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 Th17:
  seq^\k is convergent implies seq is convergent & lim seq = lim( seq^\k)
proof
  set seq0=seq^\k;
  assume
A1: seq^\k is convergent;
  per cases by A1,MESFUNC5:def 11;
  suppose
    seq0 is convergent_to_finite_number;
    hence thesis by Th16;
  end;
  suppose
A2: seq0 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 <= seq.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 <= seq0.m by A2,MESFUNC5:def 9;
      take n1=n+k;
      hereby
        let m be Nat;
        assume
A4:     n1 <= m;
        k <= n+k by NAT_1:11;
        then reconsider mk = m-k as Element of NAT by A4,INT_1:5,XXREAL_0:2;
A5:     seq0.(m-k) = seq.(mk+k) by NAT_1:def 3;
        n+k-k <= m-k by A4,XREAL_1:9;
        hence g <= seq.m by A3,A5;
      end;
    end;
    then
A6: seq is convergent_to_+infty by MESFUNC5:def 9;
    hence
A7: seq is convergent by MESFUNC5:def 11;
    lim seq0 = +infty by A1,A2,MESFUNC5:def 12;
    hence thesis by A6,A7,MESFUNC5:def 12;
  end;
  suppose
A8: seq0 is convergent_to_-infty;
    for g be Real st g<0 ex n be Nat st for m be Nat st n<=m holds
    seq.m <= g
    proof
      let g be Real;
      assume g < 0;
      then consider n be Nat such that
A9:   for m be Nat st n<=m holds seq0.m <= g by A8,MESFUNC5:def 10;
      take n1=n+k;
      hereby
        let m be Nat;
        assume
A10:    n1 <= m;
        k <= n+k by NAT_1:11;
        then reconsider mk = m-k as Element of NAT by A10,INT_1:5,XXREAL_0:2;
A11:    seq0.(m-k) = seq.(mk+k) by NAT_1:def 3;
        n+k-k <= m-k by A10,XREAL_1:9;
        hence seq.m <= g by A9,A11;
      end;
    end;
    then
A12: seq is convergent_to_-infty by MESFUNC5:def 10;
    hence
A13: seq is convergent by MESFUNC5:def 11;
    lim seq0=-infty by A1,A8,MESFUNC5:def 12;
    hence thesis by A12,A13,MESFUNC5:def 12;
  end;
end;
