Real;
      assume 0<g;
      then consider n be Nat such that
A3:   for m be Nat st n <= m holds g <= seq.m by A2,MESFUNC5:def 9;
      take n;
      hereby
        let m be Nat such that
A4:     n <= m;
        m <= m+k by NAT_1:11;
        then n <= m+k by A4,XXREAL_0:2;
        then g <=seq.(m+k) by A3;
        hence g <= seq0.m by NAT_1:def 3;
      end;
    end;
    then
A5: seq0 is convergent_to_+infty by MESFUNC5:def 9;
    hence
A6: seq0 is convergent by MESFUNC5:def 11;
    lim seq = +infty by A1,A2,MESFUNC5:def 12;
    hence thesis by A5,A6,MESFUNC5:def 12;
  end;
  suppose
A7: seq is convergent_to_-infty;
    for g be Real st g < 0 ex n be Nat st for m be Nat st n <= m
    holds seq0.m <= g
    proof
      let g be Real;
      assume g < 0;
      then consider n be Nat such that
A8:   for m be Nat st n <= m holds seq.m <= g by A7,MESFUNC5:def 10;
      take n;
      hereby
        let m be Nat such that
A9:     n <= m;
        m <= m+k by NAT_1:11;
        then n <= m+k by A9,XXREAL_0:2;
        then seq.(m+k) <= g by A8;
        hence seq0.m <= g by NAT_1:def 3;
      end;
    end;
    then
A10: seq0 is convergent_to_-infty by MESFUNC5:def 10;
    hence
A11: seq0 is convergent by MESFUNC5:def 11;
    lim seq = -infty by A1,A7,MESFUNC5:def 12;
    hence thesis by A10,A11,MESFUNC5:def 12;
  end;
end;
